1. introduction.pdfs.semanticscholar.org/bbd4/667ab31ac2a3b32fc7f... · on the degree of mixed...

ON THE DEGREE OF MIXED POLYNOMIAL MATRICES∗

KAZUO MUROTA†

SIAM J. MATRIX ANAL. APPL. c© 1998 Society for Industrial and Applied MathematicsVol. 20, No. 1, pp. 196–227

Abstract. The mixed polynomial matrix, introduced as a convenient mathematical tool for thedescription of physical/engineering dynamical systems, is a polynomial matrix of which the coeffi-cients are classified into fixed constants and independent parameters. The valuated matroid, inventedby Dress and Wenzel [Appl. Math. Lett., 3 (1990), pp. 33–35], is a combinatorial abstraction of thedegree of minors (subdeterminants) of a polynomial matrix. We discuss a number of implicationsof the recent developments in the theory of valuated matroids in the context of polynomial matrixtheory. In particular, we apply the valuated matroid intersection theorem to the analysis of thedegree of the determinant of a mixed polynomial matrix to obtain a novel duality identity togetherwith an efficient algorithm.

Key words. combinatorial matrix theory, degree of determinant, mixed matrix, polynomialmatrix, valuated matroid

AMS subject classifications. 05C50, 68Q40, 90C27

PII. S0895479896311438

1. Introduction. Matrices consisting of polynomials or rational functions playfundamental roles in various branches in engineering (Gohberg, Lancaster, and Rod-man [22]). For example, in dynamical system theory (Rosenbrock [51], Vidyasagar[59]), a linear time-invariant system is described by a polynomial matrix called thesystem matrix (the Laplace transform of the state-space equations) or by a rationalfunction matrix called the transfer function matrix. Therefore, it is often the case thatthe degrees of minors (subdeterminants) of such a matrix have essential engineeringsignificance (see section 2). The objective of this paper is to contribute to the combi-natorial theory of matrices (Brualdi and Ryser [3], Edmonds [12]) by investigating thecombinatorial aspects of the degree of minors of a polynomial/rational matrix usingrecent results on valuated matroids.

Let A(s) = (Aij(s)) be an m × n rational function matrix with Aij(s) beinga rational function in s with coefficients from a certain field F (typically the realnumber field R). Denote by R and C the row set and the column set of A. In thispaper we are interested in the highest degree of a minor (subdeterminant) of order kof A(s):

δk = δk(A) = maxdegs detA[I, J ] | |I| = |J | = k,(1.1)

where A[I, J ] denotes the submatrix of A with row set I ⊆ R and column set J ⊆ C,and the degree of a rational function f(s) = p(s)/q(s) (with p(s) and q(s) beingpolynomials) is defined by degs f(s) = degs p(s) − degs q(s). By convention we putdegs(0) = −∞.

Define δ : 2R × 2C → Z ∪ −∞ and ω : 2R∪C → Z ∪ −∞ by

δ(I, J) = degs detA[I, J ] (I ⊆ R, J ⊆ C),(1.2)

ω(B) = δ(R \B,C ∩B) (B ⊆ R ∪ C),(1.3)

∗Received by the editors November 1, 1996; accepted for publication (in revised form) December5, 1997; published electronically September 23, 1998.

http://www.siam.org/journals/simax/20-1/31143.html†Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

([email protected]).

196

DEGREE OF MIXED POLYNOMIAL MATRIX 197

where δ(∅, ∅) = ω(R) = 0 and δ(I, J) = −∞ unless |I| = |J |. A combinatorialproperty of the function ω (equivalently, of δ) has been abstracted by Dress andWenzel [10], [11] as the concept of valuated matroid. A valuated matroid is a pairM = (V, ω) of a finite set V and a function ω : 2V → R∪ −∞ such that B = B ⊆V | ω(B) 6= −∞ is nonempty and that the following exchange property holds:

(MV) For B,B′ ∈ B and u ∈ B−B′, there exists v ∈ B′−B such that B−u+v ∈B, B′ + u− v ∈ B, and

ω(B) + ω(B′) ≤ ω(B − u + v) + ω(B′ + u− v).

The function ω of (1.3) arising from a rational function matrix A(s) defines avaluated matroid M = (V, ω) with V = R ∪ C and

B = B ⊆ R ∪ C | A[R \B,C ∩B] is nonsingular.(1.4)

It has turned out that valuated matroids afford a nice combinatorial frameworkto which the optimization algorithms established for matroids generalize naturally(see Welsh [60], White [61] for matroid theory; Faigle [15], Fujishige [18], Lawler[32] for combinatorial optimization on matroids; and Iri [27], Murota [34], Recski[49] for application of matroids). Variants of greedy algorithms work for maximizinga matroid valuation, as has been shown by Dress and Wenzel [10] as well as byDress and Terhalle [7, 8, 9] and Murota [39]. (These greedy-type algorithms aresimilar to, but not the same as, those in Korte, Lovasz, and Schrader [31].) Theweighted matroid intersection problem has been extended by Murota [41, 42] to thevaluated matroid intersection problem with natural extensions of optimality criteriaand algorithms. The essence of the present paper is an application of these results onthe valuated matroid intersection problem to mixed polynomial matrices (or rather, itwas the analysis of mixed polynomial matrices that had motivated the present authorto investigate the valuated matroid intersection problem).

The concept of mixed polynomial matrix was introduced by Murota [34] (seealso Murota and Iri [45]) as a convenient mathematical tool for the description ofphysical/engineering (linear time-invariant) dynamical systems. Let K be a subfieldof a field F . A polynomial matrix A(s) over F (i.e., Aij(s) ∈ F [s]) is called a mixedpolynomial matrix with respect to F /K if

A(s) = Q(s) + T (s) =

N∑

k=0

skQk +

N∑

k=0

skTk(1.5)

for some integer N ≥ 0, where(MP-Q) Qk (k = 0, 1, . . . , N) are matrices over K , and(MP-T) Tk (k = 0, 1, . . . , N) are matrices over F such that the set of their nonzero

entries is algebraically independent over K .The assumption (MP-T) means that we can regard the nonzero entries of Tk (k =0, 1, . . . , N) as independent parameters.

Example 1.1. For an illustration of the definition above, here is a concrete exampleof 2 × 3 mixed polynomial matrix:

A(s) =

c1 c2 c3r1 s3 + 1 s2 + α1 α2s + 1r2 s2 + α3 s 0

198 KAZUO MUROTA

with

Q(s) =s3 + 1 s2 1s2 s 0

, T (s) = 0 α1 α2sα3 0 0

.

Here we assume α1, α2, α3 to be algebraically independent over Q (field of rationalnumbers). Then we may take K = Q and F = Q(α1, α2, α3) (field of rationalfunctions in α1, α2, α3 over Q).

For a nonsingular mixed polynomial matrix A(s) = Q(s) + T (s) we have thefollowing identity:

degs detA = max|I|=|J|

I⊆R,J⊆C

degs detQ[I, J ] + degs detT [R− I, C − J ],(1.6)

which can be derived from (MP-Q), (MP-T), and the Laplace expansion of deter-minants (see Theorem 4.1 in section 4). The right-hand side of this identity in-

volves a maximization over all pairs (I, J), the number of which is ( |R|+|C||R|

), too

large for an exhaustive search for maximization. Fortunately, however, the functionsδQ(I, J) = degs detQ[I, J ] and δT (I, J) = degs detT [I, J ] enjoy a nice combinatorialproperty, each defining a valuated matroid, as explained above. Moreover, this max-imization problem can be rewritten as a valuated matroid intersection problem, forwhich efficient (polynomial-time) algorithms have been developed.

This approach to the computation of degs detA extends to the computation ofδk(A) for a specified k. It is one of the main objectives of this paper to describe aconcrete procedure for efficiently computing δk(A) for a mixed polynomial matrix A.

This paper is organized as follows. In section 2 we justify our interest in δkby explaining significances of δk in engineering context. In section 3 we summarizerelevant facts on valuated matroids and indicate their implications in engineeringapplications. Sections 4 and 5 compose the main part of the paper. We derive newidentities for δk and determine the coefficient of the highest degree term of detA withreference to the combinatorial canonical form of a layered mixed matrix. In section 5we describe the algorithm for δk together with a worked-out example.

Remark 1.1. Already in [33, 34] (before the invention of valuated matroids), thepresent author was interested in the degree of the determinant of a mixed polynomialmatrix and designed an efficient algorithm by making use of the results on the matroidintersection problem. It was possible to avoid valuated matroids because of a strongerassumption imposed on Q(s):(MP-Q′) Every nonvanishing subdeterminant of Q(s) is a monomial in s over K .It was discussed at full length that this stronger assumption could be justified inengineering applications for a physical reason that can be categorized as a kind ofdimensional analysis. See also an expository article [40].

Remark 1.2. The problem of computing δk(A) has attracted considerable researchinterest. See Bujakiewicz [4], Commault, Dion, and Perez [6], Hovelaque, Commault,and Dion [26], Reinschke [50], Suda, Wan, and Ueno [52], Svaricek [53, 54], and van derWoude [65] for graph-theoretic approaches; Murota [37, 38] and Iwata, Murota, andSakuta [30] for combinatorial relaxation algorithms based on graph-theoretic methods;Murota and van der Woude [47] for a matroid-theoretic approach; and Dress andTerhalle [7, 8] and Murota [39] for valuated matroid-theoretic approaches. A recentpaper of Iwata and Murota [29] affords a combinatorial relaxation algorithm basedon the results of the present paper.


2. Degree of subdeterminants. In this section we dwell on the significance ofδk(A) = max|I|=|J|=k degs detA[I, J ] in engineering context in order to motivate andjustify our present interest in δk. The reader may go on to section 3 as the subsequenttechnical developments are independent of this section.

2.1. Kronecker form of a matrix pencil. A linear time-invariant dynamicalsystem can be expressed most naturally in a descriptor form

Ex(t) = Fx(t) + Gu(t), y(t) = Hx(t)(2.1)

with “state” x(t) ∈ RN , input u(t), and output y(t). When described in the frequencydomain, the coefficient matrix is given by

(F − sE G

H O

).

Here it is natural to assume that sE−F is a regular pencil, i.e., that det(sE−F ) 6= 0,while E can be singular. Then δN (sE −F ) represents the dynamical degree [25] (thenumber of independent initial conditions that can be imposed), which is equal to thedimension of the equivalent state-space equations.

For an n×n regular matrix pencil A(s) in general, more detailed information canbe obtained from the sequence δk(A) (k = 1, 2, . . . , n). As is well known (Gantmacher[19]), a regular pencil can be brought into the Kronecker form by means of “strictequivalence” as follows.

Theorem 2.1 (Kronecker form). Assume that F is an algebraically closed fieldof characteristic zero (e.g., F = C (complex numbers)), and let A(s) be an n × nregular pencil. There exist nonsingular constant matrices P and Q such that

PA(s)Q = block-diag (sIm0+ B;Nm1

(s), Nm2(s), . . . , Nmb

(s)) ,(2.2)

where

m1 ≥ m2 ≥ · · · ≥ mb ≥ 1, m0 + m1 + · · · + mb = n,

B is an m0 ×m0 constant matrix, and Nm(s) denotes an m ×m bidiagonal matrixdefined by

Nm(s) =

1 s1 s

. . .. . .. . . s

1

.

The matrices Nmk(s) (k = 1, . . . , b) are called the nilpotent blocks, and the number

m1 = max1≤k≤b mk is the index of nilpotency.Apart from the matrix B, the structural indices of the Kronecker form, i.e., the

integers b, m0, m1, . . . ,mb, can be determined from δk(A) (k = 1, 2, . . . , n) by

b = n− maxk | δk(A) − δk−1(A) = 1(2.3)

(where b = n if such k does not exist) and by

mk =

δn(A) (k = 0),δn−k(A) − δn−k+1(A) + 1 (k = 1, . . . , b).

(2.4)

Formulae (2.3) and (2.4) can be derived easily from (2.2) (cf. Murota [38]).

200 KAZUO MUROTA

In the literature of numerical analysis a system of equations consisting of a mixtureof differential and algebraic relations is often abbreviated to DAE. For a linear time-invariant DAE in general, say Ax = b with A = A(s) being a nonsingular polynomialmatrix in s, the index is defined by1

ν(A) = maxi, j

degs(A−1)ji + 1.

Here it should be clear that each entry (A−1)ji of A−1 is a rational function in s. Analternative expression is

ν(A) = δn−1(A) − δn(A) + 1.

When degs Aij = 0 or 1 for all (i, j) with Aij 6= 0, as in (2.1), the index ν(A) agreeswith the index of nilpotency of A as a matrix pencil; namely, we have ν(A) = m1.

The solution x to Ax = b is of course given by x = A−1b, and therefore ν(A)− 1equals the highest order of the derivatives of the input b that can possibly appear inthe solution x. As such, a high index indicates the difficulty in numerical solution ofthe DAE, and sometimes even the inadequacy in mathematical modeling. See Brenan,Campbell, and Petzold [2], Gear [20, 21], Hairer and Wanner [23], Ungar, Kroner, andMarquardt [57] for more about the index of DAE.

2.2. Smith–McMillan form at infinity of a rational matrix. A rationalfunction f(s) is called proper if degs f(s) ≤ 0. We call a matrix a proper rationalmatrix if its entries are proper rational functions. A square proper rational matrix iscalled biproper if it is invertible and its inverse is a proper rational matrix. A properrational matrix is biproper if and only if its determinant is of degree zero.

Since the proper rational functions form a Euclidean ring, any proper rationalmatrix can be brought into the Smith form (Newman [48]), which is sometimes referredto as the structure at infinity in the control literature. From this we see further thatany rational matrix can be brought into the Smith–McMillan form at infinity, asstated below (Verghese and Kailath [58]).

Theorem 2.2 (Smith–McMillan form at infinity). Let A(s) be a rational functionmatrix. There exist biproper matrices U(s) and V (s) such that

U(s)A(s)V (s) =

(Γ(s) OO O

),

where

Γ(s) = diag (st1 , . . . , str ),

r = rankA(s), and tk = tk(A) (k = 1, . . . , r) are integers with t1 ≥ · · · ≥ tr. Further-more, tk can be expressed in terms of the minors of A as

tk(A) = δk(A) − δk−1(A) (k = 1, . . . , r),(2.5)

where δk(A) is defined by (1.1) and δ0(A) = 0 by convention.The integers tk (k = 1, . . . , r), uniquely determined by (2.5), are referred to as the

contents at infinity (Verghese and Kailath [58]). If they are positive, tk (k = 1, . . . , r)

1The definition of index given here applies only to linear time-invariant DAE systems. Index canbe defined for more general systems, and two kinds are distinguished in the literature, differentialindex and perturbation index, which coincide with each other for linear time-invariant DAE systems.


are the orders of the poles at infinity; if negative, −tk (k = 1, . . . , r) are the orders ofthe zeroes at infinity.

A (proper) rational function matrix typically appears as the transfer functionmatrix of a linear time–invariant dynamical system. The transfer function matrix ofthe descriptor system (2.1) is given by

P (s) = H(sE − F )−1G,(2.6)

provided that det(sE−F ) 6= 0 (while E can be singular). The Smith–McMillan format infinity of P (s) has control-theoretic significances (Commault and Dion [5], Hautus[24], Svaricek [54], Verghese and Kailath [58]).

In such a case it is desirable to express the Smith–McMillan form at infinity ofP (s) directly from the matrices E, F , G, and H, without referring to the entries ofP (s) explicitly. From the well-known formula

det

(F − sE G′

H ′ O

)= det(F − sE) · det[−H ′(F − sE)−1G′],

where H ′ denotes a submatrix of H with k rows and G′ is a submatrix of G with kcolumns, it follows that

δk(P ) = δN+k(A; I0, J0) − δN (F − sE),(2.7)

where

A(s) =

(F − sE G

H O

),

I0 and J0 are, respectively, the row and column sets corresponding to the N × Nnonsingular submatrix F − sE, and

δN+k(A; I0, J0) = maxdegs detA[I, J ] | I ⊇ I0, J ⊇ J0, |I| = |J | = N + k

means the highest degree of a minor of order N + k that contains row set I0 andcolumn set J0. Note that δN+k(A; I0, J0) = δN+k(A) − 2Nd for a sufficiently largeinteger d and

A(s) =

(diag (sd, . . . , sd) O

O I

)(F − sE G

H O

)(diag (sd, . . . , sd) O

O I

).

2.3. Causal splitting for autoregressive models. In the behavioral approachof Willems [62, 63] to dynamical systems, no a priori distinction is made between in-puts and outputs in the description of a dynamical system, but they are distinguishedonly a posteriori in view of the causality implied by the description. The maximumdegree of determinants plays an important role in this connection.

To be specific, let (wj(t) | t = 0, 1, 2, . . .) (j = 1, . . . , n) be n sequences, eachindexed by Z+ = t ∈ Z | t ≥ 0. We consider here an autoregressive (AR) model,in which we assume that they are subject to a system of m homogeneous differenceequations

n∑

j=1

Nij∑

k=1

Aijkwj(t + k) = 0 (i = 1, . . . ,m)

202 KAZUO MUROTA

with constant coefficients Aijk. Denoting by s the backward time shift, i.e., s ·wj(t) = wj(t + 1), the above equation can be rewritten as A(s)w(t) = 0 with

A(s) = (∑Nij

k=1 Aijksk | i = 1, . . . ,m; j = 1, . . . , n) and w(t) = (wj(t) | j = 1, . . . , n).

The variables wj are called external variables, which are to be divided into twoparts, inputs and outputs, so that (in the Z-domain) the outputs can be computedfrom the inputs as well as their initial values using proper transfer matrices. Thenumber of outputs is equal to the rank of A(s), say r, and consequently, that ofinputs is n − r. Such a splitting of n external variables into inputs and outputs isnamed a causal splitting by van der Woude [66].

In case r = m, a causal splitting is tantamount to a splitting of the columnset CA of A(s) into two disjoint sets B and CA − B in such a way that A[RA, B]is nonsingular and A[RA, B]−1A[RA, CA − B] is proper, where RA denotes the rowset of A(s). It is easy to see that A[RA, B]−1A[RA, CA − B] is proper if and onlyif degs detA[RA, B] is maximized by B. Also in the general case, finding a causalsplitting amounts to finding a submatrix A[I,B] with |I| = |B| = r that has themaximum value of degs detA[I,B], as follows.

Theorem 2.3 (van der Woude [64]). (B,CA−B) is a causal splitting if and onlyif there exists I ⊆ RA such that |I| = |B| = r and degs detA[I,B] = δr, where δr isdefined by (1.1).

Let us call B ⊆ CA a dynamical base of a polynomial matrix A(s) if (B,CA −B)is a causal splitting. In section 3.1 we will see that the family of dynamical bases ofa given matrix possesses a nice combinatorial property, forming the basis family of amatroid.

3. Valuated matroid. In this section we summarize relevant facts on valuatedmatroids and discuss their implications for polynomial/rational matrices.

3.1. Definition. As is already mentioned in the introduction, a valuated ma-troid is a pair M = (V, ω) of a finite set V and a function ω : 2V → R ∪ −∞ suchthat

B = B ⊆ V | ω(B) 6= −∞(3.1)

is nonempty and that the following exchange property holds:(MV) For B,B′ ∈ B and u ∈ B−B′, there exists v ∈ B′−B, such that B−u+v ∈ B,

B′ + u− v ∈ B, and

ω(B) + ω(B′) ≤ ω(B − u + v) + ω(B′ + u− v).

If this is the case, B satisfies the following simultaneous exchange property:(SE) For B,B′ ∈ B and u ∈ B−B′, there exists v ∈ B′−B such that B−u+ v ∈ B,

B′ + u− v ∈ B,and accordingly B forms the basis family of a matroid. Therefore, we can alternativelysay that a valuated matroid is a triple M = (V,B, ω), where (V,B) is a matroid(defined in terms of the basis family) and ω : B → R is a function satisfying (MV).

Our interest in valuated matroids originates from the fact that a rational functionmatrix A(s) defines a valuated matroid. Let A(s) be an m×n matrix of rank m witheach entry being a rational function in a variable s, and let MA = (CA,BA) denotethe (linear) matroid [60, 61] defined on the column set CA of A(s) by the linearindependence of the column vectors. Namely,

BA = B ⊆ CA | A[RA, B] is nonsingular,(3.2)


where RA denotes the row set of A. Then ωA : BA → Z defined by

ωA(B) = degs detA[RA, B] (B ∈ BA)(3.3)

satisfies the exchange axiom (MV) above (see Dress and Wenzel [11] for the proof).The valuated matroid explained in Introduction (cf., (1.3), (1.4)) is a variant of thisconstruction.

3.2. Maximization in a valuated matroid. Let M = (V,B, ω) be a valuatedmatroid. For B ∈ B, u ∈ B, and v ∈ V −B we define

ω(B, u, v) = ω(B − u + v) − ω(B).(3.4)

The first lemma is most fundamental, showing the local optimality implies the globaloptimality.

Lemma 3.1 (Dress and Wenzel [10, 11]). Let B ∈ B. Then ω(B) ≥ ω(B′) forany B′ ⊆ V if and only if

ω(B, u, v) ≤ 0 for any u ∈ B and v ∈ V −B.(3.5)

When applied to a rational function matrix, this lemma yields the following. Itis remarked that the second statement below is implicit in Willems [62] and van derWoude [64] (cf., Theorem 2.3).

Lemma 3.2. For MA = (CA,BA, ωA) associated with a rational matrix A(s) ofrow-full rank, we have the following.

(1) For B ∈ BA,

ωA(B, u, v) = degs(A[RA, B]−1A[RA, CA −B])uv (u ∈ B, v ∈ CA −B).

(The right-hand side designates the degree of the (u, v) entry of the rational matrixA[RA, B]−1A[RA, CA −B].)

(2) B ⊆ CA maximizes ωA if and only if A[RA, B]−1A[RA, CA − B] is a properrational matrix.

For p : V → R we define ω[p] : 2V → R ∪ −∞ (or B → R) by

ω[p](B) = ω(B) +∑

p(u) | u ∈ B.(3.6)

M[p] = (V,B, ω[p]) is again a valuated matroid, called a similarity transformation ofM. For MA = (CA,BA, ωA) associated with a rational matrix A(s), this operationcorresponds to multiplying a diagonal matrix diag (s; p) = diag (sp1 , . . . , spn) from theright.

The following theorem characterizes a valuated matroid as a family of matroids.The “only if” part is immediate from (MV), as observed by Dress and Wenzel [11].

Theorem 3.3 (Murota [43, 44]). Let ω : 2V → R∪−∞ be a function such thatB = B ⊆ V | ω(B) 6= −∞ forms the basis family of a matroid on V . Then (V,B, ω)is a valuated matroid if and only if for any p : V → R the set of the maximizers ofω[p] forms the basis family of a matroid on V .

As explained in section 2.3, a causal splitting for A(s) consists of finding a baseB ∈ BA such that A[RA, B]−1A[RA, CA − B] is proper. We have named such B adynamical base. Combination of Lemma 3.2 and Theorem 3.3 reveals that the set ofdynamical bases indeed forms the basis family of a matroid on CA. This observation

204 KAZUO MUROTA

suggests a promising research direction toward optimization on and enumeration ofdynamical bases using general results obtained in matroid theory [15, 18, 32].

Lemma 3.4. For MA = (CA,BA, ωA) associated with a rational matrix A(s)of row-full rank, the matroid defined by the maximizers of ωA agrees with the linearmatroid defined on CA by a constant matrix

A∗B = lim

s→∞A[RA, B]−1A[RA, CA],

where B is a maximizer of ωA. (The matroid defined on CA by A∗B is independent of

the choice of B.)Proof. The proof is immediate from Lemma 3.2.Let R and C be disjoint finite sets and δ : 2R × 2C → R ∪ −∞ be a map such

that

δ(I, J) = ω((R− I) ∪ J) (I ⊆ R, J ⊆ C)(3.7)

for some valuated matroid (R∪C,ω) with ω(R) 6= −∞. Such triple (R,C, δ) is calleda valuated bimatroid in [39]. Define

S = (I, J) | |I| = |J |, I ⊆ R, J ⊆ C,

r = max|I| | ∃(I, J) ∈ S : δ(I, J) 6= −∞,

Sk = (I, J) | |I| = |J | = k, I ⊆ R, J ⊆ C (0 ≤ k ≤ r),

δk = maxδ(I, J) | (I, J) ∈ Sk (0 ≤ k ≤ r),

Mk = (I, J) ∈ Sk | δ(I, J) = δk (0 ≤ k ≤ r).

Note that δ(∅, ∅) 6= −∞, and δ(I, J) 6= −∞ only if (I, J) ∈ S.Theorem 3.5 (Murota [39]). The sequence (δ0, δ1, . . . , δr) is concave, i.e.,

δk−1 + δk+1 ≤ 2δk (1 ≤ k ≤ r − 1).

Theorem 3.6 (Murota [39]). For any (Ik, Jk) ∈ Mk with 1 ≤ k ≤ r − 1, thereexist (Il, Jl) ∈ Ml (0 ≤ l ≤ r, l 6= k) such that

(∅ =) I0 ⊆ I1 ⊆ · · · ⊆ Ik−1 ⊆ Ik ⊆ Ik+1 ⊆ · · · ⊆ Ir,

(∅ =) J0 ⊆ J1 ⊆ · · · ⊆ Jk−1 ⊆ Jk ⊆ Jk+1 ⊆ · · · ⊆ Jr.

Theorem 3.6 justifies the following incremental greedy algorithm for computingδk for k = 0, 1, . . . , r. This algorithm involves O(r|R| |C|) evaluations of δ to computethe whole sequence (δ0, δ1, · · · , δr).

Greedy algorithm for δk (k = 1, 2, . . .).I0 := ∅; J0 := ∅;for k := 1, 2, . . . do.

Find i ∈ R−Ik−1, j ∈ C−Jk−1 that maximizes δ(Ik−1+i, Jk−1+j)and put Ik := Ik−1 + i, Jk := Jk−1 + j and δk := δ(Ik, Jk).

The iteration stops when δ(Ik, Jk) = −∞, and then r = k − 1. See [39] for anotheralgorithm to compute (δ0, δ1, . . . , δr).

Given a rational matrix A(s) we can naturally define a valuated bimatroid (RA,CA, δA) by

δA(I, J) = degs detA[I, J ] (I ⊆ RA, J ⊆ CA).(3.8)


The associated valuated matroid in (3.7) is given by ωA

of (3.3) for an m× (m + n)

matrix A = (Im A). In this special case the nesting property stated in Theorem 3.6has been observed, though in a slightly weaker form, by Svaricek [53], [54, Satz 6.23]along with the greedy algorithm above.

3.3. Valuated independent assignment problem. The valuated indepen-dent assignment problem is defined as follows. Suppose we are given a bipartite graphG = (V +, V −;E), valuated matroids M+ = (V +,B+, ω+) and M− = (V −,B−, ω−),and a weight function w : E → R.

Valuated independent assignment problem.Find a matching M(⊆ E) that maximizes

Ω(M) ≡ w(M) + ω+(∂+M) + ω−(∂−M)(3.9)

subject to the constraint

∂+M ∈ B+, ∂−M ∈ B−,(3.10)

where ∂+M (resp., ∂−M) denotes the set of vertices in V + (resp., V −) incident to M .A matching M satisfying the constraint (3.10) is called an independent assignment.Clearly the two matroids must have the same rank for the feasibility of this problem.

The above problem reduces to the independent assignment problem of Iri andTomizawa [28] if the valuations are trivial with ω±(B) = 0 for B ∈ B±, and reducesfurther to the conventional assignment problem (cf., e.g., Lawler [32]) if the matroids

are trivial or free with B± = 2V±

.The following theorem gives an optimality criterion in (1), referring to the ex-

istence of a “potential” function, whereas its reformulation in (2) reveals its dualitynature. This is a natural extension of the corresponding result [28] for the ordinary(independent) assignment problem.

Theorem 3.7 (Murota [41]). (1) An independent assignment M in G is optimalfor the valuated independent assignment problem (3.9)–(3.10) if and only if there existsa “potential” function p : V + ∪ V − → R such that

(i) w(a) − p(∂+a) + p(∂−a)

≤ 0 (a ∈ E),= 0 (a ∈ M).

(ii) ∂+M is a maximum-weight base of M+ with respect to ω+[p+],(iii) ∂−M is a maximum-weight base of M− with respect to ω−[−p−], where p±

is the restriction of p to V ±, and ω+[p+] (resp., ω−[−p−]) is the similarity transfor-mation defined in (3.6); namely,

ω+[p+](B+) = ω+(B+) +∑

p(u) | u ∈ B+ (B+ ⊆ V +),

ω−[−p−](B−) = ω−(B−) −∑

p(u) | u ∈ B− (B− ⊆ V −).

(2)

maxM

Ω(M) | M : independent assignment

= minp

max(ω+[p+]) + max(ω−[−p−]) | w(a) − p(∂+a) + p(∂−a) ≤ 0 (a ∈ E).

(3) If ω+, ω−, and w are all integer-valued, the potential p in (1) and (2) can bechosen to be integer-valued.

206 KAZUO MUROTA

(4) Let p be a potential that satisfies (i)–(iii) above for some (optimal) independentassignment M = M0. An independent assignment M ′ is optimal if and only if itsatisfies (i)–(iii) (with M replaced by M ′).

Remark 3.1 Just as the weighted matroid intersection problem may be regardedas being equivalent to the independent assignment problem, the following problem isequivalent to the valuated independent assignment problem (see [41] for other equiv-alent problems).

Valuated matroid intersection problem.Given a pair of valuated matroids M1 = (V,B1, ω1) and M2 =(V,B2, ω2) defined on a common ground set V , and a weight func-tion w : V → R, find a common base B ∈ B1 ∩ B2 that maximizesw(B) + ω1(B) + ω2(B).

The optimality criterion of Theorem 3.7, when adapted to this problem, gives a gen-eralization of the well-known optimality criterion [13, 14, 16, 17, 32] for the weightedmatroid intersection problem.

The duality result in Theorem 3.7 above admits a linear algebraic interpretationfor a triple matrix product as follows, although the author is not yet aware of itsengineering applications.

Theorem 3.8. Assume that a matrix product A(s) = Q1(s)T (s)Q2(s) is non-singular, where Q1(s) (resp., Q2(s)) is a k ×m (resp., n× k) rational matrix over afield K , and T (s) is an m×n rational matrix over an extension field F (⊇ K ) suchthat the set of the coefficients is algebraically independent over K . Then there existk × k nonsingular rational matrices S1(s), S2(s) and diagonal matrices diag (s; p) =diag (sp1 , . . . , spm), diag (s; q) = diag (sq1 , . . . , sqn) with p ∈ Zm and q ∈ Zn such that

degs detA = degs detS1 + degs detS2

and the matrices

Q1(s) = S1(s)−1

Q1(s) diag (s; p),

T (s) = diag (s;−p)T (s) diag (s;−q),

Q2(s) = diag (s; q)Q2(s)S2(s)−1

are all proper. Note that S1(s)−1

A(s)S2(s)−1

= Q1(s)T (s)Q2(s).Proof. First, by the Cauchy–Binet formula, we have

detA =∑

|I|=|J|=k

±detQ1[∗, I] · detT [I, J ] · detQ2[J, ∗],

where Q1[∗, I] designates the k × k submatrix of Q1 with column set I and thewhole row set, and similarly for Q2[J, ∗]. There is no numerical cancellation in thesummation above by virtue of the assumed algebraic independence of the coefficientsin T (s), and hence

degs detA = max|I|=|J|=k

degs detQ1[∗, I] + degs detT [I, J ] + degs detQ2[J, ∗].

Next, consider a valuated independent assignment problem defined as follows.The vertex sets V + and V − are the row set and the column set of T (s), respectively,and E = (i, j) | Tij(s) 6= 0. The valuated matroids attached to V + and V − arethose defined by Q1(s) and Q2(s) as in (3.3), and the weight wij of an edge (i, j) ∈ E


is defined by wij = degs Tij(s). Note that the maximum value of∑

(i,j)∈M wij over

all matchings M with I = ∂+M and J = ∂−M is equal to degs detT [I, J ].

Then we see from the above identity that degs detA is equal to the maximumvalue of Ω(M) over all independent assignment M . Let M be an optimal independentassignment and put I = ∂+M and J = ∂−M . Let p : V + ∪ V − → Z be the potentialin Theorem 3.7, and define p ∈ Zm and q ∈ Zn by pi = pi for i ∈ V + and qj = −pj forj ∈ V −. Define S1 = Q1[∗, I] diag (s; pI) and S2 = diag (s; qJ)Q2[J, ∗], where pI ∈ ZI

is the restriction of p to I and similarly for qJ ∈ ZJ .

The conditions (i), (ii) and (iii) in Theorem 3.7(1), coupled with Lemma 3.2,imply the properness of T (s), Q1(s), and Q2(s), respectively.

Remark 3.2. The close relationship between the triple matrix product and theindependent assignment problem through the Binet-Cauchy formula was first observedby Tomizawa and Iri [55, 56]. To be more precise, the rank of A = Q1TQ2 wasexpressed in [55] as the maximum size of an independent matching, whereas thedegree of the determinant of A(s) = Q1T (s)Q2 with constant matrices Qi (i = 1, 2)was represented in [56] as the optimal value of an independent assignment. Ourpresent contribution lies in an extension to the more general case with polynomialmatrices Qi(s) (i = 1, 2) by means of valuated matroids, and also in an explicitstatement concerning the transformation into proper matrices.

4. Degree of mixed polynomial matrix.

4.1. Mixed polynomial matrix. Let K be a subfield of a field F . A matrixA over F (i.e., Aij ∈ F ) is called a mixed matrix with respect to F /K if

A = Q + T,(4.1)

where

(M-Q) Q is a matrix over K (i.e., Qij ∈ K ), and(M-T) T is a matrix over F (i.e., Tij ∈ F ) such that the set of its nonzero entries is

algebraically independent over K .

A mixed matrix A of (4.1) is called a layered mixed matrix (or an LM matrix) ifthe nonzero rows of Q and T are disjoint. In other words, A is an LM matrix if it canbe put into the following form with a permutation of rows:

A =

(QT

)=

(QO

)+

(OT

),(4.2)

where Q and T satisfy (M-Q) and (M-T) above, respectively.

Though an LM-matrix is a special case of mixed matrix, the class of LM matricesis as general as the class of mixed matrices both in theory and in application. Withan m× n mixed matrix A = Q + T we associate a (2m) × (m + n) LM matrix

A =

(QT

)=

(Im Q

−diag (t1, . . . , tm) T

),(4.3)

where diag (t1, . . . , tm) is a diagonal matrix with “new” variables t1, . . . , tm(∈ F ).Such transformation often works in the analysis of a mixed matrix by way of an LM-

208 KAZUO MUROTA

matrix. For example, if we are interested in the rank of A, we may instead computethe rank of A and use the relation rankA = rankA−m.

A polynomial matrix A(s) over F (i.e., Aij(s) ∈ F [s]) is called a mixed polynomialmatrix with respect to F /K if

A(s) = Q(s) + T (s) =

N∑

k=0

skQk +

N∑

k=0

skTk(4.4)

for some integer N ≥ 0, where(MP-Q) Qk (k = 0, 1, . . . , N) are matrices over K , and(MP-T) Tk (k = 0, 1, . . . , N) are matrices over F such that the set of their nonzero

entries is algebraically independent over K .A mixed polynomial matrix with respect to F /K is a mixed matrix with respect to

F (s)/K (s). Also note that A(s) =∑N

k=0 skAk with Ak = Qk + Tk and that Ak, for

each k, is a mixed matrix with respect to F /K .A mixed polynomial matrix A(s) of (4.4) is called a layered mixed polynomial

matrix (or an LM-polynomial matrix ) if the nonzero rows of Q(s) and T (s) are disjoint,that is, if it looks like

A(s) =

(Q(s)T (s)

),(4.5)

where Q(s) and T (s) satisfy (MP-Q) and (MP-T) above, respectively. We denoteby RQ and RT the row sets of Q(s), and T (s), respectively, whereas the columnsets of A(s), Q(s), and T (s), are identified and denoted by C. We put mQ = |RQ|,mT = |RT |, n = |C|. Obviously, an LM-polynomial matrix with respect to F /K isan LM-matrix with respect to F (s)/K (s).

The concepts of (layered) mixed (polynomial) matrices were introduced in Murotaand Iri [45], Murota, Iri, and Nakamura [46], and Murota [34] as mathematical toolsfor the structural/combinatorial analysis of engineering systems. See [36] and [40] forsurveys; the former deals with mathematical properties of (layered) mixed matriceswhile the latter explains engineering motivations.

4.2. Basic identities. We present basic identities concerning the degree of thedeterminant of (layered) mixed polynomial matrices, which are easy to derive from(MP-Q), (MP-T), and the Laplace expansion of determinants. They will be upgradedin section 5.1 to novel identities of deeper mathematical content.

Recall that R and C denote the row set and the column set of A(s), respectively,and Q[I, J ], e.g., denotes the submatrix of Q with row set I and column set J .

Theorem 4.1 (Murota [33]). For a square mixed polynomial matrix A(s) =Q(s) + T (s),

degs detA = max|I|=|J|

I⊆R,J⊆C

degs detQ[I, J ] + degs detT [R− I, C − J ].(4.6)

(For a singular matrix A both sides are equal to −∞.)Proof. By the Laplace expansion [19] we see

detA =∑

|I|=|J|

±detQ[I, J ] · detT [R− I, C − J ].


Since the degree of a sum is bounded by the maximum degree of a summand, weobtain

degs detA ≤ max|I|=|J|

degs(detQ[I, J ] · detT [R− I, C − J ])

= max|I|=|J|

degs detQ[I, J ] + degs detT [R− I, C − J ],

where the inequality turns into an equality provided the highest-degree terms do notcancel one another. The algebraic independence of the nonzero coefficients in T (s)ensures this.

The above theorem immediately yields a similar identity for an LM-polynomialmatrix A(s) = (Q(s)

T (s)). Recall that RQ and RT denote the row sets of Q(s) and T (s),

respectively, and C denotes the column sets of A(s), Q(s), and T (s).

Theorem 4.2. For a square LM-polynomial matrix A(s) = (Q(s)T (s)

),

degs detA = maxJ⊆C

degs detQ[RQ, J ] + degs detT [RT , C − J ].(4.7)

(For a singular matrix A both sides are equal to −∞.)In what follows we focus on an LM-polynomial matrix A(s) and consider a variant

of δk(A). Namely, for A(s) = (Q(s)T (s)

) we define

δLMk (A) = max

I, Jdegs detA[RQ ∪ I, J ] |(4.8)

I ⊆ RT , J ⊆ C, |I| = k, |J | = mQ + k,

where 0 ≤ k ≤ min(mT , n − mQ). It should be clear that δLMk (A) designates the

highest degree of a minor of order mQ + k with row set containing RQ, and thatδLMk (A) = −∞ if there exists no (I, J) that satisfies the conditions on the right-hand

side of (4.8). By substituting (4.7) into (4.8) we obtain

δLMk (A) = max

I, J, Bdegs detQ[RQ, B] + degs detT [I, J −B] |(4.9)

I ⊆ RT , B ⊆ J ⊆ C, |I| = k, |J | = mQ + k, |B| = mQ.

We prefer to work with δLMk (A) for an LM-polynomial matrix A(s) rather than to

deal directly with δk(A) for a mixed polynomial matrix A(s) = Q(s) + T (s). This isbecause (i) any algorithm for δLM

k can be used to compute δk(A) for a general mixedpolynomial matrix A(s) (as explained below), and (ii) our algorithm description ismuch simpler for δLM

k .

The reduction of δk(A) for A(s) = Q(s)+T (s) to δLMk (A) with an LM-polynomial

matrix A(s) is similar to the transformation (4.3) of a mixed matrix to another LM-matrix. Given an m× n mixed polynomial matrix A(s) = Q(s) + T (s) we consider a(2m) × (m + n) LM-polynomial matrix

A(s) =

(Q(s)

T (s)

)=

(diag (sd1 , . . . , sdm) Q(s)

−diag (t1sd1 , . . . , tmsdm) T (s)

),(4.10)

where t1, . . . , tm (∈ F ) are “new” variables, and

di = maxj∈CA

degs Qij(s) (i ∈ RA),(4.11)

210 KAZUO MUROTA

where RA and CA denote the row set and the column set of A(s), and hence those of

Q(s). The following lemma reveals the relation between δk(A) and δLMk (A).

Lemma 4.3. Let A(s) be a mixed polynomial matrix and A(s) be the associatedLM-polynomial matrix defined by (4.10). Then we have

δk(A) = δLMk (A) −

m∑

i=1

di.

Proof. Define

A(s) =

( RA CA

RQ diag (sd1 , . . . , sdm) Q(s)RT −diag (sd1 , . . . , sdm) T (s)

),(4.12)

which is obtained from A(s) by putting ti = 1 (i = 1, . . . ,m). It is obtained from

A(s) also by dividing the (m + i)th row by ti (i = 1, . . . ,m) and redefining Tij(s)/tito be Tij(s). The latter fact implies δLM

k (A) = δLMk (A). (Here it should be clear that

δLMk (A) is defined similarly to (4.8), although this is a slight abuse of notation since

A(s) is not an LM-polynomial matrix.)We denote the row sets (column sets) of A, Q, and T by RA, RQ, and RT (by CA,

CQ, and CT ), respectively, where RQ and RT (CQ and CT ) have natural one-to-onecorrespondence with RA (with CA). Also we denote the row sets and the column sets

of A, by R = RQ ∪RT and C = RA ∪ CA, as indicated in (4.12).If J ⊇ RA, we have

degs det A[RQ ∪ I, J ] = degs detA[I, J ∩ CA] +

m∑

i=1

di (I ⊆ RT ).

Hence, taking the maximum of this expression over all I and J with |I| = |J |−mQ = kand J ⊇ RA, we see that δk(A) +

∑m

i=1 di is equal to

maxdegs det A[RQ ∪ I, J ] | I ⊆ RT , RA ⊆ J ⊆ C, |I| = k, |J | = mQ + k.

It remains to be shown that the extra constraint “J ⊇ RA” can be removed with-out affecting the maximum value. Fix I ⊆ RT and let J ⊆ RA∪CA be a maximizer ofdegs det A[RQ∪I, J ] satisfying J ⊇ RA. We claim that A[RQ∪I, J ]−1A[RQ∪I, CA\J ]is a proper rational matrix. This claim implies, by Lemma 3.2, that J is an optimumsolution to the maximization problem without the constraint “J ⊇ RA.”

The claim can be proven as follows. Denoting by IQ and IA the copies of I in RQ

and RA, respectively, we partition the matrix A[RQ ∪ I,RA ∪ CA] as

A[RQ ∪ I,RA ∪ CA] =

RA ∩ IA RA \ IA CA ∩ J CA \ J

RQ ∩ IQ D1 O Q11 Q12

RQ \ IQ O D2 Q21 Q22

RT ∩ I −D1 O T11 T12

with the obvious shorthand notations D1, Q11, T11, etc. for the relevant submatricesof diag (sd1 , . . . , sdm), Q(s), T (s), etc. By row transformations we obtain the following


sequence of matrices:

D1 O Q11 Q12

O D2 Q21 Q22

−D1 O T11 T12

⇒

D1 O Q11 Q12

O D2 Q21 Q22

O O A11 A12

(Aij = Qij + Tij)

⇒

D1 O Q11 Q12

O D2 Q21 Q22

O O I A11−1A12

⇒

D1 O O Q12 −Q11A11−1A12

O D2 O Q22 −Q12A11−1A12

O O I A11−1A12

⇒

I O O D1−1[Q12 −Q11A11

−1A12]O I O D2

−1[Q22 −Q12A11−1A12]

O O I A11−1A12

.

This shows

A[RQ ∪ I, J ]−1A[RQ ∪ I, CA \ J ] =

CA \ J

RA B1(s)

CA ∩ J B2(s)

with

B1(s) = diag (s−d1 , . . . , s−dm)Q[RQ, CA \ J ] −Q[RQ, CA ∩ J ]B2(s),

B2(s) = A[I, CA ∩ J ]−1A[I, CA \ J ].

Here B2(s) is a proper rational matrix (i.e., each entry has deg ≤ 0) by the choiceof J, and diag (s−d1 , . . . , s−dm)Q[RQ, CA] is also proper by the definition (4.11) of di.

Therefore, A[RQ ∪ I, J ]−1A[RQ ∪ I, CA \ J ] is a proper rational matrix.

Example 4.1. For the 2× 3 mixed polynomial matrix A(s) in Example 1.1we haved1 = 3, d2 = 2, and

A(s) =

r1 r2 c1 c2 c3rQ1 s3 s3 + 1 s2 1rQ2 s2 s2 s 0

rT1 −t1s3 0 α1 α2s

rT2 −t2s2 α3 0 0

.

It is easy to verify that

δ1(A) = degs detA[r1, c1] = 3,

δ2(A) = degs detA[r1, r2, c1, c3] = 3,

whereas

δLM1 (A) = degs det A[rQ1, rQ2, rT1, r1, r2, c1] = 3 + 5,

δLM2 (A) = degs det A[rQ1, rQ2, rT1, rT2, r1, r2, c1, c3] = 3 + 5.

212 KAZUO MUROTA

4.3. Reduction to valuated independent assignment. We describe howthe computation of δLM

k (A) for an LM-polynomial matrix A(s) = (Q(s)T (s)

) can be re-

duced to solving a valuated independent assignment problem. We denote by MQ =(CQ,BQ, ωQ) the valuated matroid associated with Q(s) as (3.2) and (3.3); namely,

BQ = B ⊆ CQ | detQ[RQ, B] 6= 0,(4.13)

ωQ(B) = degs detQ[RQ, B] (B ∈ BQ).(4.14)

Here and henceforth CQ = jQ | j ∈ C denotes a disjoint copy of the column set C ofA (with jQ ∈ CQ denoting the copy of j ∈ C), whereas RQ and RT mean, as before,the row sets of Q(s) and T (s), respectively; |RQ| = mQ, |RT | = mT , and |C| = n.

We consider a valuated independent assignment problem defined on a bipartitegraph G = (V +, V −;E) with V + = RT ∪ CQ, V − = C, and E = ET ∪ EQ, where

ET = (i, j) | i ∈ RT , j ∈ C, Tij(s) 6= 0, EQ = (jQ, j) | j ∈ C.

The valuated matroids M+ = (V +,B+, ω+) and M− = (V −,B−, ω−) attached to V +

and V − are defined by

B+ = B+ ⊆ V + | B+ ∩ CQ ∈ BQ, |B+ ∩RT | = k,

B− = B− ⊆ V − | |B−| = mQ + k

and

ω+(B+) = ωQ(B+ ∩ CQ) (B+ ∈ B+),

ω−(B−) = 0 (B− ∈ B−).

The weight wij of an edge (i, j) ∈ E is defined by

wij =

degs Tij(s) ((i, j) ∈ ET ),0 ((i, j) ∈ EQ).

(4.15)

Note the dependence of M± on k as well as the independence of G and w of k.We then have the following characterization of δLM

k (A) in terms of the optimalvalue of the valuated independent assignment problem. Recall the notation Ω(M) of(3.9) for the value of an independent assignment M .

Theorem 4.4. For an LM-polynomial matrix A(s) = (Q(s)T (s)

) and an integer k

with 0 ≤ k ≤ min(mT , n −mQ), δLMk (A) of (4.8) coincides with the optimal value of

the valuated independent assignment problem defined above. That is,

δLMk (A) = maxΩ(M) | M : independent assignment,(4.16)

where the right-hand side is defined to be −∞ if there exists no independent assign-ment M .

Proof. Define

∆(I, J,B) = degs detQ[RQ, B] + degs detT [I, J −B],(4.17)

which is the function to be maximized in the expression (4.9) for δLMk (A). By virtue

of the algebraic independence of the nonzero coefficients in T (s), the second term,degs detT [I, J−B], is equal to the maximum weight (with respect to wij) of a match-ing of size |I| = |J −B| in the bipartite graph (RT , C;ET ) that covers I and J −B.


Fig. 1. Graph G (©: arcs in M , B = x1Q, x3Q).

Given (I, J,B) with |I| = k and ∆(I, J,B) > −∞, we can construct an independentassignment M such that

I = ∂+(M ∩ ET ), J = ∂−M, B = ∂+(M ∩ EQ),(4.18)

and that M ∩ ET is a maximum weight k-matching in the graph (RT , C;ET ) thatcovers I and J−B. Note that detQ[RQ, B] 6= 0 and |I| = k if and only if B∪I ∈ B+.Moreover, ω+(B ∪ I) = degs detQ[RQ, B] by the definition, and therefore we have∆(I, J,B) = Ω(M). Conversely, an independent assignment M with Ω(M) maximumdetermines (I, J,B), as above, for which ∆(I, J,B) = Ω(M) holds true. Hence themaximum value of ∆(I, J,B) is equal to that of Ω(M).

Example 4.2. The valuated independent assignment problem associated with a4 × 5 LM-polynomial matrix

A(s) =

x1 x2 x3 x4 x5

s3 0 s3 + 1 s2 10 s2 s2 s 0

f1 −t1s3 0 0 α1 α2s

f2 0 −t2s2 α3 0 0

(4.19)

with k = 2 is illustrated in Fig. 1. This matrix is essentially the same as A(s) inExample 4.1, but the columns and the rows are now indexed as C = x1, x2, x3, x4, x5and RT = f1, f2 and accordingly CQ = x1Q, x2Q, x3Q, x4Q, x5Q. An optimalindependent assignment M = (f1, x5), (f2, x2), (x1Q, x1), (x3Q, x3) is marked by ©.We have I = ∂+(M∩ET ) = f1, f2, J = ∂−M = x1, x2, x3, x5, B = ∂+(M∩EQ) =x1Q, x3Q ∈ BQ, ωQ(B) = 5, w(M) = 1 + 2 = 3, and therefore Ω(M) = 5 + 3 = 8,which agrees with δLM

2 (A) = 8. In Section 5.2 we will come back to this example toexplain the algorithm.

4.4. Novel identities. The basic identities on the degree of subdeterminantspresented in section 4.2 are upgraded here to novel identities of duality nature. They

214 KAZUO MUROTA

are obtained from the duality result (Theorem 3.7) on the valuated independent as-signment problem, i.e., the optimality criterion involving a potential function. Besidesthis, the potential will be used extensively also in the algorithm in section 5, and fur-thermore it will play a crucial role in section 4.5 in determining the coefficient of thehighest-degree term of detA(s).

Consider the valuated independent assignment problem associated with A(s),introduced in section 4.3. Let M be an optimal independent assignment, (I, J,B) bedefined by (4.18), and p : RT ∪ C ∪ CQ → Z be a potential function guaranteed inTheorem 3.7.

We may assume that p(jQ) = p(j) for j ∈ C (where jQ ∈ CQ denotes the copyof j ∈ C). To see this, first note that p(jQ) ≥ p(j) for j ∈ C and the equality holdsif (jQ, j) ∈ M . For j ∈ C with (jQ, j) 6∈ M , we can redefine p(jQ) to p(j), since(jQ, j) is the only arc going out of jQ and its weight wjQj is zero. Define q ∈ ZRT

and p ∈ ZC by

qi = p(i) (i ∈ RT ), pj = −p(j) (j ∈ C).(4.20)

Then the conditions (i)–(iii) in Theorem 3.7(1) are expressed as follows:

degs Tij(s) ≤ qi + pj ((i, j) ∈ ET ),(4.21)

degs Tij(s) = qi + pj ((i, j) ∈ M ∩ ET ),(4.22)

ωQ[−p](B) = maxB′∈BQ

ωQ[−p](B′),(4.23)

q(I) = max|I′|=k

q(I ′),(4.24)

p(J) = max|J′|=mQ+k

p(J ′),(4.25)

where q(I) =∑

i∈ qi and p(J) =∑

j∈J pj . These conditions imply

δLMk (A) = degs detQ[RQ, B] + degs detT [I, J −B](4.26)

= ωQ(B) + q(I) + p(J −B)

= ωQ[−p](B) + q(I) + p(J)

= maxB′∈BQ

ωQ[−p](B′) + max|I′|=k

q(I ′) + max|J′|=mQ+k

p(J ′).

Thus we obtain the following theorem.Theorem 4.5. For an LM-polynomial matrix A(s) = (Q(s)

T (s)) and an integer k

such that δLMk (A) > −∞, the following identity holds true:

δLMk (A) = min

qi+pj≥degs Tij

[max|I|=k

q(I) + max|J|=mQ+k

p(J) + maxB∈BQ

ωQ[−p](B)

],(4.27)

where the minimum is taken over all q ∈ ZRT , p ∈ ZC satisfying qi + pj ≥ degs Tij

for all (i, j).Proof. Let (I, J,B) be as above. For any (q′, p′) with q′i + p′j ≥ degs Tij (∀(i, j)),

we have

δLMk (A) = degs detQ[RQ, B] + degs detT [I, J −B]

≤ ωQ(B) + q′(I) + p′(J −B)

= ωQ[−p′](B) + q′(I) + p′(J)

≤ maxB′∈BQ

ωQ[−p′](B′) + max|I′|=k

q′(I ′) + max|J′|=mQ+k

p′(J ′),


whereas the inequalities turn into equalities for (q′, p′) = (q, p), as in (4.26).With q, p, and B above, we can transform the matrix A(s) to another LM-

polynomial matrix that is somehow canonical with respect to δLMk . Let S(s) =

(Q[RQ, B] · diag (s;−pB))−1, where pB is the restriction of p to B, and define

A(s) =

(Q(s)

T (s)

)=

(S(s) OO diag (s;−q)

)·A(s) · diag (s;−p).(4.28)

The conditions (4.21)–(4.23) mean that

A(s) =

B J −B C − J

RQ ImQQ′

2(s) Q′3(s)

I T ′1(s) T •

2 (s) T ′3(s)

RT − I T ′′1 (s) T ′′

2 (s) T ′′3 (s)

(4.29)

is a proper matrix, in which T •2 (s) admits a transversal consisting of entries of degree

zero. Obviously,

δLMk (A) = max

|B′|=mQ

degs det Q[RQ, B′] + max

|I′|=|J′|=kdegs det T [I ′, J ′],(4.30)

in which all the three terms are equal to zero. From this we obtain the followingtheorem.

Theorem 4.6. For an LM-polynomial matrix A(s) = (Q(s)T (s)

) and an integer k

such that δLMk (A) > −∞, there exist p ∈ ZC and q ∈ ZRT such that

A(s) =

(ImQ

OO diag (s;−q)

)·A(s) · diag (s;−p),

Q(s) = Q(s) · diag (s;−p),

T (s) = diag (s;−q) · T (s) · diag (s;−p)

satisfy

δLMk (A) = max

|B|=mQ

degs det Q[RQ, B] + max|I|=|J|=k

degs det T [I, J ].(4.31)

Here we can impose on A an additional condition:

δLMk (A) = δLM

k (A) − max|I|=k

q(I) − max|J|=mQ+k

p(J).(4.32)

It should be clear that A(s) = ( Q(s)T (s)

), which is also an LM-polynomial matrix.

Proof. The first identity (4.31) follows from (4.30). The second identity (4.32) is

due to (4.26) combined with δLMk (A) = δLM

k (A) + degs detS−1 = ωQ[−p](B).Example 4.3. We illustrate the above argument for the LM-polynomial matrix

A(s) of (4.19) with k = 2. The vectors p ∈ ZC and q ∈ ZRT of (4.20) are given byp = (−1,−1,−3,−4,−3) and q = (4, 3). Accordingly we have

A(s) =

x1 x2 x3 x4 x5

s4 0 s6 + s3 s6 s3

0 s3 s5 s5 0

f1 −t1 0 0 α1 α2

f2 0 −t2 α3 0 0

216 KAZUO MUROTA

for which (4.31) holds true with δLM2 (A) = 9 = 9 + 0. Recall from Example 4.3 that

I = f1, f2, J = x1, x2, x3, x5, B = x1Q, x3Q. The matrix A(s) of (4.28) is equalto

A(s) =

x1 x3 x5 x2 x4

1 0 1s

−s3−1s3

−1s

0 1 0 1s2

1

f1 −t1 0 α2 0 α1

f2 0 α3 0 −t2 0

.(4.33)

In section 5.2 we will come back to this example and explain how the vectors p andq can be found (see the variable p in Fig. 4, in particular).

As corollaries to Theorem 4.6 we obtain the following two theorems, which shouldbe compared to Theorem 4.2 and Theorem 4.1, respectively.

Theorem 4.7. For a nonsingular LM-polynomial matrix A(s) = (Q(s)T (s)

), there

exists p ∈ ZC such that

A(s) = A(s) · diag (s;−p), Q(s) = Q(s) · diag (s;−p), T (s) = T (s) · diag (s;−p)

satisfy

degs det A = max|B|=|RQ|

degs det Q[RQ, B] + max|J|=|RT |

degs det T [RT , J ],(4.34)

where it should be clear that A(s) = ( Q(s)T (s)

), which is also an LM-polynomial matrix.

Proof. Apply Theorem 4.6 with k = mT = n −mQ to obtain p ∈ ZC . The rowtransformation by diag (s;−q) is not necessary in the case of k = mT .

Theorem 4.8. For a nonsingular mixed polynomial matrix A(s) = Q(s) + T (s),there exist pR ∈ ZR and pC ∈ ZC such that

A(s) = diag (s;−pR) ·A(s) · diag (s;−pC),

Q(s) = diag (s;−pR) ·Q(s) · diag (s;−pC),

T (s) = diag (s;−pR) · T (s) · diag (s;−pC)

satisfy

degs det A = max|I|=|J|

I⊆R, J⊆C

degs det Q[I, J ] + max|I|=|J|

I⊆R, J⊆C

degs det T [R− I, C − J ],(4.35)

where it should be clear that A(s) = Q(s) + T (s), which is also a mixed polynomialmatrix.

Proof. Apply Theorem 4.7 to the associated LM-polynomial matrix (4.10) to ob-tain p ∈ ZR∪C . Denote by pR and pC the restrictions of p to R and to C, respectively.Then put pR = d − pR and pC = pC , where d ∈ ZR is the vector of exponents in(4.10).

4.5. Leading coefficient. For a nonsingular LM-polynomial matrix A(s) withrespect to F /K , detA(s) is a polynomial in s with coefficients from F . If we denoteby T the set of nonzero coefficients in T (s), we see that the coefficients in detA(s)are polynomials in T over K . Recall that T ⊆ F and T is algebraically independentover K (cf. (MP-T) in section 4.1).


In this section we are interested in the leading coefficient (= the coefficient ofthe highest-degree term) of detA(s), which we denote by η(T ) ∈ K [T ] (= ring ofpolynomials in T over K ). Namely,

η(T ) = l.c.(detA(s)) = lims→∞

s−δn(A) detA(s),

where l.c.(·) means the leading coefficient of a polynomial in s, and n is the size of thematrix. The following argument shows, among others, that we can determine whichvariables of T appear in η(T ) by means of arithmetic operations in K (s) withoutinvolving T .

Recall A(s) of (4.28), where I = RT and J = C, and define a constant matrix

A∗ = lims→∞

A(s),(4.36)

which is a nonsingular LM-matrix with respect to F /K . It is noted that the matrix

A∗ depends on the choice of (q, p,B). Since

detA(s) = detQ[RQ, B] · det A(s) · sq(RT )+p(C−B), det A(s) = det A∗ + o(1),

where o(1) denotes a term that vanishes as s → ∞, we see that

η(T ) = c · det A∗, c = l.c.(detQ[RQ, B]) ∈ K .(4.37)

For an LM-matrix in general, a block-triangular canonical form is known to ex-ist (Murota [34], Murota, Iri, and Nakamura [46]). The canonical form is calledthe combinatorial canonical form (CCF) of the LM-matrix, and can be computed inO(n3 log n) time with arithmetic operations in the subfield. Furthermore, it is known[35] that the factorization of the determinant of an LM-matrix is given through theirreducible diagonal components of its CCF. See [36] for a survey on the CCF.

Using these general results we see the following.1. A variable t ∈ T appears in η(T ) if and only if t is contained in an irreducible

diagonal block of the CCF of the LM-matrix A∗. Hence, once A(s) is known,the set of variables of T that are contained in η(T ) can be computed inO(n3 log n) time with arithmetic operations in K .

2. The irreducible factors of η(T ), as a polynomial in T over K , are given by

the determinants of the irreducible components in the CCF of A∗. Hence,once A(s) is known, the irreducibility of η(T ) in K [T ] can be determined inO(n3 log n) time with arithmetic operations in K .

Example 4.4. Consider the 4× 4 submatrix of the LM-polynomial matrix A(s) of(4.19) with column set J = x1, x2, x3, x5. We have T = t1, t2, α2, α3. By directcalculation we see

detA[RA, J ] = −α2t2s8 − t1t2s

7 − α2α3s6 − t1α3s

5,

and therefore η(T ) = −α2t2. On the other hand, the matrix A∗ (cf. (4.33)) and itsCCF (which happens to be triangular) are given by

A∗ =

x1 x2 x3 x5

1 −1 0 00 0 1 0

f1 −t1 0 0 α2

f2 0 −t2 α3 0

, CCF =

x5 x1 x2 x3

f1 α2 −t1 0 0

1 −1 0

f2 −t2 α3

1

.

218 KAZUO MUROTA

We have det A∗ = −α2t2, in agreement with η(T ) (up to a constant factor). Thevariables, α2 and t2, appearing in the diagonal blocks agree with those variablescontained in η(T ).

5. Algorithm.

5.1. Algorithm description. In section 4.3 we have explained how to reducethe computation of δLM

k (A) to solving a valuated independent assignment problem.Here we will provide an algorithm for δLM

k (A) by adapting the augmenting algorithmof Murota [42] for a general valuated independent assignment problem. Our algorithmcomputes δLM

k (A) successively for k = 0, 1, 2, . . . , kmax, where kmax is the maximumk with δLM

k (A) > −∞.As described in section 4.3, the associated valuated independent assignment prob-

lem is defined on the bipartite graph G = (V +, V −;E) = (RT ∪ CQ, C;ET ∪ EQ),where CQ is a disjoint copy of C (with jQ ∈ CQ denoting the copy of j ∈ C), and

ET = (i, j) | i ∈ RT , j ∈ C, Tij(s) 6= 0, EQ = (jQ, j) | j ∈ C.

The algorithm maintains a pair (M,B) of a matching M ⊆ ET ∪ EQ and a baseB ∈ BQ (⊆ 2CQ) that maximizes

Ω′′(M,B) ≡ w(M) + ωQ(B) = w(M ∩ ET ) + ωQ(B)(5.1)

subject to the constraint that ∂+(M ∩EQ) = B and M is of a specified size. We put

MT = M ∩ ET , MQ = M ∩ EQ.

With reference to (M,B) it constructs an auxiliary directed graph G = G(M,B) =

(V , E) with vertex set V = RT ∪CQ ∪C and arc set E = ET ∪EQ ∪E+ ∪M, where

E+ = (iQ, jQ) | iQ ∈ B, jQ ∈ CQ −B,B − iQ + jQ ∈ BQ,

M = a | a ∈ M (a: reorientation of a).

It should be emphasized that the arcs in E+ have both ends in CQ and that the arcsin M are directed from C to RT ∪ CQ, i.e., ∂+M ⊆ C and ∂−M ⊆ RT ∪ CQ. Weput

MT = a ∈ M | ∂−a ∈ RT = a | a ∈ MT ,

MQ = a ∈ M | ∂−a ∈ CQ = a | a ∈ MQ.

We define the entrance S+ ⊆ V and the exit S− ⊆ V by

S+ = RT − ∂+MT = RT − ∂−MT , S− = C − ∂−M = C − ∂+M.

Note that no vertex in CQ belongs to the entrance S+.

We define the arc length γ = γ(M,B) : E → Z by

γ(a) =

−degs Tij(s) (a = (i, j) ∈ ET ),degs Tij(s) (a = (j, i) ∈ M

T ),−ωQ(B, iQ, jQ) (a = (iQ, jQ) ∈ E+),0 (a ∈ EQ ∪M

Q),

(5.2)


where ωQ(B, iQ, jQ) = ωQ(B− iQ + jQ)−ωQ(B), compatibly with the notation (3.4).By Lemma 3.2 we can compute ωQ(B, iQ, jQ) by means of pivoting operations onQ(s), namely, for P (s) = S(s)Q(s) with S(s) = Q[RQ, B]−1 we have ωQ(B, iQ, jQ) =degs Pij(s).

Suppose there is a shortest path in G(M,B) from the entrance S+ to the exit S−

with respect to the arc length γ, and let L be (the set of arcs on) a shortest pathfrom S+ to S− having the smallest number of arcs. Then we can update (M,B) to(M,B) by

M = M − a ∈ M | a ∈ L ∩M + (L ∩ (ET ∪ EQ)),(5.3)

B = B − ∂+a | a ∈ L ∩ E+ + ∂−a | a ∈ L ∩ E+.(5.4)

In fact, M is obviously a matching with ∂+(M ∩ EQ) = B and |M | = |M | + 1, andfurthermore, it can be shown [42] that B ∈ BQ and (M,B) maximizes Ω′′(M,B)under these constraints.

Our algorithm for δLMk (A) repeats finding a shortest path and updating (M,B)

as follows.Outline of the algorithm.

Starting from a maximum-weight base B ∈ BQ with respect to ωQ

and the corresponding matching M = (jQ, j) | jQ ∈ B, repeat(i)–(ii) below:(i) Find a shortest path L having the smallest number of arcs (from

S+ to S− in G(M,B) with respect to the arc length γ(M,B)).[Stop if there is no path from S+ to S−.]

(ii) Update (M,B) according to (5.3) and (5.4).An initial base B of maximum value of ωQ can be found by the greedy algorithmdescribed in section 3.2. At each stage of this algorithm it holds that δLM

k (A) =Ω′′(M,B) for k = |M | −mQ and that (I, J,B) defined by (4.18) gives the maximumin the expression (4.9) of δLM

k (A).Just as the primal-dual algorithm for the ordinary minimum-cost flow problem

and the independent assignment problem, the algorithm outlined above can be mademore efficient by the explicit use of a potential function on the auxiliary graph G =(V , E). To this end we maintain p : V → Z that satisfies

−degs Tij(s) + p(i) − p(j) ≥ 0 ((i, j) ∈ ET ),(5.5)

−degs Tij(s) + p(i) − p(j) = 0 ((i, j) ∈ MT ),(5.6)

p(jQ) − p(j) ≥ 0 (j ∈ C),(5.7)

p(jQ) − p(j) = 0 ((jQ, j) ∈ MQ),(5.8)

−ωQ(B, iQ, jQ) + p(iQ) − p(jQ) ≥ 0 ((iQ, jQ) ∈ E+),(5.9)

p(i) − p(k) ≥ 0 (i ∈ RT , k ∈ S+),(5.10)

p(k) − p(j) ≥ 0 (j ∈ C, k ∈ S−).(5.11)

It is remarked that the existence of such p implies the optimality of (M,B) withrespect to Ω′′ of (5.1). In fact, for any (M ′, B′) with |M ′| = |M | and ∂+(M ′ ∩EQ) =B′ we have

w(M ′) + ωQ(B′) = wp(M′) + ωQ[pQ](B′) + p(∂+M ′

T ) − p(∂−M ′)

≤ ωQ[pQ](B) + p(∂+MT ) − p(∂−M)

= w(M) + ωQ(B),

220 KAZUO MUROTA

where M ′T = M ′∩ET , wp(a) = w(a)−p(∂+a)+p(∂−a), and pQ denotes the restriction

of p to CQ. Note that wp(M′) ≤ wp(M) = 0 by (5.5)–(5.8), ωQ[pQ](B′) ≤ ωQ[pQ](B)

by (5.9) and Lemma 3.1, p(∂+M ′T ) ≤ p(∂+MT ) by (5.10), and p(∂−M ′) ≥ p(∂−M)

by (5.11).Initially, we have MT = ∅, S+ = RT , and S− = C, and therefore we can put

p(i) = maxk∈RT ,j∈C

degs Tkj(s) (i ∈ RT ), p(j) = p(jQ) = 0 (j ∈ C)(5.12)

to meet the conditions (5.5)–(5.11). In general steps, p is updated to

p(v) = p(v) + ∆p(v) (v ∈ V )(5.13)

based on the length ∆p(v) of the shortest path from S+ to v with respect to themodified arc length

γp(a) = γ(a) + p(∂+a) − p(∂−a) ≥ 0 (a ∈ E),(5.14)

where the nonnegativity of γp is due to (5.5)–(5.11). It can be shown [42] that psatisfies the conditions (5.5)–(5.11).

To compute ωQ(B, iQ, jQ) we use two matrices (or two-dimensional arrays) P =P (s) and S = S(s), as well as a vector (or one-dimensional array) base. The array Prepresents an mQ × n matrix of rational functions in s over K , where P = Q at thebeginning of the algorithm (Step 1 below). The other array S is an mQ ×mQ matrixof rational functions in s over K , which is set to the unit (identity) matrix in Step 1.Variable base is a vector of size mQ, which represents a mapping (correspondence):RQ → C ∪ 0. We also use a scalar (integer-valued) variable δQ to compute ωQ(B).

The following algorithm computes δLMk (A) for k = 0, 1, 2, . . . , kmax, as well as the

value of kmax, where kmax = −1 by convention, if rankQ(s) < mQ.Algorithm for δLM

k (A) (k = 0, 1, 2, . . . , kmax).Step 1: (Initialize)

M := ∅; B := ∅; δQ := 0;base[i] := 0 (i ∈ RQ); P [i, j] := Qij (i ∈ RQ, j ∈ C);S := unit matrix of order mQ;p[i] := maxk∈RT , j∈C degs Tkj(i ∈ RT ); p[j] := p[jQ] := 0 (j ∈ C) (cf. (5.12)).

Step 2: (Find B ∈ BQ that maximizes ωQ)While |B| < mQ do

Find (h, j) that maximizes degs P [h, j]s.t. base[h] = 0, jQ 6∈ B, and P [h, j] 6= 0;If there exists no such (h, j), then stop with kmax := −1;B := B + jQ; δQ := δQ + degs P [h, j]; M := M + (jQ, j);base[h] := j; w := 1/P [h, j];P [h, l] := w × P [h, l] (l ∈ C); S[h, l] := w × S[h, l] (l ∈ RQ);P [m, l] := P [m, l] − P [m, j] × P [h, l] (h 6= m ∈ RQ, l ∈ C);S[m, l] := S[m, l] − P [m, j] × S[h, l] (h 6= m ∈ RQ, l ∈ RQ) ;

k := 0.Step 3: (Construct the auxiliary graph G(M,B))

δLMk (A) := δQ +

∑

(i,j)∈M∩ET

degs Tij ;

S+ := RT − ∂+(M ∩ ET ); S− := C − ∂−M ; M := a | a ∈ M;E+ := (iQ, jQ) | h ∈ RQ, jQ ∈ B,P [h, j] 6= 0, i = base[h];


γ(a) :=

−degs Tij(s) (a = (i, j) ∈ ET )degs Tij(s) (a = (j, i) ∈ M

T )−degs P [h, j] (a = (iQ, jQ) ∈ E+, base[h] = i)0 (a ∈ EQ ∪M

Q)

(cf. (5.2))where M

T = a | a ∈ M ∩ ET , MQ = a | a ∈ M ∩ EQ;

γp(a) := γ(a) + p(∂+a) − p(∂−a) (a ∈ E). (cf. (5.14))Step 4: (Augment M along a shortest path)

For each v ∈ V compute the length ∆p(v) of the shortest path from S+ to v

in G(M,B) with respect to the modified arc length γp;If there is no path from S+ to S− (including the case where S+ = ∅ orS− = ∅), then stop with kmax := k;Let L (⊆ E) be (the set of arcs on) a shortest path, having the smallestnumber of arcs, from S+ to S− with respect to the modified arc length γp;M := M − a ∈ M | a ∈ L ∩M + (L ∩ (ET ∪ EQ)) ; k := k + 1;

p[v] := p[v] + ∆p(v) (v ∈ V ); (cf. (5.13))For all (iQ, jQ) ∈ L ∩ E+ (in the order from S+ to S− along L) do thefollowing:

Find h such that i = base[h];B := B − iQ + jQ; δQ := δQ + degs P [h, j];base[h] := j; w := 1/P [h, j];P [h, l] := w × P [h, l] (l ∈ C); S[h, l] := w × S[h, l] (l ∈ RQ);P [m, l] := P [m, l] − P [m, j] × P [h, l] (h 6= m ∈ RQ, l ∈ C);S[m, l] := S[m, l] − P [m, j] × S[h, l] (h 6= m ∈ RQ, l ∈ RQ) ;

Go to Step 3.

The shortest path in Step 4 can be found in time linear in the size of the graphG, which is O((|R|+ |C|)2), by means of the standard graph algorithms; see, e.g., [1].

For the updates of P in Steps 2 and 4, the algorithm assumes the availabilityof arithmetic operations on rational functions in a single variable s over the subfieldK . It is emphasized that no arithmetic operations are done on the T -part, so thatno rational function operations involving coefficients in T (which are independentsymbols) are needed.

The updates of P are the standard pivoting operations [19], the total numberof which is bounded by O(|R|2|C| kmax). Note that pivoting operations are requiredfor each arc (iQ, jQ) ∈ L ∩ E+ (see Step 4). The sparsity of P should be taken intoaccount in actual implementations of the algorithm.

The matrix S(s) gives the inverse of Q[RQ, B], which is often useful (see, e.g., theproof of Theorem 4.6). When S(s) is not needed, it may simply be eliminated fromthe computation without any side effect.

5.2. Example. The algorithm above is illustrated here for the 4 × 5 LM-poly-nomial matrix A(s) of (4.19):

A(s) =

x1 x2 x3 x4 x5

s3 0 s3 + 1 s2 10 s2 s2 s 0

f1 −t1s3 0 0 α1 α2s

f2 0 −t2s2 α3 0 0

.(5.15)

222 KAZUO MUROTA

We work with a 2 × 5 matrix P (s), a 2 × 2 matrix S(s), a vector base of size 2, andanother vector p of size 12.

The flow of computation is traced below.Step 1. M := ∅; B := ∅; δQ := 0;

(base, P, S) := r1 0r2 0

,

x1 x2 x3 x4 x5

r1 s3 0 s3 + 1 s2 1r2 0 s2 s2 s 0

, 1 00 1

;

p :=f1 f2 x1 x2 x3 x4 x5 x1Q x2Q x3Q x4Q x5Q

3 3 0 0 0 0 0 0 0 0 0 0.

Step 2. (h, j) := (r1, x1); B := x1Q, δQ := 3; M := (x1Q, x1);

(base, P, S) := r1 x1

r2 0

,

x1 x2 x3 x4 x5

r1 1 0 s3+1s3

1s

1s3

r2 0 s2 s2 s 0

, 1s3

00 1

;

(h, j) := (r2, x2); B := x1Q, x2Q, δQ := 5; M := (x1Q, x1), (x2Q, x2);

(base, P, S) := r1 x1

r2 x2

,

x1 x2 x3 x4 x5

r1 1 0 s3+1s3

1s

1s3

r2 0 1 1 1s

0

, 1s3

00 1

s2

;

k := 0.Step 3. δLM

0 (A) := 5; S+ := f1, f2; S− := x3, x4, x5;

M := (x1, x1Q), (x2, x2Q);E+ := (x1Q, x3Q), (x1Q, x4Q), (x1Q, x5Q), (x2Q, x3Q), (x2Q, x4Q);γ and γp are given in G(0) of Fig. 2. (See G(0) in Fig. 2.)

Step 4.

∆p :=f1 f2 x1 x2 x3 x4 x5 x1Q x2Q x3Q x4Q x5Q

0 0 0 1 0 1 2 0 1 0 1 3;

There exists a path from S+ to S−;L := (f1, x1), (x1, x1Q), (x1Q, x3Q), (x3Q, x3);M := (f1, x1), (x2Q, x2), (x3Q, x3); k := 1;


3 3 0 1 0 1 2 0 1 0 1 3;

(iQ, jQ) := (x1Q, x3Q) ∈ L ∩ E+; h := r1; B := x3Q, x2Q, δQ := 5;

(base, P, S) := r1 x3

r2 x2

,

x1 x2 x3 x4 x5

r1s3

s3+1 0 1 s2

s3+11

s3+1

r2 − s3

s3+1 1 0 1s(s3+1)

−1s3+1

,1

s3+1 0

−1s3+1

1s2

.


Fig. 2. Graph G(0) (© : arcs in M, B = x1Q, x2Q, S+ = f1, f2, S− = x3, x4, x5).

Fig. 3. Graph G(1) (© : arcs in M, B = x2Q, x3Q, S+ = f2, S− = x4, x5).

224 KAZUO MUROTA

Fig. 4. Graph G(2) (©: arcs in M, B = x1Q, x3Q, S+ = ∅, S− = x4).

Step 3. δLM1 (A) := 5 + 3 = 8; S+ := f2; S

− := x4, x5;M :=(x1, f1), (x2, x2Q), (x3, x3Q);E+ :=(x2Q, x1Q), (x2Q, x4Q), (x2Q, x5Q), (x3Q, x1Q), (x3Q, x4Q), (x3Q, x5Q);γ and γp are given in G(1) of Fig. 3. (See G(1) in Fig. 3.)

Step 4.

∆p :=f1 f2 x1 x2 x3 x4 x5 x1Q x2Q x3Q x4Q x5Q

1 0 1 0 3 3 1 1 0 3 3 1;

There exists a path from S+ to S−;L := (f2, x2), (x2, x2Q), (x2Q, x1Q), (x1Q, x1), (x1, f1), (f1, x5);M := (f1, x5), (f2, x2), (x1Q, x1), (x3Q, x3); k := 2;


4 3 1 1 3 4 3 1 1 3 4 4;

(iQ, jQ) := (x2Q, x1Q) ∈ L ∩ E+; h := r2; B := x3Q, x1Q, δQ := 5;

(base, P, S) := r1 x3

r2 x1

,

x1 x2 x3 x4 x5

r1 0 1 1 1s

0

r2 1 −s3−1s3

0 −1s4

1s3

, 0 1s2

1s3

−s3−1s5

.

Step 3. δLM2 (A) := 5 + 3 = 8; S+ := ∅; S− := x4;M := (x5, f1), (x2, f2), (x1, x1Q), (x3, x3Q);E+ := (x1Q, x2Q), (x1Q, x4Q), (x1Q, x5Q), (x3Q, x2Q), (x3Q, x4Q);γ and γp are given in G(2) of Fig. 4. (See G(2) in Fig. 4.)

Step 4. There exists no path from S+ (= ∅) to S−;Stop with kmax := 2.


Acknowledgments. The author expresses his thanks to Satoru Iwata for dis-cussion and to Vincent Hovelaque, Akiyoshi Shioura, and Jacob van der Woude forcomments.

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