complexity of choosing subsets from color sets
TRANSCRIPT
DISCRETE MATHEMATICS
ELSEVIER Discrete Mathematics 191 (1998) 139 148
Complexity of choosing subsets from color sets 1
J. K r a t o c h v f l a, Zs. T u z a b, M. V o ig t c'*
a Department of Applied Mathematics, Charles University, Prague, Czech Republic b Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary
c Institut fiir Mathematik, Technische Universitdt llmenau, Postfaeh 0565 98684, Ilmenau, German),
Received 11 June 1996; revised 11 March 1997; accepted 23 December 1997
Abstract
We raise and investigate the algorithmic complexity of the following problem. Given a graph G = (V,E) and p-element sets L(v) for its vertices v C V such that IL(u)t3 L(v)l >1 p + r for all edges uv C E, do there exist q-element subsets C(v)C L(v) with C(u)A C(v)= 0 for all uv E E? Here p,q, r are positive integers, p ~>q and p + r >~2q. We characterize precisely which triples (p, q, r) admit a polynomial solution, and for which ones the problem is NP-complete. Moreover, it is shown that for some restricted ranges of p and r with respect to q, the existence of subsets C(v) C L(v) for every collection {Z(v) lv ~ V}, is closely related to Turfin's problem on uniform hypergraphs. @ 1998 Elsevier Science B.V. All rights reserved
Keywords: List colorings; Set choosability; Complexity; Satisfiability
1. Introduction
Let p , q , r be natural numbers, and G = ( V , E ) a graph with vertex set V and edge set E. A (p,r)-assignment o f G is a col lec t ion
Lf = {L(v) ] v ~ V}
of lists (color sets) assigned to the vertices such that
[L(v)l = p for all v C V,
and
IL(u) U L(v)] >>. p + r for all uv E E.
l Parts of the research were done during visits of various subsets of the authors to Technical University Ilmenau and to Charles University Prague. * Corresponding author. E-mail: [email protected].
0012-365X/98/$19.00 Copyright @ 1998 Elsevier Science B.V. All rights reserved PH S001 2-365X(98)00 1 01-0
140 .L Kratochvil et al./Discrete Mathematics 191 (1998) 13~148
An ~q~-admissible q-set colorin9 of G is a collection
<e = {C(v) I v c v}
such that
C(v)CL(v), IC(v)l =q for all vE V,
and
C(u) n C(v) = 0 for all uv E E.
The graph G is said to be (p,q,r)-choosable if it admits a q-set coloring for every (p, r)-assignment 5 a.
For r = 0 and q = 1, we obtain the concept of p-choosable graphs, introduced by Vizing [19] and independently by Erd6s et al. [4] in the 1970s. The case r = 0 for arbitrary q is termed (p,q)-choosability in the literature.
There are lots of interesting problems which lead to this type of restricted colorings (see e.g. [13,14]). In the last decade, the choosability of graphs and related concepts such as list colorings [4] and precoloring extensions [2,8, 11] grew into an important research field in graph theory [3, 10]. Detailed references concerning these topics can be found in [10] and in the forthcoming extensive survey [16].
The algorithmic aspects of list colorings have been investigated in several papers [4,12,7,9]. In particular, the structure of 2-choosable (i.e., (2, 1,0)-choosable) graphs can be characterized [4], and, as a consequence, they are easily recognizable in linear time. On the other hand, already the problem of (3, 1,0)-choosability is //P-complete [4, 7]. For a given set LP of lists, (2, I, 0)-list coloring is decidable in linear time, while (3, 1,0)-list coloring is NP-complete already on planar bipartite graphs of maximum degree three [12], and also on complete bipartite graphs [9].
In this note, we analyze the complexity of the general problem with parameters p,q,r. In Section 2 we give a complete characterization of all triples (p ,q , r ) for which the existence of a q-set coloring can be decided in polynomial time for every pair (G,L~) of a graph G and its (p,r)-assignment 5 a. The necessary and sufficient condition is that either p = r > i q or q + 1 >ip>iq>>-r should hold (Theorem 2.2).
Section 3 presents a partial result on (p, q, r)-choosability, giving a sufficient condi- tion for its solvability in linear time (Theorem 3.1). In that case, the close relationship between q-set choosability and a Tur~in-type function in extremal hypergraph theory is pointed out. One of the consequences is that, for some values of p, q, r, choosability is much easier to decide than colorability with respect to a given list assignment.
2. The complexity of (p,q,r)-list colorings
In this section we study the following decision problem parametrized by (fixed) integers p, q, r:
J. Kratochvll et al./Discrete Mathematics 191 (1998) 139-148 141
00000000000~ 00000000000 0000000000 0 0 0 0 ~ 00000 00000000 0000000 0000
1 ~"
Fig. I. Complexity of (p,q,r)-list coloring - • = polynomial, o - NP-complete. (Illustration for q = 6.)
(p, q, r)-iist coloring ((p, q, r ) -LC)
Instance: A graph G and a collection of lists, Lf = {L(u) u E V(G)}, such that
(i) [L(u)l = p for every u c V(G), and
(ii) [L(u) U L(v)[ >~ p + r for every edge uv C E(G). Question: Is there an LP-admissible q-set coloring o f G, 1.e., a collection c~= {C(u)l
u E V(G)} such that
(i) C(u)C_ L(u) for every u,
(ii) [C(u)[ = q for every u, and (iii) C(u) A C(v) = 0 for every edge uv E E(G)? Note that the question only makes sense for p>~q (if p < q , then no admissible
coloring exists) and p>~r (if p < r , then there are no inputs satisfying both (i) and
(ii)). The following lemma is also straightforward, by definition. For a concise notation,
the symbol cx means that there is a polynomial transformation (a function, computable
in polynomial time) which transforms every instance I o f the first problem to an
instance I ' o f the second problem such that I is feasible if and only if I ' is feasible
(cf. [6]).
Lemma 2.1. For every r' <~r, and for every p and q, ( p ,q , r ) -LC ~ (p,q ,r ' ) -LC.
The complete discussion of the complexity o f the (p,q,r)-list coloring problem is
given in the following theorem and illustrated in Fig. 1.
Theorem 2.2. The (p,q,r)-l ist coloring problem is
(i) NP-complete for p>~ max {q + 2 , r + 1} and
(ii) solvable in linear time for p = r>~q and for q <~ p<~q + 1.
Proof (ii). The problem is trivial for p = r, because then any two adjacent lists o f the
input are disjoint, and the answer is yes without computation. The case p = q is also
142 J. Kratochvll et al. ID&crete Mathematics 191 (1998) 139-148
easy, since C(u)=L(u) must hold then for every u E V(G), and therefore the instance is feasible if and only if the lists of admissible colors of adjacent vertices are disjoint.
The only interesting case is p = q + 1, r = 0. (Note that once we prove this particular case, part (ii) of the theorem will be proved via Lemma 2.1).
Given an instance of (q+ 1, q, 0)-LC, we introduce p I V(G)I boolean variables x(u, a) (uE V(G), aEL(u)) . We will encode the boolean values so that x(u,a)=true iff
a E C(u), and construct a formula • so that ¢ is satisfiable if and only if (G, ~ )
admits a q-set coloring. For every edge uv E E(G) and every color a E L(u)AL(v), we se t ~u,v,a=-nx(u,a) V-~x(v,a). For every u c V(G) and every two colors a, bCL(u), we set ~u,~,b =x(u, a)Vx(u, b). The subformulas • ..... control the disjointness of C(u) and C(v), while the subformulas 45 .... b guarantee that L(u) - C(u) has at most one element. Therefore, the 2-formula
~'= A ¢~,~,~ A A a~ .... b uvCE(G),aEL(u)nL(v) uEV(G),a,b@L(u)
is satisfiable if and only (G, £,e) admits a q-set coloring. Since the length of q~ is linear in E, the assertion (ii) follows by the well known fact that 2-SATISFIABILITY is solvable in linear time. []
The NP-completeness part of Theorem 2.2 will be proved via an auxiliary problem called (2.5, 1, 1.5)-list coloring: (2.5, 1, 1.5)-list coloring ((2.5, 1, 1.5)-LC) Instance: A graph G and a collection of lists, ~,e = {L(u) [ u E V(G)}, such that
(i) 2~<[L(u)[~<3 for every uE V(G), and
(ii) [L(u) fq L(v)[ ~< 1 for every edge uv E E(G). Question: Is there an ~-admissible coloring of G, i.e., a collection oK= {C(u)[u C
V(G)} such that
(i) C(u)C L(u) for every u, (ii) [C(u)[ = 1 for every u, and
(iii) C(u) M C ( v ) = 0 for every edge uvEE(G)?
Lemma 2.3. The problem (2.5, 1, 1 .5)-LC/s NP-complete.
Proof. It suffices to show that 3-SAT (3( (2.5, 1, 1.5)-LC. Given a formula • with a set C of clauses over a set X of variables as an instance of 3-SAT1SFIABILITY, w e consider the graph
G = ( C u X , { x e l x ~ c c C V ~ x ~ c c C})
and lists defined as
L(x) = {x,x'}, x e X
and
L(c) = { x ' l x ~ c} u {x I -~x ~ c}.
J. Kratochvil et al./Discrete Mathematics 191 (1998) 139 148 143
Obviously, (G, ~ ) is an instance of (2.5, 1, 1.5)-LC, and it is feasible if and only if ~b is satisfiable. (The truth valuation to be taken is x = t r u e i f fx is colored by {x}.)
A graph G ( u , b ) equipped with a (p , r ) -ass ignment , and with a specified vertex u and a color b E L(u) , is called a ( p, q, r ) -chooser if it is a legal and admissible instance
of ( p , q , r ) - L C and every admissible q-set coloring (g satisfies b E C(u) .
Lemma 2.4. I f p >~ q + 2, p >~ r + l, and i f a (p , q, r ) -chooser exists , then (2.5, 1, 1.5)- L C ~ ( p , q , r ) - L C .
Proof. Let (G, LP) be an instance of (2.5, 1,1.5)-list color. We build a graph G' equipped with lists U by a series of local adjustments. If a vertex ui E V ( G ) is asso-
ciated with a list o f size s = s ( i ) , we consider p - s( i ) new colors bi, l,bi,2 . . . . . bi.q i,
Fi. 1, ri,2 . . . . . ?'i, p-q s ( i ) + l and p - q - s ( i ) + 1 disjoint copies of the chooser G(ui./ , ri./), j = 1,2 . . . . . p - q - s ( i ) + 1. The new list for ui will be U ( u i ) = L ( u i ) U {bi.l,b,,2 . . . . .
bi, q - 1 , ri, l,?'i,2 . . . . . ri, p - q s ( i ) + l } , and ui will be adjacent to ui, j , j = 1,2 . . . . . lgp_q s ' ( t ) - -1 ,
This construction is performed for every vertex ui ~ V(G) , where the color sets
{bi, l, bi.2 . . . . . bi ,q-l ,r i . | ,r i .2 . . . . . ri, p-q-.~(i)+l} and {bk,~,bk.2 . . . . . bLq- l , rL i , rk .2 . . . . . r~,p_,t_,~!+l } are disjoint for each pair ui, uk of adjacent vertices. The pair ( G ' , S ' )
obtained in this way is an instance of ( p , q , r ) - L C . In any admissible q-set color- ing ~' o f G t, r,.j E C(ui, j ) ( j = 1,2 . . . . . p - q - s ( i ) + 1), and hence C(ui ) C L(ui) U
{bi, l,bi,2 . . . . . bi, q - l } for every ui E V (G) . It follows that IC(ui) N L(ui)[ >~ 1 for ever)' ui, and hence c~, induces a proper coloring of G. The other direction (i.e., that (G' , Y ' )
is feasible if (G, Lt') is feasible) is trivial. []
Lemma 2,5. A (p, q, p - 1)-chooser ex is t s f o r ever), p > q >~ 1.
Proof, Below, i , j , k denote integers and h = ( h l , h 2 . . . . . hp) is a vector of length p whose coordinates take values from {q + 1,q + 2 . . . . . p}. The graph G ( u , b ) will have the vertex set
V ( G ( u , b ) ) ={u} U {ui, i I q + l <<,i<~p, 1 <~j<~p}
U{Ui, h [q + l <<,i<<,p, h E {q + l , q + 2 . . . . . p}P},
associated with lists
= {b = b , , b2 . . . . . be},
L ( u i . i ) = { b i , b ( i , j , 2 ) . . . . . b ( i , j , p ) } , q + l < ~ i < ~ p , l<~j<~p,
L ( u i . h ) = { b ( i , k , h k ) l l < ~ k < ~ p } , q + l<~i<<,p, h E { q + l , q + 2 . . . . . p}~',
and edges
E ( G ( u , b ) ) = {uui, j l q + 1 <~i<~p, 1 <~j<~p}
U{ui, jui, h I q + 1 <~ i ~ p, 1 <~j <~ p, h E {q + 1, q + 2 . . . . . p }P} .
144 J. Kratochvil et al./Discrete Mathematics 191 (1998) 139-148
Note that elements denoted differently are different. The reader may easily verify that G(u,b) with the constructed list assignment is an instance of the (p,q, p - 1)-LC
problem.
The lists admit a proper q-set coloring; e.g., the subsets
C ( u ) : { b l , b 2 , . . . , b q } ,
C(ui, j ) : {b i , b(i,j, 2) . . . . . b(i,j,q)}, q+ l<~i<~p, l<~j<~p,
C(ui, h)={b(i , l ,hl) . . . . . b(i,q, hq)}, q+ l<~i<<,p, h E { q + l ,q+ 2 , . . . ,p} p
satisfy the requirements. To show that b E C(u) for any coloring cg, one may argue as follows. I f b q~ C(u),
then there must be at least one k>~q + 1 such that bk E C(u). Then for j = 1,2 . . . . . p,
bkf~ C(uk, j) and for every j = 1,2 . . . . , p , there is an hj>~q + 1 such that b(k, Lhj)E C(uk, j). It follows that the vertex U~,(h,,h2,...,h~) cannot obtain any color. []
Proof of Theorem 2.2.(i). It is now a straightforward application of Lemmas 2.3-2.5. []
3. Complexity of (p,q,r)-choosabifity
Again, let p, q, r be fixed integers. In this section we investigate the following deci-
sion problem: (p, q, r)-choosability Instance: A graph G=(V,E). Question: Is G (p, q, r)-choosable?
The main result of this section is
Theorem 3.1. I f (1) 2r>>,p and 4q>3r+p, or (2) 2r<_p and 4 q > 2 p + r ,
then (p,q,r)-choosability is solvable in O(n) time.
First, we will prove that under the assumptions of Theorem 3.1 every (p,q,r)- choosable graph is a star forest, i.e. each component of the graph is a star (a tree which does not contain P4, the path consisting of 4 vertices and 3 edges). Thus, the choosability of G depends on the degree of the stars only. Lemma 3.4 gives then the complete charaterization of stars which are (p,q,r)-choosable.
Lemma 3.2. I f (1) 2r>~p and 4q>3r+p, or (2) 2r<~p and 4q>2p+r,
then the path P4 is not (p,q,r)-choosable.
.L Kratochvil et al./Discrete Mathematics 191 (1998) 139-148 145
ProoL Denote the vertices of P4 by Vl, /)2, /)3, v4 (in this order along the path). We
construct two types of list assignments, depending on the relationship between p and r.
In each case, the lists will be intervals. Let L~ denote the list assigned to the vertex v,.
Case 1. 2r >~ p and 4 q > 3 r + p
Define
c , -- {1 . . . . . p} ,
L2 = { r + l . . . . . r + p } , Ls - - { 2 r + l . . . . . 2 r + p } ,
L4 - - {3r + 1 . . . . . 3r + p}.
Notice that IL, I = p and ILi UL/I = p ÷ r for all i , j E {1 . . . . . 4} and i = j + 1. Suppose
Ci has been selected from Li and ICi[ = q. We consider the interval I = { p + 1 . . . . . 3r}.
First, we ask about the minimum number of elements of C2 included in this interval.
Since 2r ~> p, no element of L2 (and hence none of C2 either) is greater than 3 r Furthermore, there are (p + 1) - (r + 1) elements of L2 smaller than p + 1. On the
other hand, we consider L1 and obtain that at least q - r elements o f the interval
{r + 1 . . . . . p} belong to C1. Thus, at most (p + 1 ) - (r + 1 ) - (q - r) = p - q elements
o f C2 are smaller than p + 1, and therefore IC2 N II >~q - ( p - q) = 2 q - p.
Analogous investigations lead to the inequality IC3 n i I >~2q-p . Since C2 and C3 arc
disjoint, we obtain 4 q - 2p ~< Ill = 3 r - p, a contradiction to the assumption 4q > 3r+p.
Case 2. 2r<~p a n d 4 q > 2 p + r
Define
L, = {1 . . . . . p} ,
L2 -- { r + 1 . . . . . r + p } ,
L3 = { p + l . . . . . 2p}, L4 = {r + p + 1 . . . . . r + 2p}.
It is easy to check that ILl I= p and ILl U L j I F p + r for all i , j C {1 . . . . . 4} and i = j + 1. Suppose Ci has been selected from Li and ICil = q. In this case, we consider the interval
I ~ = { p + 1 . . . . . p + r}. Again, we obtain that at least 2q - p elements of C2 and also
of C3 are contained in F . Consequently, we have 4q - 2p ~< II'1 = r, a contradiction to the assumption 4q > 2p + r.
Lemma 3.3. I f
(1) 2r>~p and q>r , or
(2) 2r <, p and 3q > p + r,
then the trianyle 1£3 is not (p,q,r)-choosable.
Proof. Denote the three lists on the vertices of/£3 by L1,L2 and L 3 .
Case 1. 2r >~ p a n d q > r
Take 6 pairwise disjoint color sets T1, T2, T3, Si, $2, $3 with I Til = 2 r - p and ISil = p - r
for i = l, 2, 3. Define L1 = T1 U SI U $2, L2 = T2 U $1 U $3 and L3 = ~ U $2 U $3. Notice
146 J. Kratochvil et al./Discrete Mathematics 191 (1998) 139-148
that IL~I= P and ILl U L j I = p +r for all i , j E { 1, 2, 3} and i C j . Furthermore, we obtain ILl u L2 u L3[= 3r which should be at least 3q, a contradiction to the assumption.
Case 2. 2r <~ p and 3q > p + r
In this case, we take 4 pairwise disjoint color sets T,S~,S2,S3 with IT[ = p - 2r and
ISil =-r for i = 1,2,3. Define Ll = TUS1 US2, L2= T U S I (..iS 3 and L3 = TUS2 US3.
Notice that IZ~ I = p and [L i (3 Lj I = P ÷ r for all i,j E { 1,2, 3 } and i ¢ j. Furthermore,
we obtain IL~ u L2 u L31 = p + r which should be at least 3q, a contradiction to the
assumption. []
Let p,q , r be positive integers satisfying the assumptions of Theorem 3.1. Since
p>~q, the assumptions of Lemma 3.3 are also fulfilled. Thus, a (p,q,r)-choosable graph G cannot contain cycles and each (tree) component has diameter at most 2.
Consequently, each component of G has to be a star, and the choosability depends on
the degree of these stars only. Let d denote the degree of the center of a given star.
For further investigations, we introduce the following Tur~in-type function:
f ( p , q , r ) is the smallest integer m for which there exists a collection ovf of m r-element subsets of a p-element set L such that every (p - q)-element subset of L contains more than p + r - 2q elements from at least one of the sets H C ~ .
Notice that f is well-defined if (and only if) 2q > p and q > r. (For general infor-
mation concerning Tur~in's problem, see e.g. [5,15].)
L e m m a 3.4. I f 2q<~p or q<<.r, then every star is (p,q,r)-choosable. I f 2q> p and
q>r, then a star of degree d is (p,q,r)-choosable i f and only i f d < f ( p , q , r ) .
P r o o f . Denote the list of the center of the star by L, the end-vertices of the star by
v/ and the corresponding lists by Li, i = 1 . . . . , d. In the case 2q ~< p, we can choose q
arbitrary colors for the center of the star. After that, we have at least p - q / > q further
colors available for each end-vertex vi. In the case q ~< r, we choose first q colors for
each end-vertex vi from Li\L. This is always possible because ]Li\L I >-r>~q. Then each selected Ci is disjoint from L, therefore we have a free choice at the center of the star.
Now, we assume 2q > p and q > r. Obviously, the "hardest" case is where IL\L~]= r for all i. Let C be the q-element subset of L chosen for the center of the star. If C
contains at least 2q - p colors from each of the r-element sets LkL~, then it contains at
most q - (2q - p) = p - q elements from L~ and we can choose q colors for every vi.
On the other hand, if there is an Li such that C contains fewer than 2q - p elements
from LkLg, then there are fewer than q colors available for vg and we cannot finish the coloring.
Thus, the existence of a q-element subset of L which contains at least 2 q - p elements
from each of the r-element subsets LkL~ is a necessary and sufficient condition for the existence of an admissible q-set coloring. Consequently, there must be a (p-q)-element subset S :=LkC such that S contains at most r - ( 2 q - p) elements from each of the
d subsets LkLi.
J. Kratoehvll et al./Discrete Mathematics 191 (1998) 139-148 147
Obviously, the existence of such a subset S can be guaranteed if and only if
d < f (p ,q , r ) . []
Combining Lemmas 3.2-3.4, we obtain Theorem 3.1, since finding the value of f requires a constant number of steps for every fixed p, q and r.
4. Concluding remarks
We have defined the general concept of (p,q,r)-list colorings of a graph for triples of positive integers p, q, r (p ~> r, p + r >~ 2q), and characterized those triples for which a polynomial-time algorithmic solution exists. The related problem where one asks whether every (p, r)-assignment {L(v) lv E V} admits a q-set coloring remains open in general. Our results show, however, that there is an infinite class of triples for which the (p, q,r)-list coloring problem (where the lists L(v) are also given) is NP-complete, while the (p,q,r)-choosability can be decided in linear time.
Beside the characterization problem of polynomial instances for (p, q, r)-choosability, it would also be interesting to see whether the known results on (p,q)-choosability - - e.g., its relationship with the fractional chromatic number, or conclusions of the kind '(p, q)-choosability implies (pm, qm)-choosability for every m > 1' - (cf. [1 ] and [17, 18], respectively) can be extended in some way to (p, q, r)-choosability, too.
Acknowledgements
Zs. Tuza acknowledges partial support of Czech research grants GACR 0194/1996 and GAUK 194/1996. Research supported in part by the Hungarian Scientific Research Fund, Grant OTKA T-016416.
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