Happy 80th Bday Dick Perfect matchings and symbolic matrices
Avi Wigderson IAS, Princeton Perfect Matchings [Edmonds-Karp 72] Jacoby 1890 [Hopcroft-Karp 72]
[Karp-Sipser 81] [Karp-Upfal-W 85] [Karp-Vazirani-Vazirani 90] sequential, parallel, on-line deterministic, randomized, Average-case, Jacoby 1890 ODEs Optimization Combinatorics Stat Physics Economics Biology CS - Unfamiliar algorithms Favorite open problems Something new Perfect Matchings V U 1 G AG Bipartite graphs G(U,V;E). |U|=|V|=n
G U V AG Bipartite graphs G(U,V;E). |U|=|V|=n Fact:G has a PM iff Per(AG)>0 [Hall 1935]G has a PM iff G has no ixj minor with i+j>n [Jacoby 1890] PM P (Augmenting paths) PMs & symbolic matrices [Edmonds67]
G U V 1 AG X31 X21 X11 X12 X13 AG[X] Of course, this is in PPM [Edmonds 67]G has a PM iff Det(AG[X]) 0( P) Symbolic matrices [Edmonds67]
L[X] X = {X1,X2, } Lij(X) = aX1+bX2+ :affine form SDIT: Is Det(L[X]) = 0 ? [Edmonds 67]SDIT P ?? [Lovasz 79] SDIT RP [Valiant 79]SDIT PIT for arithmetic formulas [Kabanets-Impagliazzo01] SDIT P circuit lower bds [Valiant 82] PM RNC !FindPM NC ?? Of course, this is in PPM FindPM RNC [KUW 85] G(U,V;E), Assume G has a PM. Let S E
rank(S) = max { |S M| : M is PM } redundant(S) = { e : rank(S+e) = rank(S) } Lemma: rank, redundant RNC(symbolic matrices) Repeat c.log n times: Pick S at random (clever) Remove red(S) Clean & Update G Analysis: Pr[ |red(S)| < |E|/10 ]< 1/10 Notes: no regret algorithm! Two sources of randomness! rank(S) = 2 red(S) = rank(S) = 1 red(S) = rank(S) = 2 red(S) = A year later MVV discovered the FindPM NC ? [Sharan-W 96]
Augmenting paths out, Augmenting cycles in G(U,V;E), Assume G has a PM. degree(G) 3(wlog) quasiPM: S E, odd degree in every vertex Find a quasiPM S Repeat c.log2n times: Find augmenting cycles C (many disjoint ones) S S C Analysis:In cubic graphs, Q is a perfect matching In deg3 graphs, Q is an induced forest Linear Algebra C C Lev-Pippenger-Valiant Approximating the Permanent
Non-negative matrix A:efficiently approximate Per(A) [Jerrum-Sinclair-Vigoda04]Probabilistic. (1+ )-approx. [Linial-Samorodnitsky-W01]Deterministic.en- approx. [Gurvits-Samorodnitsky14] Deterministic. 2n- approx. Open:Deterministic.(1+ )-approx. [Sinkhorn 64] Iterations: make A doubly stochastic exp(n.999) Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) 1 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) 1 1/3 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) 3/7 1/7 1 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) 1 1/15 7/15 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) 1/2 1 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) R(A) = diag(row sums)-1C(A) = diag(column sums)-1 Repeat n2 times: Normalize rows A R(A)A Normalize cols A A C(A) 1 1/2 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) R(A) = diag(row sums)-1C(A) = diag(column sums)-1 Repeat n2 times: Normalize rows A R(A)A Normalize cols A A C(A) 1 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) R(A) = diag(row sums)-1C(A) = diag(column sums)-1 Repeat n2 times: Normalize rows A R(A)A Normalize cols A A C(A) 1 1/2 1/3 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) R(A) = diag(row sums)-1C(A) = diag(column sums)-1 Repeat n2 times: Normalize rows A R(A)A Normalize cols A A C(A) 6/11 3/11 3/5 2/11 2/5 1 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) R(A) = diag(row sums)-1C(A) = diag(column sums)-1 Repeat n2 times: Normalize rows A R(A)A Normalize cols A A C(A) 1 15/48 33/48 10/87 22/87 55/87 Scaling algorithm [LSW 01]
A non-negative matrix. Try making it doubly stochastic. (e.g. the adjacency matrix A=AG of a bipartite graph G) R(A) = diag(row sums)-1C(A) = diag(column sums)-1 Repeat n2 times: Normalize rows A R(A)A Normalize cols A A C(A) (C(A)=I) Test if R(A) I (up to 1/n) Yes: Per(A) > 0. No:Per(A) = 0. Analysis: Progress measure = Permanent 1 Grows by (1+1/n) (AMGM inequality) Initially > e-n (van der Waerden conj) 1 Quantum scaling alg [Gurvits 04]
L=L[X]=iLixi symbolic matrix (Li integer matrices) completely positive quantum operator. Try making it doubly stochastic. R(L) = (iLiLit)-1/2C(L) = (iLitLi)-1/2 Repeat nc times: Normalize rows L R(L)L Normalize cols L L C(L) Test if R(L) I (up to 1/n) Yes: QuantPer(L) > 0. No:QuantPer(L) = 0. Analysis: Progress measure = Qantum Permanent Grows by (1+1/n) (AMGM inequality) Initially ?? (large if Det(L[X]) 0 ) What does this Algorithm do ?? ifDet(L[X]) 0 Det(L[X]) = 0 Matching in the dark [Gurvits 04]
1 AG X31 X21 X11 X12 X13 AG[X] Of course, this is in PPM Can be NPC toinvert B->A (in matroid intersection) Open over finite fields Invertible left, right & variable change Det(A[X]) 0iffDet(B[Y]) 0 [Gurvits]Quantum algorithm determines if G has PM B[Y] What does Gurvits alg do?? [Garg-Gurvits-Olivera-W 15]
Linninteger matrices L=L[X]=i Lixi nnsymbolic matrix SDIT: Is Det(L) 0 ? [GGOW 15] NC-SDIT P(Previously EXP) Non-commutative algebra Is L invertible when in the skew field? (NC PIT for rational functions) Invariant Theory Is (L1,Lm) nullcone of L-R gp action? Quantum Information Theory Is the completely positive quantum operator defined by (L1,Lm)rank decreasing? Optimization Matroid intersection, (in the dark) Linninteger matrices L(k)=L(k)[X]=i LiXiXi kk nknksymbolic matrix NC-SDIT: k Det(L(k)) 0 ? ANALITIC Even EXP bd for NC-SDIT is nontrivial Best upper bound on k is exponential (but this is essential to our analysis giving polytime alg [H,D,IQS] kSymbolic Determinant / PIT in P What does NC-SDIT P imply? Even EXP bd for NC-SDIT is nontrivial Best upper bound on k is exponential (but this is essential to our analysis giving polytime alg Happy 80th Bday Dick