1. Happy 80th B’day
Dick

2. Perfect matchings and symbolic matrices
Avi Wigderson
IAS, Princeton

3. 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…

4. Perfect Matchings V U 1 G AG Bipartite graphs G(U,V;E). |U|=|V|=nG 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)

5. PMs & symbolic matrices [Edmonds‘67]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)

6. Symbolic matrices [Edmonds‘67]
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-Impagliazzo‘01] SDIT  P  circuit lower bds
[Valiant ‘82] PM  RNC ! FindPM  NC ??
Of course, this is in PPM

7. 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

8. 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 deg3 graphs, Q is an induced forest
Linear
Algebra C C
Lev-Pippenger-Valiant

9. Approximating the Permanent
Non-negative matrix A: efficiently approximate Per(A)
[Jerrum-Sinclair-Vigoda’04] Probabilistic. (1+ )-approx.
[Linial-Samorodnitsky-W’01] Deterministic. en- approx.
[Gurvits-Samorodnitsky‘14] Deterministic. 2n- approx.
Open: Deterministic. (1+ )-approx.
[Sinkhorn ‘64] Iterations: make A doubly stochastic
 exp(n.999)

10. 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

11. 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

12. 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

13. 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

14. 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

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)
R(A) = diag(row sums)-1 C(A) = diag(column sums)-1
Repeat n2 times:
Normalize rows A  R(A)A
Normalize cols A  A C(A) 1
1/2

16. 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)-1 C(A) = diag(column sums)-1
Repeat n2 times:
Normalize rows A  R(A)A
Normalize cols A  A C(A) 1

17. 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)-1 C(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

18. 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)-1 C(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

19. 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)-1 C(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

20. 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)-1 C(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

21. 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)-1/2 C(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 ??
if Det(L[X])  0
 Det(L[X]) = 0

22. 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])  0 iff Det(B[Y])  0
[Gurvits] “Quantum” algorithm determines if G has PM
B[Y]

23. What does Gurvits alg do?? [Garg-Gurvits-Olivera-W ‘15]
Li nn integer matrices
L=L[X]=i Lixi
nn symbolic 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)
Li nn integer matrices
L(k)=L(k)[X]=i LiXi Xi kk
nknk symbolic 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] k<exp(n2)

24. Homework Perfect Matching in NC (1+ )-approx of the Permanent in P
Symbolic 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

25. Happy 80th B’day
Dick