1. Monomer-dimer model and a new deterministic approximation algorithm for computing a permanent of a 0,1 matrix
David Gamarnik
MIT
Joint work with
Dmitriy Katz (MIT)
March, 2007

2. Talk Outline Permanents. Background and algorithmic challenges.
Monomer-dimer model, correlation decay and deterministic counting algorithm.
Our results: (1§ )n deterministic approximation algorithm
Polynomial (PTAS) for constant degree expanders
eO(n2/3 log3 n) for general graphs.
Conclusions

3. Permanent. Background and algorithms
When Perm(A)= # full matchings in G
-- bi-partite graph corresponding to A
Notation: M(k) - # of k-matchings
Perm(G)=M(n)

4. Permanent. Background and algorithms
Ryser [1963]. (n2n) time exact algorithm.
Kasteleyn [1961]. (n3) exact algorithm for planar graphs.
Valiant [1979]. Permanent is in #P complexity class.

5. Permanent. Background and algorithms
Randomized algorithms:
basis of our approach …
Broder [1988]. Proposed MCMC algorithm.
Jerrum and Sinclair [1989]. FPRAS when M(n-1)/M(n)=O(Poly(n))
Jerrum and Vazirani [1996] approx. algorithm for an arbitrary graph
Barvinok [1999] factor polynomial time approx. algorithm, general matrix.
Jerrum, Sinclair and Vigoda [2003]. FPRAS, general matrix.

6. Permanent. Background and algorithms
Deterministic algorithms:
Linial, Samorodnitsky and Wigderson, [2000] approximation algorithm.
Reduction from matrix scaling problem and van der Waerden’s conjecture.
Better factor (k/(k-1))kn for small row/column sums =k, using strengthened van der Waerden conjecture. Schrijver [1998], Gurvits [2006]

7. Permanent. Background and algorithms
Deterministic algorithms: this work
Theorem approximation factor algorithm,
Running time:
Poly(n) when the graph is a constant degree expander
for general bi-partite graphs (0,1 matrices)

8. Monomer-dimer model. Correlation decay and deterministic approximation algorithm
Input: graph G
Goal: compute partition function corresponding to partial matchings

9. Monomer-dimer model. Correlation decay and deterministic approximation algorithm
Theorem. (Bayati, G, Katz, Nair and Tetali [2006]) Model exhibits a correlation decay on a computation tree. This leads to
deterministic FPTAS for computing for constant degree graphs.
algorithm
for arbitrary graph, constant
Precise running time:

10. Monomer-dimer model. Correlation decay and deterministic approximation algorithm
Theorem. (Bayati, G, Katz, Nair and Tetali [2006]) Model exhibits a correlation decay on a computation tree. This leads to
Heilman and Lieb [1972], van den Berg [1996]
Spatial decay of correlation.

11. Monomer-dimer model. Correlation decay and deterministic approximation algorithm
Related works:
Weitz [2005]. Independent sets.
Bandyopahdyay and G [2005]. Ind sets and colorings.
G and Katz [2006]. Colorings and MRF.

12. Permanents. Idea: use for approximating Perm(G)
we need this …
this we can compute …
Algorithm:
Make “large”.
Compute using BGKNT.

13. Permanents. Idea: use for approximating Perm(G)
we need this …
this we can compute …
Analysis: control

14. Permanents. Algorithm and analysis.
Definition. a-expander.

15. Permanents. Part I, “large”  -expansion.
Proposition. Ratio of k-matchings to (k+1)-matchings.
As a result,
What good is it ?..

16. Permanents. Part I, “large”  -expansion.
Say
Then
Solution: “kill” with large

17. Permanents. Part I, “large”  -expansion.
Proposition.
Proof: variation of alternating path argument from Jerrum & Vazirani [96].
Claim: there exists (n-k)/2 ways
of finding an alternating path with length d with
resulting in (k+1)-matching.

18. Permanents. Part I, “large”  -expansion.
Proposition.
Proof: variation of alternating path argument from Jerrum & Vazirani [96].
Claim: each (k+1)-matching can be obtained from
at most
k-matchings – the number of length-d
alternating paths.
This provides a bound on M(k)/M(k+1)

19. Permanents. Part II, “small” expansion.
More precise bound (skipping log terms)
When we need
Running time from BGKNT:
But what to do for smaller expansion?..

20. Permanents. Part II, “small” expansion.
Lemma. (Jerrum & Vazirani [96]). If the expansion of the graph
is less than then a “counter-example” can be found

21. Permanents. Part II, “small” expansion.
Solve the problem recursively:
The number of sub-problems:

22. Further questions:
Better algorithms for permanent through better algorithms for matchings
Deterministic algorithms for other counting problems (bin-packing, volume of a polytope, etc.)