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David Gamarnik MIT Joint work with Dmitriy Katz ( MIT ) March, 2007

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. Talk Outline. Permanents. Background and algorithmic challenges.

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David Gamarnik MIT Joint work with Dmitriy Katz ( MIT ) March, 2007

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  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. basis of our approach … Permanent. Background and algorithms Randomized algorithms: • 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. • 2. 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. this we can compute … we need this … • Algorithm: • Make “large”. • Compute using BGKNT. Permanents. Idea: use for approximating Perm(G)

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

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