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. 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

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)

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.

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.

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]

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)

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

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:

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.

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.

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

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

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

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

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

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.

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)

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?..

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

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

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