A deterministic approximation algorithm for computing the permanent of a 0, 1 matrix

David Gamarnik, Dmitriy Katz

Abstract: We consider the problem of computing the permanent of a 0,1<math><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn></math> n by n matrix. For a class of matrices corresponding to constant degree expanders we construct a deterministic polynomial time approximation algorithm to within a multiplicative factor (1+ϵ)n<math><msup is="true"><mrow is="true"><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo is="true">+</mo><mi is="true">ϵ</mi><mo stretchy="false" is="true">)</mo></mrow><mi is="true">n</mi></msup></math>, for arbitrary ϵ>0<math><mi is="true">ϵ</mi><mo is="true">&gt;</mo><mn is="true">0</mn></math>. This is an improvement over the best known approximation factor en<math><msup is="true"><mi is="true">e</mi><mi is="true">n</mi></msup></math> obtained in Linial, Samorodnitsky and Wigderson (2000) [9], though the latter result was established for arbitrary non-negative matrices. Our results use a recently developed deterministic approximation algorithm for counting partial matchings of a graph (Bayati, Gamarnik, Katz, Nair and Tetali (2007) [2]) and Jerrum–Vazirani method (Jerrum and Vazirani (1996) [8]) of approximating permanent by near perfect matchings.