Parameterized and exact computation : 7th international symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012 : proceedings

International audience7th International Symposium, IPEC 2012, Ljubljana, Slovenia, September 12-14, 2012. Proceeding

Parity Separation: A Scientifically Proven Method for Permanent Weight Loss

Given an edge-weighted graph G, let PerfMatch(G) denote the weighted sum over all perfect matchings M in G, weighting each matching M by the product of weights of edges in M. If G is unweighted, this plainly counts the perfect matchings of G. In this paper, we introduce parity separation, a new method for reducing PerfMatch to unweighted instances: For graphs G with edge-weights -1 and 1, we construct two unweighted graphs G1 and G2 such that PerfMatch(G) = PerfMatch(G1) - PerfMatch(G2). This yields a novel weight removal technique for counting perfect matchings, in addition to those known from classical #P-hardness proofs. We derive the following applications: 1. An alternative #P-completeness proof for counting unweighted perfect matchings. 2. C=P-completeness for deciding whether two given unweighted graphs have the same number of perfect matchings. To the best of our knowledge, this is the first C=P-completeness result for the "equality-testing version" of any natural counting problem that is not already #P-hard under parsimonious reductions. 3. An alternative tight lower bound for counting unweighted perfect matchings under the counting exponential-time hypothesis #ETH. Our technique is based upon matchgates and the Holant framework. To make our #P-hardness proof self-contained, we also apply matchgates for an alternative #P-hardness proof of PerfMatch on graphs with edge-weights -1 and 1.Comment: 14 page

Graph Isomorphism in Quasipolynomial Time Parameterized by Treewidth

We extend Babai's quasipolynomial-time graph isomorphism test (STOC 2016) and develop a quasipolynomial-time algorithm for the multiple-coset isomorphism problem. The algorithm for the multiple-coset isomorphism problem allows to exploit graph decompositions of the given input graphs within Babai's group-theoretic framework. We use it to develop a graph isomorphism test that runs in time $n^{\operatorname{polylog}(k)}$ where $n$ is the number of vertices and $k$ is the minimum treewidth of the given graphs and $\operatorname{polylog}(k)$ is some polynomial in $\operatorname{log}(k)$. Our result generalizes Babai's quasipolynomial-time graph isomorphism test.Comment: 52 pages, 1 figur