KAIST Discrete Math Seminar

A celebrated conjecture of Sidorenko and Erdős–Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs with the same order and edge density. This conjecture has attracted considerable interest over the last decade and is now known to hold for a broad range of bipartite graphs, with the overall trend saying that a graph satisfies the conjecture if it can be built from simple building blocks such as trees in a certain recursive fashion.Our contribution here, which goes beyond this paradigm, is to show that the conjecture holds for any bipartite graph H with bipartition A∪B where the number of vertices in B of degree k satisfies a certain divisibility condition for each k. As a corollary, we have that for every bipartite graph H with bipartition A∪B, there is a positive integer p such that the blow-up H_A^p formed by taking p vertex-disjoint copies of H and gluing all copies of A along corresponding vertices satisfies the conjecture. Joint work with David Conlon.

Problems such as Vertex Cover and Multiway Cut have been well-studied in parameterized complexity. Cygan et al. 2011 drastically improved the running time of several problems including Multiway Cut and Almost 2SAT by employing LP-guided branching and aiming for FPT algorithms parameterized above LP lower bounds. Since then, LP-guided branching has been studied in depth and established as a powerful technique for parameterized algorithms design.

In this talk, we make a brief overview of LP-guided branching technique and introduce the latest results whose parameterization is above even stronger lower bounds, namely μ(I)=2LP(I)-IP(dual-I). Here, LP(I) is the value of an optimal fractional solution and IP(dual-I) is the value of an optimal integral dual solution. Tutte-Berge formula for Maximum Matching (or equivalently Edmonds-Gallai decomposition) and its generalization Mader’s min-max formula are exploited to this end. As a result, we obtain an algorithm running in time 4^k-μ(I)for multiway cut and its generalizations, where k is the budget for a solution.

This talk is based on a joint work with Yoichi Iwata and Yuichi Yoshida from NII.