Integrality Gap of the Hypergraphic Relaxation of Steiner Trees: a short proof of a 1.55 upper bound

Recently Byrka, Grandoni, Rothvoss and Sanita (at STOC 2010) gave a 1.39-approximation for the Steiner tree problem, using a hypergraph-based linear programming relaxation. They also upper-bounded its integrality gap by 1.55. We describe a shorter proof of the same integrality gap bound, by applying some of their techniques to a randomized loss-contracting algorithm

Diversities and the Geometry of Hypergraphs

The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into $\ell_1$. Here we show that this theory can be generalized considerably to encompass Steiner tree packing problems in both graphs and hypergraphs. Instead of the theory of $\ell_1$ metrics and minimal distortion embeddings, the parallel is the theory of diversities recently introduced by Bryant and Tupper, and the corresponding theory of $\ell_1$ diversities and embeddings which we develop here.Comment: 19 pages, no figures. This version: further small correction

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

List colouring hypergraphs and extremal results for acyclic graphs

We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes. We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every $s$, every STS of high enough order is not $s$-list-colourable. The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability $p$ where the random graph $G_{n,p}$ asymptotically almost surely contains every tree of bounded degree. This $p$ is nearly optimal in terms of the maximum degree of the trees that we embed. Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph

Hamilton cycles in 5-connected line graphs

A conjecture of Carsten Thomassen states that every 4-connected line graph is hamiltonian. It is known that the conjecture is true for 7-connected line graphs. We improve this by showing that any 5-connected line graph of minimum degree at least 6 is hamiltonian. The result extends to claw-free graphs and to Hamilton-connectedness