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

Improving the integrality gap for multiway cut

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of $k$ terminal nodes, and the goal is to partition the node set of the graph into $k$ non-empty parts each containing exactly one terminal so that the total weight of the edges crossing the partition is minimized. The multiway cut problem for $k\ge 3$ is APX-hard. For arbitrary $k$, the best-known approximation factor is $1.2965$ due to [Sharma and Vondr\'{a}k, 2014] while the best known inapproximability factor is $1.2$ due to [Angelidakis, Makarychev and Manurangsi, 2017]. In this work, we improve on the lower bound to $1.20016$ by constructing an integrality gap instance for the CKR relaxation. A technical challenge in improving the gap has been the lack of geometric tools to understand higher-dimensional simplices. Our instance is a non-trivial $3$-dimensional instance that overcomes this technical challenge. We analyze the gap of the instance by viewing it as a convex combination of $2$-dimensional instances and a uniform 3-dimensional instance. We believe that this technique could be exploited further to construct instances with larger integrality gap. One of the ingredients of our proof technique is a generalization of a result on \emph{Sperner admissible labelings} due to [Mirzakhani and Vondr\'{a}k, 2015] that might be of independent combinatorial interest.Comment: 28 page

New Approximability Results for the Robust k-Median Problem

We consider a robust variant of the classical $k$-median problem, introduced by Anthony et al. \cite{AnthonyGGN10}. In the \emph{Robust $k$-Median problem}, we are given an $n$-vertex metric space $(V,d)$ and $m$ client sets $\set{S_i \subseteq V}_{i=1}^m$. The objective is to open a set $F \subseteq V$ of $k$ facilities such that the worst case connection cost over all client sets is minimized; in other words, minimize $\max_{i} \sum_{v \in S_i} d(F,v)$. Anthony et al.\ showed an $O(\log m)$ approximation algorithm for any metric and APX-hardness even in the case of uniform metric. In this paper, we show that their algorithm is nearly tight by providing $\Omega(\log m/ \log \log m)$ approximation hardness, unless ${\sf NP} \subseteq \bigcap_{\delta >0} {\sf DTIME}(2^{n^{\delta}})$. This hardness result holds even for uniform and line metrics. To our knowledge, this is one of the rare cases in which a problem on a line metric is hard to approximate to within logarithmic factor. We complement the hardness result by an experimental evaluation of different heuristics that shows that very simple heuristics achieve good approximations for realistic classes of instances.Comment: 19 page

Exploiting Resolution-based Representations for MaxSAT Solving

Most recent MaxSAT algorithms rely on a succession of calls to a SAT solver in order to find an optimal solution. In particular, several algorithms take advantage of the ability of SAT solvers to identify unsatisfiable subformulas. Usually, these MaxSAT algorithms perform better when small unsatisfiable subformulas are found early. However, this is not the case in many problem instances, since the whole formula is given to the SAT solver in each call. In this paper, we propose to partition the MaxSAT formula using a resolution-based graph representation. Partitions are then iteratively joined by using a proximity measure extracted from the graph representation of the formula. The algorithm ends when only one partition remains and the optimal solution is found. Experimental results show that this new approach further enhances a state of the art MaxSAT solver to optimally solve a larger set of industrial problem instances

Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for $\gamma$-stable Max Cut instances with $\gamma \geq c\sqrt{\log n}\log\log n$ for some absolute constant $c > 0$. Our algorithm is robust: it never returns an incorrect answer; if the instance is $\gamma$-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not $\gamma$-stable. We prove that there is no robust polynomial-time algorithm for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$, where $\alpha_{SC}$ is the best approximation factor for Sparsest Cut with non-uniform demands. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with $\ell_2^2$ triangle inequalities) is integral if $\gamma \geq D_{\ell_2^2\to \ell_1}(n)$, where $D_{\ell_2^2\to \ell_1}(n)$ is the least distortion with which every $n$ point metric space of negative type embeds into $\ell_1$. On the negative side, we show that the SDP relaxation is not integral when $\gamma < D_{\ell_2^2\to \ell_1}(n/2)$. Moreover, there is no tractable convex relaxation for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$. That suggests that solving $\gamma$-stable instances with $\gamma =o(\sqrt{\log n})$ might be difficult or impossible. Our results significantly improve previously known results. The best previously known algorithm for $\gamma$-stable instances of Max Cut required that $\gamma \geq c\sqrt{n}$ (for some $c > 0$) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability.Comment: 24 page