On different Versions of the Exact Subgraph Hierarchy for the Stable Set Problem

Let $G$ be a graph with $n$ vertices and $m$ edges. One of several hierarchies towards the stability number of $G$ is the exact subgraph hierarchy (ESH). On the first level it computes the Lov\'{a}sz theta function $\vartheta(G)$ as semidefinite program (SDP) with a matrix variable of order $n+1$ and $n+m+1$ constraints. On the $k$-th level it adds all exact subgraph constraints (ESC) for subgraphs of order $k$ to the SDP. An ESC ensures that the submatrix of the matrix variable corresponding to the subgraph is in the correct polytope. By including only some ESCs into the SDP the ESH can be exploited computationally. In this paper we introduce a variant of the ESH that computes $\vartheta(G)$ through an SDP with a matrix variable of order $n$ and $m+1$ constraints. We show that it makes sense to include the ESCs into this SDP and introduce the compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems favorable as the SDP is smaller. However, we prove that the bounds based on the ESH are always at least as good as those of the CESH. In computations sometimes they are significantly better. We also introduce scaled ESCs (SESCs), which are a more natural way to include exactness constraints into the smaller SDP and we prove that including an SESC is equivalent to including an ESC for every subgraph

Sum-of-Squares Certificates for Vizing's Conjecture via Determining Gr\"obner Bases

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs $G$ and $H$ is at least the product of the domination numbers of $G$ and $H$. Recently Gaar, Krenn, Margulies and Wiegele used the graph class $\mathcal{G}$ of all graphs with $n_\mathcal{G}$ vertices and domination number $k_\mathcal{G}$ and reformulated Vizing's conjecture as the problem that for all graph classes $\mathcal{G}$ and $\mathcal{H}$ the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of $k_\mathcal{G}$, $n_\mathcal{G}$, $k_\mathcal{H}$, and $n_\mathcal{H}$. In this paper, we consider their approach for $k_\mathcal{G} = k_\mathcal{H} = 1$. For this case we are able to derive the unique reduced Gr\"obner basis of the Vizing ideal. Based on this, we deduce the minimum degree $(n_\mathcal{G} + n_\mathcal{H} - 1)/2$ of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes $\mathcal{G}$ and $\mathcal{H}$ with $n_\mathcal{G} + n_\mathcal{H} -1 = d$ for general $d$, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes $\mathcal{G}$ and $\mathcal{H}$ with $k_\mathcal{G}=k_\mathcal{H}=1$ and $n_\mathcal{G} + n_\mathcal{H} \leq 15$.Comment: 36 pages, 2 figure

Relationship of kk-Bend and Monotonic ℓ\ell-Bend Edge Intersection Graphs of Paths on a Grid

If a graph $G$ can be represented by means of paths on a grid, such that each vertex of $G$ corresponds to one path on the grid and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge, then this graph is called EPG and the representation is called EPG representation. A $k$-bend EPG representation is an EPG representation in which each path has at most $k$ bends. The class of all graphs that have a $k$-bend EPG representation is denoted by $B_k$. $B_\ell^m$ is the class of all graphs that have a monotonic (each path is ascending in both columns and rows) $\ell$-bend EPG representation. It is known that $B_k^m \subsetneqq B_k$ holds for $k=1$. We prove that $B_k^m \subsetneqq B_k$ holds also for $k \in \{2, 3, 5\}$ and for $k \geqslant 7$ by investigating the $B_k$-membership and $B_k^m$-membership of complete bipartite graphs. In particular we derive necessary conditions for this membership that have to be fulfilled by $m$, $n$ and $k$, where $m$ and $n$ are the number of vertices on the two partition classes of the bipartite graph. We conjecture that $B_{k}^{m} \subsetneqq B_{k}$ holds also for $k\in \{4,6\}$. Furthermore we show that $B_k \not\subseteq B_{2k-9}^m$ holds for all $k\geqslant 5$. This implies that restricting the shape of the paths can lead to a significant increase of the number of bends needed in an EPG representation. So far no bounds on the amount of that increase were known. We prove that $B_1 \subseteq B_3^m$ holds, providing the first result of this kind

A Computational Study of Exact Subgraph Based SDP Bounds for Max-Cut, Stable Set and Coloring

The "exact subgraph" approach was recently introduced as a hierarchical scheme to get increasingly tight semidefinite programming relaxations of several NP-hard graph optimization problems. Solving these relaxations is a computational challenge because of the potentially large number of violated subgraph constraints. We introduce a computational framework for these relaxations designed to cope with these difficulties. We suggest a partial Lagrangian dual, and exploit the fact that its evaluation decomposes into several independent subproblems. This opens the way to use the bundle method from non-smooth optimization to minimize the dual function. Finally computational experiments on the Max-Cut, stable set and coloring problem show the excellent quality of the bounds obtained with this approach.Comment: arXiv admin note: substantial text overlap with arXiv:1902.0534

Exact solution approaches for the discrete α\alpha-neighbor pp-center problem

The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances

Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

A graph $G$ is called an edge intersection graph of paths on a grid if there is a grid and there is a set of paths on this grid, such that the vertices of $G$ correspond to the paths and two vertices of $G$ are adjacent if and only if the corresponding paths share a grid edge. Such a representation is called an EPG representation of $G$. $B_{k}$ is the class of graphs for which there exists an EPG representation where every path has at most $k$ bends. The bend number $b(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k$. $B_{k}^{m}$ is the subclass of $B_k$ containing all graphs for which there exists an EPG representation where every path has at most $k$ bends and is monotonic, i.e. it is ascending in both columns and rows. The monotonic bend number $b^m(G)$ of a graph $G$ is the smallest natural number $k$ for which $G$ belongs to $B_k^m$. Edge intersection graphs of paths on a grid were introduced by Golumbic, Lipshteyn and Stern in 2009 and a lot of research has been done on them since then. In this paper we deal with the monotonic bend number of outerplanar graphs. We show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize in terms of forbidden subgraphs the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$. As a consequence we show that for any maximal outerplanar graph and any cactus a (monotonic) EPG representation with the smallest possible number of bends can be constructed in a time which is polynomial in the number of vertices of the graph

Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization

Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs. Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.Comment: 24 pages, 8 figure

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.Comment: arXiv admin note: substantial text overlap with arXiv:2111.0682

Efficient implementation of SDP relaxatios for the stable set problem

Elisabeth GaarAlpen-Adria-Universität Klagenfurt, Dissertation, 2018OeBB(VLID)437724