We study the natural extended-variable formulation for the disjunction of n+1 polytopes in ℝ^d . We demonstrate that the convex hull 𝒟 in the natural extended-variable space ℝ^d+n is given by full optimal big-M lifting (i) when d≤ 2 (and that it is not generally true for d≥ 3 ), and also (ii) when the polytopes are all axis-aligned hyper-rectangles. We give further results on the polyhedral structure of 𝒟 , emphasizing the role of full optimal big-M lifting.
We present a family of integer programming formulations for the maximum cut problem. These formulations encode the incidence vectors of the cuts of a connected graph by employing a subset of the odd-cycle inequalities that relate to a spanning tree, and they require only the corresponding edge variables to be integral explicitly. They so describe sufficient restrictions of the classic integer linear program by Barahona and Mahjoub. In addition, we characterize according formulations comprising facet-defining inequalities only. Trade-offs and comparisons to prevalent formulations concerning size and relaxation strength are subject to an experimental study.
We study the single pair capacitated network design problem and the budget constrained max flow problem on undirected series-parallel graphs. These problems were well studied on directed series-parallel graphs, but little is known in the context of undirected graphs. The major difference between the cases is that the source and sink of the problem instance do not necessarily coincide with the terminals of the underlying series-parallel graph in the undirected case, thus creating certain complications. We provide pseudopolynomial time algorithms to solve both of the problems and provide an FPTAS for the budget constrained max flow problem. We also provide some extensions, arguing important cases when the problems are polynomial-time solvable, and describing a series-parallel gadget that captures an edge upgrade version of the problems.
In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.
Given a connected undirected graph G = ( V , E ) , let G [ S ] be the subgraph of G induced by the set of vertices S ⊆ V . The Chordless Cycle Problem (CCP) consists in finding a subset S ⊆ V of maximum cardinality such that G [ S ] is a chordless cycle. We present a Quadratically Constrained reformulation for the CCP, derive a Semidefinite Programming (SDP) relaxation for it and solve that relaxation by Lagrangian Relaxation (LR). Compared to previously available dual bounds, our SDP bounds resulted to be quite strong. We then introduce a hybrid algorithm involving two combined phases: the LR scheme, which acts as a warm starter for a Branch-and-cut (BC) algorithm that follows it. In short, the LR algorithm allows us to formulate a finite set of SDP cuts that can be used to retrieve the SDP bounds in a Linear Programming relaxation for the CCP. Such cuts are not ready to be used by the BC as they are formulated in an extended variable space. Thus, the BC projects them back onto the original space of variables and separates them by solving a Linear Program. On dense input graphs, our proposed BC algorithm, in its current preliminary state of development, already outperforms its competitors in the literature.
