Extended Formulations for Independence Polytopes of Regular Matroids
The ancillary file Corrigendum.pdf contains an explanation why our proof does not work.
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arXiv subjects
Matthias Walter
Publications and source records attributed to Matthias Walter.
The Almost-Disjoint 2-Path Decomposition Problem
We consider the problem of decomposing a given (di)graph into paths of length 2 with the additional restriction that no two such paths may have more than one vertex in common. We establish its NP-hardness by a reduction from 3-SAT, characterize (di)graph classes for which the problem can be be reduced to the Stable-set problem on claw-free graphs and describe a dynamic program for solving it for series-parallel digraphs.
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Complete Description of Matching Polytopes with One Linearized Quadratic Term for Bipartite Graphs
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic matching problems with a single linearized quadratic term. We provide a complete irredundant inequality description, which settles a conjecture by Klein (Ph.D. thesis, TU Dortmund, 2015). In addition, we also derive facetness and separation results for the polytopes. The completeness proof is based on a geometric relationship to a matching polytope of a nonbipartite graph. Using standard techniques, we finally extend the result to capacitated b-matchings.
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Extended Formulations for Radial Cones
This paper studies extended formulations for radial cones at vertices of polyhedra, where the radial cone of a polyhedron $ P $ at a vertex $ v \in P $ is the polyhedron defined by the constraints of $ P $ that are active at $ v $. Given an extended formulation for $ P $, it is easy to obtain an extended formulation of comparable size for each its radial cones. On the contrary, it is possible that radial cones of $ P $ admit much smaller extended formulations than $ P $ itself. A prominent example of this type is the perfect-matching polytope, which cannot be described by subexponential-size extended formulations (Rothvoß 2014). However, Ventura & Eisenbrand (2003) showed that its radial cones can be described by polynomial-size extended formulations. Moreover, they generalized their construction to $ V $-join polyhedra. In the same paper, the authors asked whether the same holds for the odd-cut polyhedron, the blocker of the $ V $-join polyhedron. We answer this question negatively. Precisely, we show that radial cones of odd-cut polyhedra cannot be described by subexponential-size extended formulations. To obtain our result, for a polyhedron $ P $ of blocking type, we establish a general relationship between its radial cones and certain faces of the blocker of $ P $.
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Parity Polytopes and Binarization
We consider generalizations of parity polytopes whose variables, in addition to a parity constraint, satisfy certain ordering constraints. More precisely, the variable domain is partitioned into $k$ contiguous groups, and within each group, we require that $x_i \geq x_{i+1}$ for all relevant $i$. Such constraints are used to break symmetry after replacing an integer variable by a sum of binary variables, so-called binarization. We provide extended formulations for such polytopes, derive a complete outer description, and present a separation algorithm for the new constraints. It turns out that applying binarization and only enforcing parity constraints on the new variables is often a bad idea. For our application, an integer programming model for the graphic traveling salesman problem, we observe that parity constraints do not improve the dual bounds, and we provide a theoretical explanation of this effect.
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A characterization of Linearizable instances of the Quadratic Traveling Salesman Problem
We consider the linearization problem associated with the quadratic traveling salesman problem (QTSP). Necessary and sufficient conditions are given for a cost matrix $Q$ of QTSP to be linearizable. It is shown that these conditions can be verified in $O(n^5)$ time. Some simpler sufficient conditions for linearization are also given along with related open problems.
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A Note on Matchings Constructed during Edmonds' Weighted Perfect Matching Algorithm
We reprove that all the matchings constructed during Edmonds' weighted perfect matching algorithm are optimal among those of the same cardinality (provided that certain mild restrictions are obeyed on the choices the algorithm makes). We conclude that in order to solve a weighted matching problem it is not needed to solve a weighted perfect matching problem in an auxiliary graph of doubled size. This result was known before, e.g., posed as an exercise in see Lawler's book from 1976, but is not present in several modern books on combinatorial optimization.
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Subgraph Polytopes and Independence Polytopes of Count Matroids
Given an undirected graph, the non-empty subgraph polytope is the convex hull of the characteristic vectors of pairs (F, S) where S is a non-empty subset of nodes and F is a subset of the edges with both endnodes in S. We obtain a strong relationship between the non-empty subgraph polytope and the spanning forest polytope. We further show that these polytopes provide polynomial size extended formulations for independence polytopes of count matroids, which generalizes recent results obtained by Iwata et al. referring to sparsity matroids. As a byproduct, we obtain new lower bounds on the extension complexity of the spanning forest polytope in terms of extension complexities of independence polytopes of these matroids.
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Simple Extensions of Polytopes
We introduce the simple extension complexity of a polytope P as the smallest number of facets of any simple (i.e., non-degenerate in the sense of linear programming) polytope which can be projected onto P. We devise a combinatorial method to establish lower bounds on the simple extension complexity and show for several polytopes that they have large simple extension complexities. These examples include both the spanning tree and the perfect matching polytopes of complete graphs, uncapacitated flow polytopes for non-trivially decomposable directed acyclic graphs, hypersimplices, and random 0/1-polytopes with vertex numbers within a certain range. On our way to obtain the result on perfect matching polytopes we generalize a result of Padberg and Rao's on the adjacency structures of those polytopes. To complement the lower bounding techniques we characterize in which cases known construction techniques yield simple extensions.
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Implementation of a Unimodularity Test
This paper describes implementation and computational results of a polynomial test of total unimodularity. The test is a simplified version of a prior method. The program also decides two related unimodularity properties. The software is available free of charge in source code form under the Boost Software License.
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