The Diameter of the Fractional Matching Polytope and Its Hardness Implications

Abstract

The (combinatorial) diameter of a polytope P ⊆ R d is the maximum value of a shortest path between a pair of vertices on the 1-skeleton of P , that is the graph where the nodes are given by the 0-dimensional faces of P , and the edges are given the 1-dimensional faces of P . The diameter of a polytope has been studied from many different perspectives, including a computational complexity point of view. In particular, [Frieze and Teng, 1994] showed that computing the diameter of a polytope is (weakly) NP-hard.In this paper, we show that the problem of computing the diameter is strongly NP-hard even for a polytope with a very simple structure: namely, the fractional matching polytope. We also show that computing a pair of vertices at maximum shortest path distance on the 1-skeleton of this polytope is an APX-hard problem. We prove these results by giving an exact characterization of the diameter of the fractional matching polytope, that is of independent interest.

Pivot Rules for Circuit-Augmentation Algorithms in Linear Optimization

“…Note that the above result is orthogonal to the NP-hardness results on the computation of the diameter of a polytope [16,27]. In fact, the hardness results in [16] and [27] rely on the existence/non existence of vertices with a certain structure, and do not provide a specific objective function to minimize over their polytopes.…”

“…Note that the above result is orthogonal to the NP-hardness results on the computation of the diameter of a polytope [16,27]. In fact, the hardness results in [16] and [27] rely on the existence/non existence of vertices with a certain structure, and do not provide a specific objective function to minimize over their polytopes. 1 We note that the hardness results and subsequent implications can be also derived if instead of the bipartite matching polytope one considers the circulation polytope [3], where the caracterization of circuits, and the correspondent hardness reduction, become easier.…”

“…Let us denote by C(P FMAT (G)) the set of circuits of P FMAT (G) with co-prime integer components. [4,27]), so it remains to be shown that all circuits belong to one of these sets. Let B denote the constraint matrix corresponding to the constraints (2).…”

“…We first characterize the circuits of the more general fractional matching polytope, i.e., the polytope given by the standard LP-relaxation for the matching problem on general graphs, in Section 3.1. This builds on the known graphical characterization of adjacency given in [27,4]. Then, we construct a reduction from the NP-hard Hamiltonian path problem in Section 3.2 1 .…”

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“…Here, by the diameter of a polyhedron, we mean the diameter of the graph made up of its vertices and edges. The largest possible diameter of a polyhedron has been studied from a number of different perspectives [4,5,6,17,25,29,30], and in particular as a function of its dimension and number of facets [15,18,22,26,28], two parameters that reflect the number of variables and the number of constraints of a linear optimization problem. In practice, the vertices of polyhedra often have rational coordinates and, up to the multiplication by an integer, these vertices are contained in the integer lattice.…”