Class Of Facet-defining Inequalities Research Articles

The Circlet Inequalities: A New, Circulant-Based, Facet-Defining Inequality for the TSP

Facet-defining inequalities of the symmetric traveling salesman problem (TSP) polytope play a prominent role in both polyhedral TSP research and state-of-the-art TSP solvers. In this paper, we introduce a new class of facet-defining inequalities, the circlet inequalities. These inequalities were first conjectured in Gutekunst and Williamson [Gutekunst SC, Williamson DP (2019) Characterizing the integrality gap of the subtour LP for the circulant traveling salesman problem. SIAM J. Discrete Math. 33(4):2452–2478] when studying the circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture from Marco Buratti in 2007. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facet-defining and compute their strength following Goemans [Goemans MX (1995) Worst-case comparison of valid inequalities for the TSP. Math. Programming 69:335–349]; they achieve the same worst case strength as the similarly circulant crown inequalities of Naddef and Rinaldi [Naddef D, Rinaldi G (1992) The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17(2):308–326] but are generally stronger. Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1650441] and by the National Science Foundation Division of Computing and Communications Foundations [Grant CCF-1908517].

Mathematical Models and Polyhedral Studies for Integral Sheet Metal Design

We deal with the optimization of the production of branched sheet metal products. New forming techniques for sheet metal give rise to a wide variety of possible profiles and possible ways of production. In particular, we show how the problem of producing a given profile geometry can be modeled as a discrete optimization problem. We provide a theoretical analysis of the model in order to improve its solution time. In this context we give the complete convex hull description of some substructures of the underlying polyhedron. Moreover, we introduce a new class of facet-defining inequalities that represent connectivity constraints for the profile and show how these inequalities can be separated in polynomial time. Finally, we present numerical results for various test instances, both real-world and academic examples.

On cut-based inequalities for capacitated network design polyhedra

In this article, we study capacitated network design problems. We unify and extend polyhedral results for directed, bidirected, and undirected link capacity models. Valid inequalities based on a network cut are known to be strong in several special cases. We show that regardless of the link model, facets of the polyhedra associated with such a cut translate to facets of the original network design polyhedra if the two subgraphs defined by the network cut are (strongly) connected. Our investigation of the facial structure of the cutset polyhedra allows to complement existing polyhedral results for the three variants by presenting facet-defining flow-cutset inequalities in a unifying way. In addition, we present a new class of facet-defining inequalities, showing as well that flow-cutset inequalities alone do not suffice to give a complete description for single-commodity, single-module cutset polyhedra in the bidirected and undirected case – in contrast to a known result for the directed case. The practical importance of the theoretical investigations is highlighted in an extensive computational study on 27 instances from the Survivable Network Design Library (SNDlib). © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 141–156 2011 © 2011 Wiley Periodicals, Inc.

Facets of the linear ordering polytope: A unification for the fence family through weighted graphs

The binary choice polytope appeared in the investigation of the binary choice problem formulated by Guilbaud and Block and Marschak. It is nowadays known to be the same as the linear ordering polytope from operations research (as studied by Grötschel, Jünger and Reinelt). The central problem is to find facet-defining linear inequalities for the polytope. Fence inequalities constitute a prominent class of such inequalities (Cohen and Falmagne; Grötschel, Jünger and Reinelt). Two different generalizations exist for this class: the reinforced fence inequalities of Leung and Lee, and independently Suck, and the stability–critical fence inequalities of Koppen. Together with the fence inequalities, these inequalities form the fence family. Building on previous work on the biorder polytope by Christophe, Doignon and Fiorini, we provide a new class of inequalities which unifies all inequalities from the fence family. The proof is based on a projection of polytopes. The new class of facet-defining inequalities is related to a specific class of weighted graphs, whose definition relies on a curious extension of the stability number. We investigate this class of weighted graphs which generalize the stability–critical graphs.

A note on the Boolean quadric polytope

We observe that a new class of facet-defining inequalities for the Boolean quadric polytope, introduced by Sherali et al. [4], can in fact be obtained from a class of facet-defining inequalities for the cut polytope via a well-known linear transformation.