Asymmetric Traveling Salesman Polytope Research Articles

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International Transactions in Operational ResearchVolume 13, Issue 2 p. 169-174 Free Access IFORS' Operational Research Hall of Fame Egon Balas, Egon BalasSearch for more papers by this author Egon Balas, Egon BalasSearch for more papers by this author First published: 09 March 2006 https://doi.org/10.1111/j.1475-3995.2006.00541.xAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Selected Original Works Balas, E., 1965. An additive algorithm for solving linear programs with zero–one variables. Operations Research 13, 517– 546. Balas, E., 1969. Machine sequencing via disjunctive graphs: an implicit enumeration algorithm. Operations Research 17, 941– 957. Balas, E., 1970. Machine sequencing: disjunctive graphs and degree-constrained subgraphs. Naval Research Logistics Quarterly 17, 1– 9. Balas, E., 1974. Disjunctive programming: properties of the convex hull of feasible points. MSRR No. 348, Carnegie Mellon University, July. Also 1998. Discrete Applied Mathematics 89, 1– 44. Balas, E., 1989. The prize collecting traveling salesman problem. Networks 19, 621– 636. Balas, E., 1995. The prize collecting traveling salesman problem: II. Networks 25, 199– 216. Balas, E., Adams, J., Zawack, D., 1988. The shifting bottleneck procedure for job shop scheduling. Management Science 34, 391– 401. Balas, E., Ceria, S., Cornuéjols, G., 1993. A lift-and-project cutting plane algorithm for mixed 0–1 programs. Mathematical Programming 58, 295– 324. Balas, E., Fischetti, M., 1993. A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets. Mathematical Programming 58, 325– 352. Balas, E., Fischetti, M., 1999. Lifted cycle inequalities for the asymmetric traveling salesman problem. Mathematics of Operations Research 24, 273– 292. Balas, E., Fischetti, M., Pulleyblank, W.R., 1995. The precedence-constrained asymmetric traveling salesman polytope. Mathematical Programming 68, 241– 265. Balas, E., Pulleyblank, W.R., 1983. The perfectly matchable subgraph polytope of a bipartite graph. Networks 13, 495– 516. Balas, E., Vazacopoulos, A., 1998. Guided local search with shifting bottleneck for job shop scheduling. Management Science 44, 262– 275. Biographical Works Balas, E., 2000. Will to Freedom: A Perilous Journey through Fascism and Communism. Syracuse University Press, Syracuse, NY. Balas, E., 2002. Some thoughts on the development of integer programming during my research career. Lecture delivered upon receiving the EURO Gold Medal, July 9, 2001, Rotterdam. European Journal of Operational Research 141, 1– 7. Volume13, Issue2March 2006Pages 169-174 ReferencesRelatedInformation

New facets of the STS polytope generated from known facets of the ATS polytope

While it had been known for a long time how to transform an asymmetric traveling salesman (ATS) problem on the complete graph with n vertices into a symmetric traveling salesman (STS) problem on an incomplete graph with 2 n vertices, no method was available for using this correspondence to derive facets of the symmetric polytope from facets of the asymmetric polytope until the work of E. Balas and M. Fischetti in [Lifted cycle inequalities for the asymmetric traveling salesman problem, Mathematics of Operations Research 24 (2) (1999) 273–292] suggested an approach. The original Balas–Fischetti method uses a standard sequential lifting procedure for the computation of the coefficient of the edges that are missing in the incomplete STS graph, which is a difficult task when addressing classes of (as opposed to single) inequalities. In this paper we introduce a systematic procedure for accomplishing the lifting task. The procedure exploits the structure of the tight STS tours and organizes them into a suitable tree structure. The potential of the method is illustrated by deriving large new classes of facet-defining STS inequalities.

Detecting symmetries by branch & cut

The NP-hard problem of finding symmetries in an abstract graph plays an important role in automatic graph drawing and other applications. In this paper, we present an exact algorithm for automorphism and symmetry detection based on the branch & cut technique. We introduce IP-models for these problems and investigate the structure of the corresponding polytopes. For automorphisms, a complete description of the polytope is derived from a given set of generators of the automorphism group. The rotation polytopes are shown to be related to the asymmetric traveling salesman polytope, while the reflection polytope is related to the matching polytope. The algorithm was implemented within the ABACUS-framework and proves to run fast in practice.

On the cycle polytope of a directed graph

We give a partial description of the cycle polytope of a directed graph G, that is, the convex hull of the incidence vectors of simple directed cycles of G. First, we show how to obtain facets of the cycle polytope from facets of the asymmetric traveling salesman polytope. This involves lifting and facet projection. We discuss several lifting procedures. Next, we obtain facets that have a different source. In particular, we give a complete characterization of the facets that cut off the origin. No such characterization is known for the cycle polytope of an undirected graph. Finally, we introduce a useful equivalence class among the inequalities defining the cycle polytope. © 2000 John Wiley & Sons, Inc.

Lifted Cycle Inequalities for the Asymmetric Traveling Salesman Problem

We investigate the family of facet defining inequalities for the asymmetric traveling salesman (ATS) polytope obtainable by lifting the cycle inequalities. We establish several properties of this family that earmark it as arguably the most important among the asymmetric inequalities for the ATS polytope known to date. In particular, we give several results that characterize the pattern of lifting coefficients and thereby facilitate the identification of facet defining inequalities. This characterization is used to investigate three new, large classes of lifted cycle inequalities that we call shell, fork and curtain inequalities. These latter inequalities have unbounded Chvátal rank. Furthermore, the pattern of lifting coefficients that we identify makes it easy to develop efficient separation routines. Finally, each member of the family is shown to have a counterpart for the symmetric TS (STS) polytope that is often new, and is obtainable by mapping the inequality for the ATS polytope into a certain face of the STS polytope and then lifting the resulting inequality into one for the STS polytope itself.

A Lower Bound for Adjacencies on the Traveling Salesman Polytope

We study adjacency of vertices on Tn, the asymmetric traveling salesman polytope of degree n. Applying a result of G. Boccara [Discrete Math., 29 (1980), pp. 105--134] to permutation groups, we show that Tn has $\Omega((n-1)(n-2)!\,^2 \log n)$ edges, implying that the degree of a vertex of Tn is $\Omega((n-2)! \log n)$. We conjecture the degree to be $\Omega((n-2)! (\log n)^k)$ for any positive integer k. We compute the density function $\delta_n$ given by the fraction of the total number of vertices adjacent to a given vertex for small values of n, and conjecture that it decreases with n.

The Graphical Asymmetric Traveling Salesman Polyhedron: Symmetric Inequalities

A present trend in the study of the symmetric traveling salesman polytope is to use, as a relaxation of the polytope, the graphical traveling salesman polyhedron ($\g$). Following a parallel approach for the asymmetric traveling salesman polytope, we define the graphical asymmetric traveling salesman problem on a general digraph $D$ and its associated polyhedron $\ga(D)$. We give basic polyhedral results and lifting theorems for $\ga(D)$ and we give a general condition for a facet-defining inequality for \g to yield a symmetric facet-defining inequality for \ga. Using this approach we show that all known major families of facet-defining inequalities of \g define facets of \ga. Finally, we discuss possible extension of these results to the asymmetric traveling salesman polytope.

All 0–1 polytopes are traveling salesman polytopes

We study the facial structure of two important permutation polytopes in\(\mathbb{R}^{n^2 } \), theBirkhoff orassignment polytopeB n , defined as the convex hull of alln×n permutation matrices, and theasymmetric traveling salesman polytopeT n , defined as the convex hull of thosen×n permutation matrices corresponding ton-cycles. Using an isomorphism between the face lattice ofB n and the lattice of elementary bipartite graphs, we show, for example, that every pair of vertices ofB n is contained in a cubical face, showing faces ofB n to be fairly special 0–1 polytopes. On the other hand, we show thatevery 0–1d-polytope is affinely equivalent to a face ofT n , ford∼logn, by showing that every 0–1d-polytope is affinely equivalent to the asymmetric traveling salesman polytope of some directed graph withn nodes. The latter class of polytopes is shown to have maximum diameter [n/3].

The prize collecting traveling salesman problem: II. Polyhedral results

AbstractThe task of developing daily schedules for a steel rolling mill has been formulated as a Prize Collecting Traveling Salesman (PCTS) Problem, in which a salesman who gets a prize for every city he visits seeks a minimum‐cost tour including enough cities to collect a required amount of prize money. This paper addresses the facial structure of the PCTS polytope, the convex hull of solutions to the PCTS problem. In an earlier paper, we generalized to the PCTS polytope the subtour elimination inequalities for the Asymmetric Traveling Salesman (ATS) polytope. Here, we give a general method for deriving a facet defining inequality for the PCTS polytope from any facet defining inequality for the ATS polytope. We apply the procedure to several well‐known families of facet inducing inequalities for the ATS polytope: comb, odd CAT, SD, clique tree, and lifted cycle inequalities. We also extend the cloning and clique lifting procedure for the ATS polytope to the PCTS polytope.

Clique tree inequalities define facets of the asymmetric traveling salesman polytope

This paper solves, in the affermative, the open question of whether the Grötschel-Pulleyblank clique tree inequalities define facets of the asymmetric traveling salesman polytope and its monotonization. This generalizes the corresponding result for symmetric polytopes. The proof makes use of a new recursive definition of clique tree, based on induction on the number of its “teeth” rather than of its “handles”.

The precedence-constrained asymmetric traveling salesman polytope

Many applications of the traveling salesman problem require the introduction of additional constraints. One of the most frequently occurring classes of such constraints are those requiring that certain cities be visited before others (precedence constraints). In this paper we study the Precedence-Constrained Asymmetric Traveling Salesman (PCATS) polytope, i.e. the convex hull of incidence vectors of tours in a precedence-constrained directed graph. We derive several families of valid inequalities, and give polynomial time separation algorithms for important subfamilies. We then establish the dimension of the PCATS polytope and show that, under reasonable assumptions, the two main classes of inequalities derived are facet inducing.

A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets

Given any family ℱ of valid inequalities for the asymmetric traveling salesman polytopeP(G) defined on the complete digraphG, we show that all members of ℱ are facet defining if the primitive members of ℱ (usually a small subclass) are. Based on this result we then introduce a general procedure for identifying new classes of facet inducing inequalities forP(G) by lifting inequalities that are facet inducing forP(G′), whereG′ is some induced subgraph ofG. Unlike traditional lifting, where the lifted coefficients are calculated one by one and their value depends on the lifting sequence, our lifting procedure replaces nodes ofG′ with cliques ofG and uses closed form expressions for calculating the coefficients of the new arcs, which are sequence-independent. We also introduce a new class of facet inducing inequalities, the class of SD (source-destination) inequalities, which subsumes as special cases most known families of facet defining inequalities.

The Fixed-Outdegree 1-Arborescence Polytope

A 1-arborescence is a spanning arborescence rooted at node 1, plus one arc incident into node 1. The 1-arborescence polytope, the convex hull of incidence vectors of 1-arborescences, is a well-known relaxation of the Asymmetric Traveling Salesman (ATS) polytope. In this paper we study the Fixed Outdegree 1-Arborescence (FDA) polytope, defined as the convex hull of incidence vectors of 1-arborescences having outdegree 1 at a prescribed subset F of nodes. The FDA polytope becomes the 1-arborescence polytope when F = ∅, and it becomes the ATS polytope when F is the entire node set. We derive several classes of valid inequalities for the FDA polytope and show that they are facet defining. This is done by using a theorem that establishes sufficient conditions for a facet defining inequality for the ATS polytope to also be facet defining for the FDA polytope. We then introduce the class of FD inequalities and show that they define facets of both the ATS polytope and the FDA polytope. Finally, we define a canonical form that facilitates rigorous comparisons between inequalities, and use it to show that the new facets are distinct from each other and from other known facets.

Facets of the Asymmetric Traveling Salesman Polytope

In this paper we consider the Asymmetric Traveling Salesman polytope, Pn, defined as the convex hull of the incidence vectors of tours in a complete digraph with n vertices, and its monotonization, P̃n. Several classes of valid inequalities for both there polytopes have been introduced in the literature which are not known to define facets of Pn or P̃n. We describe a general technique which can often be used to prove that a given inequality defines a facet of Pn. The method is applied to prove that the so-called Dk+, Dk−, C3, comb and C2 inequalities define facets of Pn, thus solving open problems. As for the monotone polytope, a necessary and sufficient condition for a facet-defining inequality of Pn to define a facet of P̃n as well, is introduced which applies to the Dk+, Dk−, C3, comb and C2 inequalities. In addition, a simple procedure for transforming any facet-defining inequality of Pn into an equivalent one which also defines a facet of P̃n, is given.