Zigzag inequalities: a new class of facet-inducing inequalities for Arc Routing Problems

In this paper we introduce a new class of facet-inducing inequalities for the Windy Rural Postman Problem and the Windy General Routing Problem. These inequalities are called Zigzag inequalities because they cut off fractional solutions containing a zigzag associated with variables with 0.5 value. Two different types of inequalities, the Odd Zigzag and the Even Zigzag inequalities, are presented. Finally, their application to other known Arc Routing Problems is discussed.     

On the graphical relaxation of the symmetric traveling salesman polytope

The Graphical Traveling Salesman Polyhedron (GTSP) has been proposed by Naddef and Rinaldi to be viewed as a relaxation of the Symmetric Traveling Salesman Polytope (STSP). It has also been employed by Applegate, Bixby, Chvátal, and Cook for solving the latter to optimality by the branch-and-cut method. There is a close natural connection between the two polyhedra. Until now, it was not known whether there are facets in TT-form of the GTSP polyhedron which are not facets of the STSP polytope as well. In this paper we give an affirmative answer to this question for n ≥ 9. We provide a general method for proving the existence of such facets, at the core of which lies the construction of a continuous curve on a polyhedron. This curve starts in a vertex, walks along edges, and ends in a vertex not adjacent to the starting vertex. Thus there must have been a third vertex on the way.      

On the orthogonal Latin squares polytope

Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine the polyhedral aspects of OLS. In particular, we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.     

Facets of the clique partitioning polytope

A subsetA of the edge set of a graphG = (V, E) is called a clique partitioning ofG is there is a partition of the node setV into disjoint setsW ¹,,W ^k such that eachW _i induces a clique, i.e., a complete (but not necessarily maximal) subgraph ofG, and such thatA = _i=1 ^k 1{uv|u, v W _i ,u v}. Given weightsw ^e for alle E, the clique partitioning problem is to find a clique partitioningA ofG such that _eA w _e is as small as possible. This problem—known to be-hard, see Wakabayashi (1986)—comes up, for instance, in data analysis, and here, the underlying graphG is typically a complete graph. In this paper we study the clique partitioning polytope of the complete graphK _n , i.e., is the convex hull of the incidence vectors of the clique partitionings ofK _n . We show that triangles, 2-chorded odd cycles, 2-chorded even wheels and other subgraphs ofK _n induce facets of. The theoretical results described here have been used to design an (empirically) efficient cutting plane algorithm with which large (real-world) instances of the clique partitioning problem could be solved. These computational results can be found in Grötschel and Wakabayashi (1989).     

Lower bounds and heuristics for the Windy Rural Postman Problem

In this paper we present several heuristic algorithms and a cutting-plane algorithm for the Windy Rural Postman Problem. This problem contains several important Arc Routing Problems as special cases and has very interesting real-life applications. Extensive computational experiments over different sets of instances are also presented.     

Facet-inducing web and antiweb inequalities for the graph coloring polytope

For a graph G and its complement , we define the graph coloring polytope P(G) to be the convex hull of the incidence vectors of star partitions of . We examine inequalities whose support graphs are webs and antiwebs appearing as induced subgraphs in G. We show that for an antiweb in G the corresponding inequality is facet-inducing for P(G) if and only if is critical with respect to vertex colorings. An analogous result is also proved for the web inequalities.     

Strengthened clique-family inequalities for the stable set polytope

The stable set polytope is a fundamental object in combinatorial optimization. Among the many valid inequalities that are known for it, the clique-family inequalities play an important role. Pêcher and Wagler showed that the clique-family inequalities can be strengthened under certain conditions. We show that they can be strengthened even further, using a surprisingly simple mixed-integer rounding argument.     

Facets of the polytope of the asymmetric travelling salesman problem with replenishment arcs

The Asymmetric Travelling Salesman Problem with Replenishment Arcs (RATSP) is a new class of problems arising from work related to aircraft routing. Given a digraph with cost on the arcs, a solution of the RATSP, like that of the Asymmetric Travelling Salesman Problem, induces a directed tour in the graph which minimises total cost. However the tour must satisfy additional constraints: the arc set is partitioned into replenishment arcs and ordinary arcs, each node has a non-negative weight associated with it, and the tour cannot accumulate more than some weight limit before a replenishment arc must be used. To enforce this requirement, constraints are needed. We refer to these as replenishment constraints.In this paper, we review previous polyhedral results for the RATSP and related problems, then prove that two classes of constraints developed in V. Mak and N. Boland [Polyhedral results and exact algorithms for the asymmetric travelling salesman problem with replenishment arcs, Technical Report TR M05/03, School of Information Technology, Deakin University, 2005] are, under appropriate conditions, facet-defining for the RATS polytope.     

On the partial order polytope of a digraph

We introduce the partial order polytope of a digraphD, defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofD. For this polytope we prove some classes of inequalities to be facet-defining and show that there is a polynomial separation algorithm for each of these classes. The results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two-chorded odd cycle inequalities. The polyhedral results concerning the partial order polytope are of interest since a cutting plane based algorithm to solve the maximum weighted transitive acyclic subdigraph problem can be used to solve the maximum weighted acyclic subdigraph problem, the maximum weighted linear ordering problem and a flexible manufacturing problem. For the acyclic subdigraph polytope we show that the separation of simplet-reinforcedk-fence-inequalities is -complete.     

Gear composition and the stable set polytope

We present a new graph composition that produces a graph G from a given graph H and a fixed graph B called gear and we study its polyhedral properties. This composition yields counterexamples to a conjecture on the facial structure of when G is claw-free.     

A new class of facets for the Latin square polytope

Latin squares of order n have a 1-1 correspondence with the feasible solutions of the 3-index planar assignment problem (3PAP_n). In this paper, we present a new class of facets for the associated polytope, induced by odd-hole inequalities.     

Relating single-scenario facets to the convex hull of the extensive form of a stochastic single-node flow polytope

We study when a facet-defining inequality for a deterministic, single-scenario subproblem is also facet-defining for the extensive form of a two-stage stochastic mixed-integer linear program (SMIP). To answer this question, we introduce a novel stochastic variant of the well-known single-node flow (SNF) polytope, and present necessary and sufficient conditions for single-scenario facet-defining inequalities to be facet-defining for the extensive form. We further demonstrate that our stochastic SNF polytope is a relaxation of a broad subclass of SMIPs, illustrating its generality.     

New results for the Directed Profitable Rural Postman Problem

We study a generalization of the Directed Rural Postman Problem where not all arcs requiring a service have to be visited provided that a penalty cost is paid if a service arc is not crossed. The problem, known as Directed Profitable Rural Postman Problem, looks for a tour visiting the selected set of service arcs while minimizing both traveling and penalty costs. We add different valid inequalities to a known mathematical formulation of the problem and develop a branch-and-cut algorithm that introduces connectivity constraints both in a “lazy” and in a standard way. We also propose a matheuristic followed by an improvement heuristic (final refinement). The matheuristic exploits information provided by a problem relaxation to select promising service arcs used to solve optimally Directed Rural Postman problems. The ex-post refinement tries to improve the solution provided by the matheuristic using a branch-and-cut algorithm. The method gets a quick convergence through the introduction of connectivity cuts that are not guaranteed to be valid inequalities, and thus may exclude integer feasible solutions.     

Lifting the knapsack cover inequalities for the knapsack polytope

Valid inequalities for the knapsack polytope have proven to be very useful in exact algorithms for mixed-integer linear programming. In this paper, we focus on the knapsack cover inequalities, introduced in 2000 by Carr and co-authors. In general, these inequalities can be rather weak. To strengthen them, we use lifting. Since exact lifting can be time-consuming, we present two fast approximate lifting procedures. The first procedure is based on mixed-integer rounding, whereas the second uses superadditivity.     

On the cut polytope

The cut polytopeP _C (G) of a graphG=(V, E) is the convex hull of the incidence vectors of all edge sets of cuts ofG. We show some classes of facet-defining inequalities ofP _C (G). We describe three methods with which new facet-defining inequalities ofP _C (G) can be constructed from known ones. In particular, we show that inequalities associated with chordless cycles define facets of this polytope; moreover, for these inequalities a polynomial algorithm to solve the separation problem is presented. We characterize the facet defining inequalities ofP _C (G) ifG is not contractible toK ₅. We give a simple characterization of adjacency inP _C (G) and prove that for complete graphs this polytope has diameter one and thatP _C (G) has the Hirsch property. A relationship betweenP _C (G) and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied.     

On the combinatorial structure of chromatic scheduling polytopes

Chromatic scheduling polytopes arise as solution sets of the bandwidth allocation problem in certain radio access networks, supplying wireless access to voice/data communication networks for customers with individual communication demands. To maintain the links, only frequencies from a certain spectrum can be used, which typically causes capacity problems. Hence it is necessary to reuse frequencies but no interference must be caused by this reuse. This leads to the bandwidth allocation problem, a special case of so-called chromatic scheduling problems. Both problems are NP-hard, and there do not even exist polynomial time algorithms with a fixed quality guarantee.As algorithms based on cutting planes have shown to be successful for many other combinatorial optimization problems, the goal is to apply such methods to the bandwidth allocation problem. For that, knowledge on the associated polytopes is required. The present paper contributes to this issue, exploring the combinatorial structure of chromatic scheduling polytopes for increasing frequency spans. We observe that the polytopes pass through various stages—emptyness, non-emptyness but low-dimensionality, full-dimensionality but combinatorial instability, and combinatorial stability—as the frequency span increases. We discuss the thresholds for this increasing “quantity” giving rise to a new combinatorial “quality” of the polytopes, and we prove bounds on these thresholds. In particular, we prove combinatorial equivalence of chromatic scheduling polytopes for large frequency spans and we establish relations to the linear ordering polytope.     

On packing and covering polyhedra of consecutive ones circulant clutters

Building on work by G. Cornuéjols and B. Novick and by L. Trotter, we give different characterizations of contractions of consecutive ones circulant clutters that give back consecutive ones circulant clutters. Based on a recent result by G. Argiroffo and S. Bianchi, we then arrive at characterizations of the vertices of the fractional set covering polyhedron of these clutters. We obtain similar characterizations for the fractional set packing polyhedron using a result by F.B. Shepherd, and relate our findings with similar ones obtained by A. Wagler for the clique relaxation of the stable set polytope of webs. Finally, we show how our results can be used to obtain some old and new results on the corresponding fractional set covering polyhedron using properties of Farey series. Our results do not depend on Lehman’s work or blocker/antiblocker duality, as is traditional in the field.     

A cutting plane algorithm for the General Routing Problem

The General Routing Problem (GRP) is the problem of finding a minimum cost route for a single vehicle, subject to the condition that the vehicle visits certain vertices and edges of a network. It contains the Rural Postman Problem, Chinese Postman Problem and Graphical Travelling Salesman Problem as special cases. We describe a cutting plane algorithm for the GRP based on facet-inducing inequalities and show that it is capable of providing very strong lower bounds and, in most cases, optimal solutions. Received: November 1998 / Accepted: September 2000?Published online March 22, 2001