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.     

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.     

Generalised 2-circulant inequalities for the max-cut problem

The max-cut problem is a fundamental combinatorial optimisation problem, with many applications. Poljak and Turzik found some facet-defining inequalities for the associated polytope, which we call 2-circulant inequalities. We present a more general family of facet-defining inequalities, an exact separation algorithm that runs in polynomial time, and some computational results.     

The node-edge weighted 2-edge connected subgraph problem: Linear relaxation, facets and separation

Let G=(V,E) be a undirected k-edge connected graph with weights c_e on edges and w_v on nodes. The minimum 2-edge connected subgraph problem, 2ECSP for short, is to find a 2-edge connected subgraph of G, of minimum total weight. The 2ECSP generalizes the well-known Steiner 2-edge connected subgraph problem. In this paper we study the convex hull of the incidence vectors corresponding to feasible solutions of 2ECSP. First, a natural integer programming formulation is given and it is shown that its linear relaxation is not sufficient to describe the polytope associated with 2ECSP even when G is series-parallel. Then, we introduce two families of new valid inequalities and we give sufficient conditions for them to be facet-defining. Later, we concentrate on the separation problem. We find polynomial time algorithms to solve the separation of important subclasses of the introduced inequalities, concluding that the separation of the new inequalities, when G is series-parallel, is polynomially solvable.     

Polyhedral studies on the convex recoloring problem

A coloring of the vertices of a graph G is convex if, for each assigned color d, the vertices with color d induce a connected subgraph of G. We address the convex recoloring problem, defined as follows. Given a graph G and a coloring of its vertices, recolor a minimum number of vertices of G, so that the resulting coloring is convex. This problem is known to be NP-hard even when G is a path. We show an integer programming formulation for the weighted version of this problem on arbitrary graphs, and then specialize it for trees. We study the facial structure of the polytope defined as the convex hull of the integer points satisfying the restrictions of the proposed ILP formulation, present several classes of facet-defining inequalities and discuss separation algorithms.     

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.     

A polyhedral study of the cardinality constrained knapsack problem

 A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas such as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous and the 0-1 variables. We use an alternative approach, in which we keep in the model only the continuous variables, and we enforce the cardinality constraint through a specialized branching scheme and the use of strong inequalities valid for the convex hull of the feasible set in the space of the continuous variables. To derive the valid inequalities, we extend the concepts of cover and cover inequality, commonly used in 0-1 programming, to this class of problems, and we show how cover inequalities can be lifted to derive facet-defining inequalities. We present three families of non-trivial facet-defining inequalities that are lifted cover inequalities. Finally, we report computational results that demonstrate the effectiveness of lifted cover inequalities and the superiority of the approach of not introducing auxiliary 0-1 variables over the traditional MIP approach for this class of problems. Received: March 13, 2003 Published online: April 10, 2003 Key Words. mixed-integer programming – knapsack problem – cardinality constrained programming – branch-and-cut     

Antiweb-wheel inequalities and their separation problems over the stable set polytopes

A stable set in a graph G is a set of pairwise nonadjacent vertices. The problem of finding a maximum weight stable set is one of the most basic ℕℙ-hard problems. An important approach to this problem is to formulate it as the problem of optimizing a linear function over the convex hull STAB(G) of incidence vectors of stable sets. Since it is impossible (unless ℕℙ=coℕℙ) to obtain a “concise” characterization of STAB(G) as the solution set of a system of linear inequalities, it is a more realistic goal to find large classes of valid inequalities with the property that the corresponding separation problem (given a point x ^*, find, if possible, an inequality in the class that x ^* violates) is efficiently solvable.?Some known large classes of separable inequalities are the trivial, edge, cycle and wheel inequalities. In this paper, we give a polynomial time separation algorithm for the (t)-antiweb inequalities of Trotter. We then introduce an even larger class (in fact, a sequence of classes) of valid inequalities, called (t)-antiweb-s-wheel inequalities. This class is a common generalization of the (t)-antiweb inequalities and the wheel inequalities. We also give efficient separation algorithms for them. Received: June 2000 / Accepted: August 2001?Published online February 14, 2002     

A Branch-and-Cut Algorithm for the Median-Path Problem

The Median-Path problem consists of locating a st-path on a network, minimizing a function of two parameters: accessibility to the path and total cost of the path. Applications of this problem can be found in transportation planning, water resource management and fluid transportation. A problem formulation based on Subtour and Variable Upper Bound (VUB) inequalities was proposed in the seminal paper by (Current, Revelle and Cohon, 1989). In this paper we introduce a tighter formulation, based on a new family of valid inequalities, named Lifted Subtour inequalities, that are proved to be facet-defining. For the class of Lifted Subtour inequalities we propose a polynomial separation algorithm. Then we introduce more families of valid inequalities derived by investigating the relation to the Asymmetric Traveling Salesman Problem (ATSP) polytope and to the Stable Set polytope. These results are used to develop a Branch-and-Cut algorithm that enables us to solve to optimality small and medium size instances in less than 2 hours of CPU time on a workstation.     

The triangle closure is a polyhedron

Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, Kannan and Schrijver who showed that the split closure is a polyhedron. Although some fairly general results were obtained by Andersen et al. (Math Oper Res 35(1):233–256, 2010) and Averkov (Discret Optimiz 9(4):209–215, 2012), some basic questions have remained unresolved. For example, maximal lattice-free triangles are the natural family to study beyond the family of splits and it has been a standing open problem to decide whether the triangle closure is a polyhedron. In this paper, we show that when the number of integer variables $m=2$ the triangle closure is indeed a polyhedron and its number of facets can be bounded by a polynomial in the size of the input data. The techniques of this proof are also used to give a refinement of necessary conditions for valid inequalities being facet-defining due to Cornuéjols and Margot (Math Program 120:429–456, 2009) and obtain polynomial complexity results about the mixed integer hull.     

Lifted inequalities for 0\mathord {-}1 mixed-integer bilinear covering sets

In this paper, we study $0\mathord {-}1$ mixed-integer bilinear covering sets. We derive several families of facet-defining inequalities via sequence-independent lifting techniques. We then show that these sets have a polyhedral structure that is similar to that of a certain fixed-charge single-node flow set. As a result, we also obtain new facet-defining inequalities for the single-node flow set that generalize well-known lifted flow cover inequalities from the integer programming literature.     

Complexity results related to monophonic convexity

The study of monophonic convexity is based on the family of induced paths of a graph. The closure of a subset X of vertices, in this case, contains every vertex v such that v belongs to some induced path linking two vertices of X. Such a closure is called monophonic closure. Likewise, the convex hull of a subset is called monophonic convex hull. In this work we deal with the computational complexity of determining important convexity parameters, considered in the context of monophonic convexity. Given a graph G, we focus on three parameters: the size of a maximum proper convex subset of G (m-convexity number); the size of a minimum subset whose closure is equal to V(G) (monophonic number); and the size of a minimum subset whose convex hull is equal to V(G) (m-hull number). We prove that the decision problems corresponding to the m-convexity and monophonic numbers are NP-complete, and we describe a polynomial time algorithm for computing the m-hull number of an arbitrary graph.     

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.An early version of this paper was presented at the Oberwolfach Conference on Combinatorial Optimization in January 1991. This research was supported in part by the National Science Foundation, Grant #DDM-8901495 and the Office of Naval Research through Contract N00014-85-K-0198.Corresponding author.The work of this author was supported by MURST, Italy.