A polyhedral approach to the alldifferent system

This paper examines the facial structure of the convex hull of integer vectors satisfying a system of alldifferent predicates, also called an alldifferent system. The underlying analysis is based on a property, called inclusion, pertinent to such a system. For the alldifferent systems for which this property holds, we present two families of facet-defining inequalities, establish that they completely describe the convex hull and show that they can be separated in polynomial time. Consequently, the inclusion property characterises a group of alldifferent systems for which the linear optimization problem (i.e. the problem of optimizing a linear function over that system) can be solved in polynomial time. Furthermore, we establish that, for systems with three predicates, the inclusion property is also a necessary condition for the convex hull to be described by those two families of inequalities. For the alldifferent systems that do not possess that property, we establish another family of facet-defining inequalities and an accompanied polynomial-time separation algorithm. All the separation algorithms are incorporated within a cutting-plane scheme and computational experience on a set of randomly generated instances is reported. In concluding, we show that the pertinence of the inclusion property can be decided in polynomial time.     

On lifted cover inequalities: A new lifting procedure with unusual properties

Lifted cover inequalities are well-known cutting planes for 0–1 linear programs. We show how one of the earliest lifting procedures, due to Balas, can be significantly improved. The resulting procedure has some unusual properties. For example, (i) it can yield facet-defining inequalities even if the given cover is not minimal, (ii) it can yield facet-defining inequalities that cannot be obtained by standard lifting procedures, and (iii) the associated superadditive lifting function is integer-valued almost everywhere.     

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.     

A branch&cut algorithm for the maximum common edge subgraph problem

We investigate the Maximum Common Edge Subgraph Problem (MCES) defined as follows. Given two graphs G and H with the same number of vertices, find a common subgraph of G and H (not necessary induced) with the maximum number of edges. This problem arises in parallel programming environments, and was first defined by Bokhari in [S. Bokhari, On the mapping problem, IEEE Trans. Comput., C-30(3), 1981]. We present a new integer programming formulation for the MCES problem and carry out a polyhedral investigation of this model. A number of valid inequalities are identified, most of which are facet-defining. We also report on computational experiments.     

The graphical relaxation: A new framework for the symmetric traveling salesman polytope

A present trend in the study of theSymmetric Traveling Salesman Polytope (STSP(n)) is to use, as a relaxation of the polytope, thegraphical relaxation (GTSP(n)) rather than the traditionalmonotone relaxation which seems to have attained its limits. In this paper, we show the very close relationship between STSP(n) and GTSP(n). In particular, we prove that every non-trivial facet of STSP(n) is the intersection ofn + 1 facets of GTSP(n),n of which are defined by the degree inequalities. This fact permits us to define a standard form for the facet-defining inequalities for STSP(n), that we calltight triangular, and to devise a proof technique that can be used to show that many known facet-defining inequalities for GTSP(n) define also facets of STSP(n). In addition, we give conditions that permit to obtain facet-defining inequalities by composition of facet-defining inequalities for STSP(n) and general lifting theorems to derive facet-defining inequalities for STSP(n +k) from inequalities defining facets of STSP(n).Partially financed by P.R.C. Mathématique et Informatique.     

n-step mingling inequalities: new facets for the mixed-integer knapsack set

The n-step mixed integer rounding (MIR) inequalities of Kianfar and Fathi (Math Program 120(2):313–346, 2009) are valid inequalities for the mixed-integer knapsack set that are derived by using periodic n-step MIR functions and define facets for group problems. The mingling and 2-step mingling inequalities of Atamtürk and Günlük (Math Program 123(2):315–338, 2010) are also derived based on MIR and they incorporate upper bounds on the integer variables. It has been shown that these inequalities are facet-defining for the mixed integer knapsack set under certain conditions and generalize several well-known valid inequalities. In this paper, we introduce new classes of valid inequalities for the mixed-integer knapsack set with bounded integer variables, which we call n-step mingling inequalities (for positive integer n). These inequalities incorporate upper bounds on integer variables into n-step MIR and, therefore, unify the concepts of n-step MIR and mingling in a single class of inequalities. Furthermore, we show that for each n, the n-step mingling inequality defines a facet for the mixed integer knapsack set under certain conditions. For n?=?2, we extend the results of Atamtürk and Günlük on facet-defining properties of 2-step mingling inequalities. For n ≥ 3, we present new facets for the mixed integer knapsack set. As a special case we also derive conditions under which the n-step MIR inequalities define facets for the mixed integer knapsack set. In addition, we show that n-step mingling can be used to generate new valid inequalities and facets based on covers and packs defined for mixed integer knapsack sets.     

A polyhedral study of triplet formulation for single row facility layout problem

The single row facility layout problem (SRFLP) is the problem of arranging n departments with given lengths on a straight line so as to minimize the total weighted distance between all department pairs. We present a polyhedral study of the triplet formulation of the SRFLP introduced by Amaral [A.R.S. Amaral, A new lower bound for the single row facility layout problem, Discrete Applied Mathematics 157 (1) (2009) 183-190]. For any number of departments n, we prove that the dimension of the triplet polytope is n(n−1)(n−2)/3 (this is also true for the projections of this polytope presented by Amaral). We then prove that several valid inequalities presented by Amaral for this polytope are facet-defining. These results provide theoretical support for the fact that the linear program solved over these valid inequalities gives the optimal solution for all instances studied by Amaral.     

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.     

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.     

Relations between facets of low- and high-dimensional group problems

One-dimensional infinite group problems have been extensively studied and have yielded strong cutting planes for mixed integer programs. Although numerical and theoretical studies suggest that group cuts can be significantly improved by considering higher-dimensional groups, there are no known facets for infinite group problems whose dimension is larger than two. In this paper, we introduce an operation that we call sequential-merge. We prove that the sequential-merge operator creates a very large family of facet-defining inequalities for high-dimensional infinite group problems using facet-defining inequalities of lower-dimensional group problems. Further, they exhibit two properties that reflect the benefits of using facets of high-dimensional group problems: they have continuous variables’ coefficients that are not dominated by those of the constituent low-dimensional cuts and they can produce cutting planes that do not belong to the first split closure of MIPs. Further, we introduce a general scheme for generating valid inequalities for lower-dimensional group problems using valid inequalities of higher-dimensional group problems. We present conditions under which this construction generates facet-defining inequalities when applied to sequential-merge inequalities. We show that this procedure yields some two-step MIR inequalities of Dash and Günlük.     

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     

A note on the Undirected Rural Postman Problem polytope

This note refers to the article by G. Ghiani and G. Laporte ``A branch-and-cut algorithm for the Undirected Rural Postman Problem', Math. Program. 87 (2000). We show that some conditions for the facet-defining property of the basic non-trivial inequalities are not sufficient and that the Rural Postman Problem polytope is more complex even when focusing on canonical inequalities only.     

Dynamic knapsack sets and capacitated lot-sizing

 A dynamic knapsack set is a natural generalization of the 0-1 knapsack set with a continuous variable studied recently. For dynamic knapsack sets a large family of facet-defining inequalities, called dynamic knapsack inequalities, are derived by fixing variables to one and then lifting. Surprisingly such inequalities have the simultaneous lifting property, and for small instances provide a significant proportion of all the facet-defining inequalities. We then consider single-item capacitated lot-sizing problems, and propose the joint study of three related sets. The first models the discrete lot-sizing problem, the second the continuous lot-sizing problem with Wagner-Whitin costs, and the third the continuous lot-sizing problem with arbitrary costs. The first set that arises is precisely a dynamic knapsack set, the second an intersection of dynamic knapsack sets, and the unrestricted problem can be viewed as both a relaxation and a restriction of the second. It follows that the dynamic knapsack inequalities and their generalizations provide strong valid inequalities for all three sets. Some limited computation results are reported as an initial test of the effectiveness of these inequalities on capacitated lot-sizing problems. Received: March 28, 2001 / Accepted: November 9, 2001 Published online: September 27, 2002 RID="★" ID="★" Research carried out with financial support of the project TMR-DONET nr. ERB FMRX–CT98–0202 of the European Union. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Present address: Electrabel, Louvain-la-Neuve, B-1348 Belgium. Key words. knapsack sets – valid inequalities – simultaneous lifting – lot-sizing – Wagner-Whitin costs     

New cutting-planes for the time- and/or precedence-constrained ATSP and directed VRP

In this paper, we introduce five classes of new valid cutting planes for the precedence-constrained (PC) and/or time-window-constrained (TW) Asymmetric Travelling Salesman Problems (ATSPs) and directed Vehicle Routing Problems (VRPs). We show that all five classes of new inequalities are facet-defining for the directed VRP-TW, under reasonable conditions and the assumption that vehicles are identical. Similar proofs can be developed for the VRP-PC. As ATSP-TW and PC-ATSP can be formulated as directed identical-vehicle VRP-TW and PC-VRP, respectively, this provides a link to study the polyhedral combinatorics for the ATSP-TW and PC-ATSP. The first four classes of these new cutting planes are cycle-breaking inequalities that are lifted from the well-known \({D^-_k}\) and \({D^+_k}\) inequalities (see Grötschel and Padberg in Polyhedral theory. The traveling salesman problem: a guided tour of combinatorial optimization, Wiley, New York, 1985). The last class of new cutting planes, the TW ₂ inequalities, are infeasible-path elimination inequalities. Separation of these constraints will also be discussed. We also present prelimanry numerical results to demonstrate the strengh of these new cutting planes.     

The Steiner tree polytope and related polyhedra

We consider the vertex-weighted version of the undirected Steiner tree problem. In this problem, a cost is incurred both for the vertices and the edges present in the Steiner tree. We completely describe the associated polytope by linear inequalities when the underlying graph is series—parallel. For general graphs, this formulation can be interpreted as a (partial) extended formulation for the Steiner tree problem. By projecting this formulation, we obtain some very large classes of facet-defining valid inequalities for the Steiner tree polytope.Research supported by Air Force contract AFOSR-89-0271 and DARPA contract DARPA-89-5-1988.