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.     

The asymptotic enumeration of rooted convex polyhedra

An asymptotic formula is obtained for the number of rooted c-nets with m vertices and n edges as m, n → ∞ with $\text{1}\text{2}\text{+ ε <}\text{n}\text{m}\text{< 2 ? ε}$ for some ε > 0.     

A porous thermoviscoelastic mixture problem: numerical analysis and computational experiments

In this paper, we study, from the numerical point of view, a porous thermoviscoelastic mixture problem. The mechanical problem is written as a linear coupled system of two hyperbolic partial differential equations for the porosities and a parabolic partial differential equation for the temperature field. An existence and uniqueness result and an energy decay property are stated. Then, fully discrete approximations are introduced by using the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. A priori error estimates are proved from which, under suitable regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximations in an academical one-dimensional example and the behaviour of the solutions in one- and two-dimensional problems.     

Reprint of: Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

The generalized assignment problem: Valid inequalities and facets

Three classes of valid inequalities based upon multiple knapsack constraints are derived for the generalized assignment problem. General properties of the facet defining inequalities are discussed and, for a special case, the convex hull is completely characterized. In addition, we prove that a basic fractional solution to the linear programming relaxation can be eliminated by a facet defining inequality associated with an individual knapsack constraint.Partial financial support under NSF grant #CCR-8812736.Partial financial support under NSF grant #DMS-8606188.     

Generalized mixed integer rounding inequalities: facets for infinite group polyhedra

We present a generalization of the mixed integer rounding (MIR) approach for generating valid inequalities for (mixed) integer programming (MIP) problems. For any positive integer n, we develop n facets for a certain (n + 1)-dimensional single-constraint polyhedron in a sequential manner. We then show that for any n, the last of these facets (which we call the n-step MIR facet) can be used to generate a family of valid inequalities for the feasible set of a general (mixed) IP constraint, which we refer to as the n-step MIR inequalities. The Gomory Mixed Integer Cut and the 2-step MIR inequality of Dash and günlük  (Math Program 105(1):29–53, 2006) are the first two families corresponding to n = 1,2, respectively. The n-step MIR inequalities are easily produced using periodic functions which we refer to as the n-step MIR functions. None of these functions dominates the other on its whole period. Finally, we prove that the n-step MIR inequalities generate two-slope facets for the infinite group polyhedra, and hence are potentially strong.      

The time dependent traveling salesman problem: polyhedra and algorithm

The time dependent traveling salesman problem (TDTSP) is a generalization of the classical traveling salesman problem (TSP), where arc costs depend on their position in the tour with respect to the source node. While TSP instances with thousands of vertices can be solved routinely, there are very challenging TDTSP instances with less than 100 vertices. In this work, we study the polytope associated to the TDTSP formulation by Picard and Queyranne, which can be viewed as an extended formulation of the TSP. We determine the dimension of the TDTSP polytope and identify several families of facet-defining cuts. We obtain good computational results with a branch-cut-and-price algorithm using the new cuts, solving almost all instances from the TSPLIB with up to 107 vertices.     

The Steiner tree problem I: Formulations,compositions and extension of facets

In this paper we give some integer programming formulations for the Steiner tree problem on undirected and directed graphs and study the associated polyhedra. We give some families of facets for the undirected case along with some compositions and extensions. We also give a projection that relates the Steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph. This is used to show that the LP-relaxation of the directed formulation is superior to the LP-relaxation of the undirected one.Corresponding author.     

The one-warehouse multi-retailer problem: reformulation, classification, and computational results

We consider the one-warehouse multi-retailer problem where a warehouse replenishes multiple retailers with deterministic dynamic demands over a horizon. The problem is to determine when and how much to order to the warehouse and retailers such that the total system-wide costs are minimized. We propose a new (combined transportation and shortest path based) integer programming reformulation for the problem in addition to the echelon stock and transportation based formulations in the literature. We analyze the strength of the LP relaxations of three formulations and show that the new formulation is stronger than others. We also show that the new and transportation based formulations are equivalent for the joint replenishment problem, where the warehouse is a crossdocking facility. We extend all formulations to the case with initial inventory at the warehouse and reveal the relation among their LP relaxations. We present our computational experiments with all formulations over a set of randomly generated test instances.     

The Steiner tree problem II: Properties and classes of facets

This is the second part of two papers addressing the study of the facial structure of the Steiner tree polyhedron. In this paper we identify several classes of facet defining inequalities and relate them to special classes of graphs on which the Steiner tree problem is known to be NP-hard.Corresponding author.The author appreciates partial support from National Science Foundation Grants Nos. DSM-8606188 and ECS 8800281.     

Deformations of hyperbolic convex polyhedra and cone-3-manifolds

The Stoker problem, first formulated in Stoker (Commun. Pure Appl. Math. 21:119–168, 1968), consists in understanding to what extent a convex polyhedron is determined by its dihedral angles. By means of the double construction, this problem is intimately related to rigidity issues for 3-dimensional cone-manifolds. In Mazzeo and Montcouquiol (J. Differ. Geom. 87(3):525–576, 2011), two such rigidity results were proven, implying that the infinitesimal version of the Stoker conjecture is true in the hyperbolic and Euclidean cases. In this second article, we show that local rigidity holds and prove that the space of convex hyperbolic polyhedra with given combinatorial type is locally parametrized by the set of dihedral angles, together with a similar statement for hyperbolic cone-3-manifolds.     

The orthogonal convex skull problem

We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n ²) algorithm is presented, which is a substantial improvement over theO(n ⁷) time algorithm for the general problem.The work of the first author was supported under a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692 and the work of the second author was partially supported by NSF Grants Nos. DCR-84-01898 and DCR-84-01633.     

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.