-
[show abstract]
[hide abstract]
ABSTRACT: We introduce the reverse Chv\'atal-Gomory rank r*(P) of an integral
polyhedron P, defined as the supremum of the Chv\'atal-Gomory ranks of all
rational polyhedra whose integer hull is P. A well-known example in dimension
two shows that there exist integral polytopes P with r*(P) equal to infinity.
We provide a geometric characterization of polyhedra with this property in
general dimension, and investigate upper bounds on r*(P) when this value is
finite. We also sketch possible extensions, in particular to the reverse split
rank.
11/2012;
-
[show abstract]
[hide abstract]
ABSTRACT: Given an integral polyhedron P and a rational polyhedron Q living in the same
n-dimensional space and containing the same integer points as P, we investigate
how many iterations of the Chv\'atal-Gomory closure operator have to be
performed on Q to obtain a polyhedron contained in the affine hull of P. We
show that if P contains an integer point in its relative interior, then such a
number of iterations can be bounded by a function depending only on n. On the
other hand, we prove that if P is not full-dimensional and does not contain any
integer point in its relative interior, then no finite bound on the number of
iterations exists.
10/2012;
-
[show abstract]
[hide abstract]
ABSTRACT: We derive a new upper bound on the diameter of a polyhedron P = {x \in R^n :
Ax <= b}, where A \in Z^{m\timesn}. The bound is polynomial in n and the
largest absolute value of a sub-determinant of A, denoted by \Delta. More
precisely, we show that the diameter of P is bounded by O(\Delta^2 n^4 log
n\Delta). If P is bounded, then we show that the diameter of P is at most
O(\Delta^2 n^3.5 log n\Delta).
For the special case in which A is a totally unimodular matrix, the bounds
are O(n^4 log n) and O(n^3.5 log n) respectively. This improves over the
previous best bound of O(m^16 n^3 (log mn)^3) due to Dyer and Frieze.
08/2011;
-
SIAM Journal on Optimization. 01/2011; 21:1594-1613.
-
SIAM J. Discrete Math. 01/2011; 25:731-732.
-
SIAM J. Discrete Math. 01/2010; 24:853-875.
-
Integer Programming and Combinatorial Optimization, 13th International Conference, IPCO 2008, Bertinoro, Italy, May 26-28, 2008, Proceedings; 01/2008
-
Integer Programming and Combinatorial Optimization, 12th International IPCO Conference, Ithaca, NY, USA, June 25-27, 2007, Proceedings; 01/2007
-
[show abstract]
[hide abstract]
ABSTRACT: We consider mixed-integer sets of the type M IX T U = {x : Ax b; xi integer, i I}, where A is a totally unimodular matrix, b is an arbitrary vector and I is a nonempty subset of the column indices of A. We show that the problem of checking nonemptiness of a set M IX T U is NP-complete when A contains at most two nonzeros per column. This is in contrast to the case when A is TU and contains at most two nonzeros per row. Denoting the set by M IX 2T U , we provide an extended formulation for the convex hull of M IX 2T U whose constraint matrix is the dual of a network matrix, and with integer right hand side vector. The size of this formulation depends on the number |F | of distinct fractional parts taken by the continuous variables in the extreme points of conv(M IX 2T U ). When this number is polynomial in the dimension of the matrix A, the formulation is of polynomial size and the optimization problem over M IX 2T U lies in P. We show that there are instances for which |F | is of exponential size, and we also give conditions under which |F | is of polynomial size. Finally we show that these results for the set M IX 2T U provide a unified framework leading to polynomial-size extended formulations for several generalizations of mixing sets and lot-sizing sets studied in the last few years.
01/2007;
-
[show abstract]
[hide abstract]
ABSTRACT: We consider here the mixing set with flows: s + xt >= bt, xt <= yt for 1 <= t <= n; s [belongs] R+exp.1+, x [belongs] R+exp.n, y [belongs] Z+exp.n. It models the "flow version" of the basic mixing set introduced and studied by Gunluk and Pochet, as well as the most simple stochastic lot-sizing problem with recourse, and more generally is a relaxation of certain mixed integer sets that arise in the study of production planning problems. We study the polyhedron obtained by convexifying the above set. Specifically we provide a system of inequalities that gives its external description and characterize its vertices and rays.
Universit� catholique de Louvain, Center for Operations Research and Econometrics (CORE), CORE Discussion Papers. 01/2005;
-
Marco Di Summa
[show abstract]
[hide abstract]
ABSTRACT: A mixed-integer program is an optimization problem where one is required to minimize a linear function over a subset defined by a system of linear inequalities, with the additional restriction that some of the variables must take an integer value. Many real-world problems can be formulated as mixed-integer programs.
Solving mixed-integer programs is difficult in general. A common approach to tackle this kind of problems exploits the fact that (under mild assumptions) the convex hull of feasible solutions is a polyhedron. When the inequalities describing such a polyhedron are known explicitly, the mixed-integer program reduces to a linear program, which is a tractable problem. Unfortunately it is usually very hard to find a linear inequality description of the convex hull of feasible solutions of a mixed-integer program. However in some cases the introduction of additional variables allows one to give a simple description of such a convex hull by means of linear inequalities in a higher dimensional space. Such a description is called an extended formulation. If an extended formulation is known that is compact (i.e. it uses a polynomial number of variables and constraints), the original mixed-integer programming problem can be solved in polynomial time by means of linear programming algorithms.
In this dissertation we study the family of mixed-integer programs whose feasible regions are defined by linear systems with totally unimodular matrices (i.e. all subdeterminants are 0, 1 or -1) having at most two nonzero entries per row. This class of problems is interesting because many instances arising e.g. in the context of production planning can be formulated as mixed-integer programs of this type.
We illustrate a technique to construct an extended formulation for any problem in this family. The approach is based on the enumeration of all possible fractional parts that the variables take at the vertices of the convex hull of the feasible region.
We then discuss the compactness of our extended formulation: we give sufficient conditions ensuring that the formulation is compact. When one of these conditions holds, the mixed-integer program can be solved in polynomial time. We also show how our technique can be successfully applied to some interesting practical problems.
Next we consider the possibility of describing the convex hull of the feasible region in the original space of definition of the problem (i.e with no additional variables). Such a formulation is found explicitly for some special cases.
Finally a possible extension is discussed: we show how a generalization of our technique can lead to a compact extended formulation for a problem that does not belong to the family introduced above.
-
[show abstract]
[hide abstract]
ABSTRACT: We consider mixed-integer sets described by system of linear inequalities in which the constraint matrix A is totally unimodular; the right-hand side is arbitrary vector; and a subset of the variables is required to be integer. We show that the problem of checking nonemptiness of a set of this type is NP-complete, even in the case in which the linear system describes mixed-integer network flows with half-integral requirement on the nodes.
-
[show abstract]
[hide abstract]
ABSTRACT: We study properties of systems of linear constraints that are minimally infeasible with respect to some subset S of constraints (i.e., systems that are infeasible but that become feasible on removal of any constraint in S). We then apply these results and a theorem of Conforti, Cornuéjols, Kapoor, and Vukovi to a class of 0, 1 matrices, for which the linear relaxation of the set-partitioning polytope LSP(A)= {x|Ax = 1, x 0} is integral. In this way, we obtain combinatorial properties of those matrices in the class that are minimal (w.r.t. taking row submatrices) with the property that the set-partitioning polytope associated with them is infeasible.
-
Marco Di Summa
[show abstract]
[hide abstract]
ABSTRACT: L'oggetto del nostro studio sono le matrici bilanciate il cui problema di set
partitioning e' privo di soluzioni e che sono costituite da un insieme minimale
di righe e di colonne rispetto a questa proprietà: le chiameremo matrici criti-
che. Per arrivare all'analisi della struttura di tali matrici, premetteremo delle
considerazioni generali sul problema di set partitioning, sulle matrici bilanciate
e su quelli che chiameremo sistemi critici di equazioni e disequazioni lineari.
Passeremo poi allo studio delle proprietà delle matrici bilanciate e critiche,
approfondendo, in particolar modo, l'esame delle matrici bilanciate e critiche
che sono 2-regolari sulle colonne (tali, cioe', che ogni loro colonna contenga
esattamente due elementi non nulli).
-
[show abstract]
[hide abstract]
ABSTRACT: For the problem of lot-sizing on a tree with constant capacities, or stochastic lot-sizing with a scenario tree, we present various reformulations based on mixing sets. We also show how earlier results for uncapacitated problems involving (Q,SQ) inequalities can be simplified and extended. Finally some limited computational results are presented.
Operations Research Letters.
