We consider the problem of maximizing a nondecreasing submodular set function over various constraint structures. Specifically, we explore the performance of the greedy algorithm, and a related variant, the locally greedy algorithm in solving submodular function maximization problems. Most classic results on the greedy algorithm and its variant assume the existence of an optimal polynomial-time … Read more
Chvatal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or 1/2, such cuts are known as {0, 1/2}-cuts. It has been proven by Caprara and Fischetti (1996) that separation of {0, 1/2}-cuts is NP-hard. In this paper, we study … Read more
The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite … Read more
We introduce a new measure of complexity of integer hulls of rational polyhedra called the small Chvatal rank (SCR). The SCR of an integer matrix A is the number of rounds of a Hilbert basis procedure needed to generate all normals of a sufficient set of inequalities to cut out the integer hulls of all … Read more
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is … Read more
We investigate the problem of sequencing jobs that have multiple components. Each component of the job needs to be processed independently on a specified machine. We derive approximate algorithms for the problem of scheduling such vector jobs to minimize their total completion time in the deterministic as well as stochastic setting. In particular, we propose … Read more
We present a new computer system, called GraPHedron, which uses a polyhedral approach to help the user to discover optimal conjectures in graph theory. We define what should be optimal conjectures and propose a formal framework allowing to identify them. Here, graphs with n nodes are viewed as points in the Euclidian space, whose coordinates … Read more
We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b’ ≤ Ax ≤ b, x ∈ Zn with b’ ≤ (AU)y ≤ b, y ∈ Zn, where U is a unimodular matrix computed via basis reduction, to make the … Read more
This paper aims at proposing tractable algorithms to find effectively good solutions to large size chance-constrained combinatorial problems. A new robust model is introduced to deal with uncertainty in mixed-integer linear problems. It is shown to be strongly related to chance-constrained programming when considering pure 0-1 problems. Furthermore, its tractability is highlighted. Then, an optimization … Read more
If $p \geq t$ are positive integers, a t-composition of p is an ordered t-tuple of positive integers summing p. If $T=(s_1, s_2, \dots, s_t)$ is a t-composition of p and W is a $p-(t-1) \times t$ matrix, call $W(T)= \sum_{k=1}^t w_{s_k k}$ the weight of the t-composition T. We show that finding a minimum … Read more