The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the permutation of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up … Read more
This paper deals with directed, bidirected, and undirected capacitated network design problems. Using mixed integer rounding (MIR), we generalize flow-cutset inequalities to these three link types and to an arbitrary modular link capacity structure, and propose a generic separation algorithm. In an extensive computational study on 54 instances from the Survivable Network Design Library (SNDlib), … Read more
Due to the high computational complexity of exact methods (e.g., integer programming) for routing and wavelength assigment in optical networks, it is beneficial to decompose the problem into a routing task and a wavelength allocation task. However, by this decomposition it is not necessarily possible to obtain a valid wavelength assignment for a given routing … Read more
We introduce orbital branching, an effective branching method for integer programs containing a great deal of symmetry. The method is based on computing groups of variables that are equivalent with respect to the symmetry remaining in the problem after branching, including symmetry which is not present at the root node. These groups of equivalent variables, … Read more
We present new valid inequalities that work in similar ways to well known cover inequalities.The differences exist in three aspects. First difference is in the generation of the inequalities. The method used in generation of the new cuts is more practical in contrast to classical cover inequalities. Second difference is the more general applicability, i.e., … Read more
We study the set $S = \{(x, y) \in \Re_{+} \times Z^{n}: x + B_{j} y_{j} \geq b_{j}, j = 1, \ldots, n\}$, where $B_{j}, b_{j} \in \Re_{+} – \{0\}$, $j = 1, \ldots, n$, and $B_{1} | \cdots | B_{n}$. The set $S$ generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and … Read more
Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it … Read more
Under the Open Shortest Path First (OSPF) protocol, traffic flow in an Internet Protocol (IP) network is routed on the shortest paths between each source and destination. The shortest path is calculated based on pre-assigned weights on the network links. The OSPF weight setting problem is to determine a set of weights such that, if … Read more
A classical theorem in Combinatorial Optimization proves the existence of fully polynomial- time approximation schemes for the knapsack problem. In a recent paper, Van Vyve and Wolsey ask whether for each 0 < epsilon ≤ 1 there exists an extended formulation for the knapsack problem, of size polynomial in the number of variables and/or 1/epsilon ... Read more
We establish that any nonconvex quadratic program having a mix of binary and continuous variables over a bounded feasible set can be represented as a linear program over the dual of the cone of copositive matrices. This result can be viewed as an extension of earlier separate results, which have established the copositive representation of … Read more