Mathieu Van Vyve

The Continuous Mixing Polyhedron

We analyze the polyhedral structure of the setsPCMIX = {( s, r, z) ?R XR+nXZ n|s +r j+z j=f j ,j = 1,..., n} andP+CMIX =PCMIXn{ s = 0}. The setP+CMIX is a natural generalization of the mixing set studied by Pochet and Wolsey [15, 16] and GA¼nlA¼k and Pochet [8] and recently has been introduced by Miller and Wolsey [12]. We introduce a new class of valid inequalities that has proven to be sufficient for describing conv( PCMIX). We give an extended formulation of sizeO( n) XO( n2) variables and constraints and indicate how to separate over conv( PCMIX) inO( n3) time. Finally, we show how the mixed integer rounding (MIR) inequalities of Nemhauser and Wolsey [14] and the mixing inequalities of GA¼nlA¼k and Pochet [8] constitute special cases of the cycle inequalities.

A General Heuristic for Production Planning Problems

We consider production planning problems with the restriction that all integer variables model setups. Since finding a feasible solution of such problems is in general NP-complete, the classical approaches have been the use of heuristics to find good feasible solutions on the one hand, or branch and cut on the other hand. In the case of the former, a dual bound is not available, and there is no guarantee of solution quality. For the latter, the accent has been on improving the dual bound and only the simplest schemes have been used to find good feasible solutions.Here we first show that such simple schemes may run into trouble, even when applied to very simple problems. This motivates the proposed heuristic, IPE, which is designed to be used within a branch-and-cut approach. We test the performance of the heuristic on various published lot-sizing and network-design problems, with and without tightened reformulations. We compare these results with other heuristics and with time truncated branch-and-bound searches. IPE appears to be the best choice for large problems with weak formulations.

Linear-programming extended formulations for the single-item lot-sizing problem with backlogging and constant capacity

Recently, several authors [8, 10] have argued for the use of extended formulations to tighten production planning models. In this work we present two linear-programming extended formulations of the constant-capacity lot-sizing problem with backlogging. The first one applies to the problem with a general cost function and has O(n3) variables and constraints. This improves on the more straightforward O(n4) Florian and Klein [2] type formulation. The second one applies when the costs satisfy the Wagner-Whitin property but it has O(n2) variables and O(n3) constraints. As a by-product, we positively answer an open question of Miller and Wolsey [4] about the tightness of an extended formulation for the continuous mixing set.

Securely Solving Simple Combinatorial Graph Problems

We investigate the problem of solving traditional combinatorial graph problems using secure multi-party computation techniques, focusing on the shortest path and the maximum flow problems. To the best of our knowledge, this is the first time these problems have been addressed in a general multi-party computation setting. Our study highlights several complexity gaps and suggests the exploration of various trade-offs, while also offering protocols that are efficient enough to solve real-world problems.

A note on the extension complexity of the knapsack polytope

We show that there are 0-1 and unbounded knapsack polytopes with super-polynomial extension complexity. More specifically, for each

Algorithms for Single-Item Lot-Sizing Problems with Constant Batch Size

n \in N

Computationally efficient MIP formulation and algorithms for European day-ahead electricity market auctions

we exhibit 0-1 and unbounded knapsack polytopes in dimension

Relaxations for two-level multi-item lot-sizing problems

n

Minimizing opportunity costs of paradoxically rejected block orders in European day-ahead electricity markets

with extension complexity

Mixing MIR inequalities with two divisible coefficients

\Omega(2^{\sqrt{n}})