Pascal Benchimol

BALANCING THE STATIONS OF A SELF SERVICE \BIKE HIRE" SYSTEM

This paper is motivated by operating self service transport systems that ourish nowa- days. In cities where such systems have been set up with bikes, trucks travel to maintain a suitable number of bikes per station. It is natural to study a version of the C-delivery TSP dened by Chalasani and Motwani in which, unlike their denition, C is part of the input: each vertex v of a graph G = (V;E) has a certain amount xv of a commodity and wishes to have an amount equal to yv (we assume that P v2V xv = P v2V yv and all quantities are assumed to be integers); given a vehicle of capacity C, nd a minimal route that balances all vertices, that is, that allows to have an amount yv of the commodity on each vertex v. This paper presents among other things complexity results, lower bounds, approximation algo- rithms, and a polynomial algorithm when G is a tree. Mathematics Subject Classication. ???, ???

Tropicalizing the Simplex Algorithm

We develop a tropical analogue of the simplex algorithm for linear programming. In particular, we obtain a combinatorial algorithm to perform one tropical pivoting step, including the computation of reduced costs, in

Improved filtering for weighted circuit constraints

O(n(m+n))

Stabilized dynamic constraint aggregation for solving set partitioning problems

time, where

Combinatorial Simplex Algorithms Can Solve Mean Payoff Games

m

Improving the held and karp approach with constraint programming

is the number of constraints and

Log-Barrier Interior Point Methods Are Not Strongly Polynomial

n

Long and winding central paths

is the dimension.

The Tropical Shadow-Vertex Algorithm Solves Mean Payoff Games in Polynomial Time on Average

We study the weighted circuit constraint in the context of constraint programming. It appears as a substructure in many practical applications, particularly routing problems. We propose a domain filtering algorithm for the weighted circuit constraint that is based on the 1-tree relaxation of Held and Karp. In addition, we study domain filtering based on an additive bounding procedure that combines the 1-tree relaxation with the assignment problem relaxation. Experimental results on Traveling Salesman Problem instances demonstrate that our filtering algorithms can dramatically reduce the problem size. In particular, the search tree size and solving time can be reduced by several orders of magnitude, compared to existing constraint programming approaches. Moreover, for medium-size problem instances, our method is competitive with the state-of-the-art special-purpose TSP solver Concorde.

Stabilized Dynamic Constraint Aggregation for Solving Set Partitioning Problems

Dynamic constraint aggregation (DCA) and dual variable stabilization (DVS) are two methods that can reduce the negative impact of degeneracy when solving linear programs. The first uses a projection to reduce the primal space whereas the second acts in the dual space. In this paper, we develop a new method, called stabilized dynamic constraint aggregation (SDCA), that combines DCA and DVS for solving set partitioning problems. It allows to fight degeneracy from both primal and dual perspectives simultaneously. To assess the effectiveness of SDCA, we report computational results obtained for highly degenerate multi-depot vehicle scheduling problem instances solved by column generation. These results indicate that SDCA can reduce the average computational time of the master problem by a factor of up to 7 with respect to the best of the two combined methods. Furthermore, they show that its performance is robust with regard to increasing levels of degeneracy in test problems.