The Maximum-Weight Stable Matching Problem: Duality and Efficiency

Given a preference system (G,≺) and an integral weight function defined on the edge set of G (not necessarily bipartite), the maximum-weight stable matching problem is to find a stable matching of (G,≺) with maximum total weight. In this paper we study this NP-hard problem using linear programming and polyhedral approaches. We show that the Rothblum system for defining the fractional stable matching polytope of (G,≺) is totally dual integral if and only if this polytope is integral if and only if (G,≺) has a bipartite representation. We also present a combinatorial polynomial-time algorithm for the maximum-weight stable matching problem and its dual on any preference system with a bipartite representation. Our results generalize Király and Pap's theorem on the maximum-weight stable-marriage problem and rely heavily on their work. © 2012 Society for Industrial and Applied Mathematics.published_or_final_versio

Stable Matching for Dynamic Ride-sharing Systems

Dynamic ride-sharing systems enable people to share rides and increase the efficiency of urban transportation by connecting riders and drivers on short notice. Automated systems that establish ride-share matches with minimal input from participants provide the most convenience and the most potential for system-wide performance improvement, such as reduction in total vehicle-miles traveled. Indeed, such systems may be designed to match riders and drivers to maximize system performance improvement. However, system-optimal matches may not provide the maximum benefit to each individual participant. In this paper we consider a notion of stability for ride-share matches and present several mathematical programming methods to establish stable or nearly-stable matches, where we note that ride-share matching optimization is performed over time with incomplete information. Our numerical experiments using travel demand data for the metropolitan Atlanta region show that we can significantly increase the stability of ride-share matching solutions at the cost of only a small degradation in system-wide performance

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics