Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph $G=(U,V,E)$ is convex if the vertices in $V$ can be linearly ordered such that for each vertex $u\in U$, the neighbors of $u$ are consecutive in the ordering of $V$. An induced matching $H$ of $G$ is a matching such that no edge of $E$ connects endpoints of two different edges of $H$. We show that in a convex bipartite graph with $n$ vertices and $m$ weighted edges, an induced matching of maximum total weight can be computed in $O(n+m)$ time. An unweighted convex bipartite graph has a representation of size $O(n)$ that records for each vertex $u\in U$ the first and last neighbor in the ordering of $V$. Given such a compact representation, we compute an induced matching of maximum cardinality in $O(n)$ time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in $O(n)$ time. If no compact representation is given, the cover can be computed in $O(n+m)$ time. All of our algorithms achieve optimal running time for the respective problem and model. Previous algorithms considered only the unweighted case, and the best algorithm for computing a maximum-cardinality induced matching or a minimum chain cover in a convex bipartite graph had a running time of $O(n^2)$

Structural restrictions in cooperation

Cooperative games with transferable utilities, or simply TU-games, refer to the situations where the revenues created by a coalition of players through cooperation can be freely distributed to the members of the coalition. The fundamental question in cooperative game theory deals with the problem of how much payoff every player should receive. The classical assumption for TU-games states that every coalition is able to form and earn the worth created by cooperation. In the literature, there are several different modifications of TU-games in order to cover the cases where cooperation among the players is restricted. The second chapter of this monograph provides a characterization of the average tree solution for TU-games where the restricted cooperation is represented by a connected cycle-free graph on the set of players. The third chapter considers TU-games for which the restricted cooperation is represented by a directed graph on the set of players and introduces the average covering tree solution and the dominance value for this class of games. Chapter four considers TU-games with restricted cooperation which is represented by a set system on the set of players and introduces the average coalitional tree solution for such structures. The last two chapters of this monograph belong to the social choice theory literature. Given a set of candidates and a set of an odd number of individuals with preferences on these candidates, pairwise majority comparison of the candidates yields a tournament on the set of candidates. Tournaments are special types of directed graphs which contain an arc between any pair of nodes. The Copeland solution of a tournament is the set of candidates that beat the maximum number of candidates. In chapter five, a new characterization of the Copeland solution is provided that is based on the number of steps in which candidates beat each other. Chapter six of this monograph is on preference aggregation which deals with collective decision making to obtain a social preference. A sophisticated social welfare function is defined as a mapping from profiles of individual preferences into a sophisticated social preference which is a pairwise weighted comparison of alternatives. This chapter provides a characterization of Pareto optimal and pairwise independent sophisticated social welfare functions

Global Constraint Catalog, 2nd Edition (revision a)

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms

Global Constraint Catalog, 2nd Edition

This report presents a catalogue of global constraints where each constraint is explicitly described in terms of graph properties and/or automata and/or first order logical formulae with arithmetic. When available, it also presents some typical usage as well as some pointers to existing filtering algorithms