153 research outputs found
The Cost of Perfection for Matchings in Graphs
Perfect matchings and maximum weight matchings are two fundamental
combinatorial structures. We consider the ratio between the maximum weight of a
perfect matching and the maximum weight of a general matching. Motivated by the
computer graphics application in triangle meshes, where we seek to convert a
triangulation into a quadrangulation by merging pairs of adjacent triangles, we
focus mainly on bridgeless cubic graphs. First, we characterize graphs that
attain the extreme ratios. Second, we present a lower bound for all bridgeless
cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic
graphs, most of which are shown to be tight. Additionally, we present tight
bounds for the class of regular bipartite graphs
Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
We consider the general problem of finding the minimum weight \bm-matching
on arbitrary graphs. We prove that, whenever the linear programming (LP)
relaxation of the problem has no fractional solutions, then the belief
propagation (BP) algorithm converges to the correct solution. We also show that
when the LP relaxation has a fractional solution then the BP algorithm can be
used to solve the LP relaxation. Our proof is based on the notion of graph
covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara
2007}.
These results are notable in the following regards: (1) It is one of a very
small number of proofs showing correctness of BP without any constraint on the
graph structure. (2) Variants of the proof work for both synchronous and
asynchronous BP; it is the first proof of convergence and correctness of an
asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete
Mathematics on March 19, 2009; accepted for publication (in revised form)
August 30, 2010; published electronically July 1, 201
Simple Combinatorial Optimisation Cost Games
In this paper we introduce the class of simple combinatorial optimisation cost games, which are games associated to {0, 1}-matrices.A coalitional value of a combinatorial optimisation game is determined by solving an integer program associated with this matrix and the characteristic vector of the coalition.For this class of games, we will characterise core stability and totally balancedness.We continue by characterising exactness and largeness.Finally, we conclude the paper by applying our main results to minimum colouring games and minimum vertex cover games.Combinatorial optimisation game;core stability;totally balancedness;largeness;exactness
Decentralised Job Matching
This paper studies a decentralised job market model where firms (academic departments) propose sequentially a (unique) position to some workers (Ph.D. candidates). Successful candidates then decide whether to accept the offers, and departments whose positions remain unfilled propose to other candidates. We distinguish between several cases, depending on whether agents’ actions are simultaneous and/or irreversible (if a worker accepts an offer he is immediately matched, and both the worker and the firm to which she is matched go out of the market). For all these cases, we provide a complete characterization of the Nash equilibrium outcomes and the Subgame Perfect equilibria. While the set of Nash equilibria outcomes contain all individually rational matchings, it turns out that in most cases considered all subgame perfect equilibria yield a unique outcome, the worker-optimal matching.Two-sided matching, Job market, Subgame perfect equilibrium, irreversibilities
On the ratio between maximum weight perfect matchings and maximum weight matchings in grids
Given a graph G that admits a perfect matching, we investigate the parameter η(G) (originally motivated by computer graphics applications) which is defined as follows. Among all nonnegative edge weight assignments, η(G) is the minimum ratio between (i) the maximum weight of a perfect matching and (ii) the maximum weight of a general matching. In this paper, we determine the exact value of η for all rectangular grids, all bipartite cylindrical grids, and all bipartite toroidal grids. We introduce several new techniques to this endeavor
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