418 research outputs found
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
We give a new method for analysing the mixing time of a Markov chain using
path coupling with stopping times. We apply this approach to two hypergraph
problems. We show that the Glauber dynamics for independent sets in a
hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and
the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the
Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4
and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the
hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19
page
On the optimality of the uniform random strategy
The concept of biased Maker-Breaker games, introduced by Chv\'atal and Erd{\H
o}s, is a central topic in the field of positional games, with deep connections
to the theory of random structures. For any given hypergraph the
main questions is to determine the smallest bias that allows
Breaker to force that Maker ends up with an independent set of . Here
we prove matching general winning criteria for Maker and Breaker when the game
hypergraph satisfies a couple of natural `container-type' regularity conditions
about the degree of subsets of its vertices. This will enable us to derive a
hypergraph generalization of the -building games, studied for graphs by
Bednarska and {\L}uczak. Furthermore, we investigate the biased version of
generalizations of the van der Waerden games introduced by Beck. We refer to
these generalizations as Rado games and determine their threshold bias up to
constant factors by applying our general criteria. We find it quite remarkable
that a purely game theoretic deterministic approach provides the right order of
magnitude for such a wide variety of hypergraphs, when the generalizations to
hypergraphs in the analogous setup of sparse random discrete structures are
usually quite challenging.Comment: 26 page
Alternative polynomial-time algorithm for Bipartite Matching
If is a bipartite graph, Hall's theorem \cite{H35} gives a condition for
the existence of a matching of covering one side of the bipartition. This
theorem admits a well-known algorithmic proof involving the repeated search of
augmenting paths. We present here an alternative algorithm, using a
game-theoretic formulation of the problem. We also show how to extend this
formulation to the setting of balanced hypergraphs
Sequential legislative lobbying
In this paper, we analyze the equilibrium of a sequential game-theoretical model of lobbying, due to Groseclose and Snyder (1996), describing a legislature that vote over two alternatives, where two opposing lobbies, Lobby 0 and Lobby 1, compete by bidding for legislatorsâ votes. In this model, the lobbyist moving first suffers from a second mover advantage and will make an offer to a panel of legislators only if it deters any credible counter-reaction from his opponent, i.e., if he anticipates to win the battle. This paper departs from the existing literature in assuming that legislators care about the consequence of their votes rather than their votes per se. Our main focus is on the calculation of the smallest budget that he needs to win the game and on the distribution of this budget across the legislators. We study the impact of the key parameters of the game on these two variables and show the connection of this problem with the combinatorics of sets and notions from cooperative game theory.Lobbying; cooperative games; noncooperative games
Radio Resource Allocation for Device-to-Device Underlay Communication Using Hypergraph Theory
Device-to-Device (D2D) communication has been recognized as a promising
technique to offload the traffic for the evolved Node B (eNB). However, the D2D
transmission as an underlay causes severe interference to both the cellular and
other D2D links, which imposes a great technical challenge to radio resource
allocation. Conventional graph based resource allocation methods typically
consider the interference between two user equipments (UEs), but they cannot
model the interference from multiple UEs to completely characterize the
interference. In this paper, we study channel allocation using hypergraph
theory to coordinate the interference between D2D pairs and cellular UEs, where
an arbitrary number of D2D pairs are allowed to share the uplink channels with
the cellular UEs. Hypergraph coloring is used to model the cumulative
interference from multiple D2D pairs, and thus, eliminate the mutual
interference. Simulation results show that the system capacity is significantly
improved using the proposed hypergraph method in comparison to the conventional
graph based one.Comment: 27 pages,10 figure
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