2,044 research outputs found
Weighted Random Popular Matchings
For a set A of n applicants and a set I of m items, we consider a problem of
computing a matching of applicants to items, i.e., a function M mapping A to I;
here we assume that each applicant provides a preference list on
items in I. We say that an applicant prefers an item p than an item q
if p is located at a higher position than q in its preference list, and we say
that x prefers a matching M over a matching M' if x prefers M(x) over M'(x).
For a given matching problem A, I, and preference lists, we say that M is more
popular than M' if the number of applicants preferring M over M' is larger than
that of applicants preferring M' over M, and M is called a popular matching if
there is no other matching that is more popular than M. Here we consider the
situation that A is partitioned into , and that each
is assigned a weight such that w_{1}>w_{2}>...>w_{k}>0m/n^{4/3}=o(1)w_{1} \geq 2w_{2}n^{4/3}/m = o(1)w_{1} \geq 2w_{2}$ has a 2-weighted popular
matching with probability 1-o(1).Comment: 13 pages, 2 figure
A role of constraint in self-organization
In this paper we introduce a neural network model of self-organization. This
model uses a variation of Hebb rule for updating its synaptic weights, and
surely converges to the equilibrium status. The key point of the convergence is
the update rule that constrains the total synaptic weight and this seems to
make the model stable. We investigate the role of the constraint and show that
it is the constraint that makes the model stable. For analyzing this setting,
we propose a simple probabilistic game that models the neural network and the
self-organization process. Then, we investigate the characteristics of this
game, namely, the probability that the game becomes stable and the number of
the steps it takes.Comment: To appear in the Proc. RANDOM'98, Oct. 199
The Complexity of Kings
A king in a directed graph is a node from which each node in the graph can be
reached via paths of length at most two. There is a broad literature on
tournaments (completely oriented digraphs), and it has been known for more than
half a century that all tournaments have at least one king [Lan53]. Recently,
kings have proven useful in theoretical computer science, in particular in the
study of the complexity of the semifeasible sets [HNP98,HT05] and in the study
of the complexity of reachability problems [Tan01,NT02].
In this paper, we study the complexity of recognizing kings. For each
succinctly specified family of tournaments, the king problem is known to belong
to [HOZZ]. We prove that this bound is optimal: We construct a
succinctly specified tournament family whose king problem is
-complete. It follows easily from our proof approach that the problem
of testing kingship in succinctly specified graphs (which need not be
tournaments) is -complete. We also obtain -completeness
results for k-kings in succinctly specified j-partite tournaments, , and we generalize our main construction to show that -completeness
holds for testing k-kingship in succinctly specified families of tournaments
for all
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