165 research outputs found
Tight bounds and conjectures for the isolation lemma
Given a hypergraph and a weight function on its vertices, we say that is isolating if there is exactly one edge
of minimum weight . The Isolation Lemma is a
combinatorial principle introduced in Mulmuley et. al (1987) which gives a
lower bound on the number of isolating weight functions. Mulmuley used this as
the basis of a parallel algorithm for finding perfect graph matchings. It has a
number of other applications to parallel algorithms and to reductions of
general search problems to unique search problems (in which there are one or
zero solutions).
The original bound given by Mulmuley et al. was recently improved by Ta-Shma
(2015). In this paper, we show improved lower bounds on the number of isolating
weight functions, and we conjecture that the extremal case is when consists
of singleton edges. When our improved bound matches this extremal
case asymptotically.
We are able to show that this conjecture holds in a number of special cases:
when is a linear hypergraph or is 1-degenerate, or when . We also
show that it holds asymptotically when
Edge-coloring linear hypergraphs with medium-sized edges
Motivated by the Erd\H{o}s-Faber-Lov\'{a}sz (EFL) conjecture for hypergraphs,
we consider the list edge coloring of linear hypergraphs. We show that if the
hyper-edge sizes are bounded between and
inclusive, then there is a list edge coloring using colors. The dependence on in the upper bound is optimal (up to the
value of )
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