213 research outputs found
The Moser-Tardos Framework with Partial Resampling
The resampling algorithm of Moser \& Tardos is a powerful approach to develop
constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this
to partial resampling: when a bad event holds, we resample an
appropriately-random subset of the variables that define this event, rather
than the entire set as in Moser & Tardos. This is particularly useful when the
bad events are determined by sums of random variables. This leads to several
improved algorithmic applications in scheduling, graph transversals, packet
routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006)
on graph transversals asymptotically, and obtain improved approximation ratios
for a packet routing problem of Leighton, Maggs, & Rao (1994)
Dually conformal hypergraphs
Given a hypergraph , the dual hypergraph of is the
hypergraph of all minimal transversals of . The dual hypergraph is
always Sperner, that is, no hyperedge contains another. A special case of
Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to
the families of maximal cliques of graphs. All these notions play an important
role in many fields of mathematics and computer science, including
combinatorics, algebra, database theory, etc. In this paper we study
conformality of dual hypergraphs. While we do not settle the computational
complexity status of recognizing this property, we show that the problem is in
co-NP and can be solved in polynomial time for hypergraphs of bounded
dimension. In the special case of dimension , we reduce the problem to
-Satisfiability. Our approach has an implication in algorithmic graph
theory: we obtain a polynomial-time algorithm for recognizing graphs in which
all minimal transversals of maximal cliques have size at most , for any
fixed
A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing
Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for
semi-algebraic -uniform hypergraphs of bounded complexity, showing that for
each the vertex set can be equitably partitioned into a bounded
number of parts (in terms of and the complexity) so that all but an
-fraction of the -tuples of parts are homogeneous. We prove that
the number of parts can be taken to be polynomial in . Our improved
regularity lemma can be applied to geometric problems and to the following
general question on property testing: is it possible to decide, with query
complexity polynomial in the reciprocal of the approximation parameter, whether
a hypergraph has a given hereditary property? We give an affirmative answer for
testing typical hereditary properties for semi-algebraic hypergraphs of bounded
complexity
Upper clique transversals in graphs
A clique transversal in a graph is a set of vertices intersecting all maximal
cliques. The problem of determining the minimum size of a clique transversal
has received considerable attention in the literature. In this paper, we
initiate the study of the "upper" variant of this parameter, the upper clique
transversal number, defined as the maximum size of a minimal clique
transversal. We investigate this parameter from the algorithmic and complexity
points of view, with a focus on various graph classes. We show that the
corresponding decision problem is NP-complete in the classes of chordal graphs,
chordal bipartite graphs, and line graphs of bipartite graphs, but solvable in
linear time in the classes of split graphs and proper interval graphs.Comment: Full version of a WG 2023 pape
Computing a maximum clique in geometric superclasses of disk graphs
In the 90's Clark, Colbourn and Johnson wrote a seminal paper where they
proved that maximum clique can be solved in polynomial time in unit disk
graphs. Since then, the complexity of maximum clique in intersection graphs of
d-dimensional (unit) balls has been investigated. For ball graphs, the problem
is NP-hard, as shown by Bonamy et al. (FOCS '18). They also gave an efficient
polynomial time approximation scheme (EPTAS) for disk graphs. However, the
complexity of maximum clique in this setting remains unknown. In this paper, we
show the existence of a polynomial time algorithm for a geometric superclass of
unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS
for intersection graphs of convex pseudo-disks
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