350 research outputs found
Covering graphs by monochromatic trees and Helly-type results for hypergraphs
How many monochromatic paths, cycles or general trees does one need to cover
all vertices of a given -edge-coloured graph ? These problems were
introduced in the 1960s and were intensively studied by various researchers
over the last 50 years. In this paper, we establish a connection between this
problem and the following natural Helly-type question in hypergraphs. Roughly
speaking, this question asks for the maximum number of vertices needed to cover
all the edges of a hypergraph if it is known that any collection of a few
edges of has a small cover. We obtain quite accurate bounds for the
hypergraph problem and use them to give some unexpected answers to several
questions about covering graphs by monochromatic trees raised and studied by
Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao,
Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi
Semantic Width and the Fixed-Parameter Tractability of Constraint Satisfaction Problems
Constraint satisfaction problems (CSPs) are an important formal framework for
the uniform treatment of various prominent AI tasks, e.g., coloring or
scheduling problems. Solving CSPs is, in general, known to be NP-complete and
fixed-parameter intractable when parameterized by their constraint scopes. We
give a characterization of those classes of CSPs for which the problem becomes
fixed-parameter tractable.
Our characterization significantly increases the utility of the CSP framework
by making it possible to decide the fixed-parameter tractability of problems
via their CSP formulations.
We further extend our characterization to the evaluation of unions of
conjunctive queries, a fundamental problem in databases. Furthermore, we
provide some new insight on the frontier of PTIME solvability of CSPs.
In particular, we observe that bounded fractional hypertree width is more
general than bounded hypertree width only for classes that exhibit a certain
type of exponential growth.
The presented work resolves a long-standing open problem and yields powerful
new tools for complexity research in AI and database theory.Comment: Full and extended version of the IJCAI2020 paper with the same titl
Optimal Distributed Covering Algorithms
We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+epsilon). Let Delta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log{Delta} / log log Delta) rounds, for constants epsilon in (0,1] and f in N^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms.
For constant values of f and epsilon, our algorithm improves over the (f+epsilon)-approximation algorithm of [Fabian Kuhn et al., 2006] whose running time is O(log Delta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(f log n) rounds, improving over the classical result of [Samir Khuller et al., 1994] that achieves a running time of O(f log^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of [Koufogiannakis and Young, 2011].
We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (f+epsilon)-approximate integral solution in O((1+f/log n)* ((log Delta)/(log log Delta) + (f * log M)^{1.01}* log epsilon^{-1}* (log Delta)^{0.01})) rounds, where f bounds the number of variables in a constraint, Delta bounds the number of constraints a variable appears in, and M=max {1, ceil[1/a_{min}]}, where a_{min} is the smallest normalized constraint coefficient. This improves over the results of [Fabian Kuhn et al., 2006] for the integral case, which combined with rounding achieves the same guarantees in O(epsilon^{-4}* f^4 * log f * log(M * Delta)) rounds
An asymptotic bound for the strong chromatic number
The strong chromatic number of a graph on
vertices is the least number with the following property: after adding isolated vertices to and taking the union with any
collection of spanning disjoint copies of in the same vertex set, the
resulting graph has a proper vertex-colouring with colours.
We show that for every and every graph on vertices with
, , which is
asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s on estimating the maximum number
of edges in a hypergraph when the (fractional) matching number is given, which
we are able to solve in some special cases using probabilistic techniques.
Based on these results, we obtain some general theorems on the minimum
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine the minimum vertex degree which
guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also
discuss an application to a problem of finding an optimal data allocation in a
distributed storage system
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with
particular emphasis on structural characterisation of the existence problem in
dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC
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