3,320 research outputs found
Hardness and Algorithms for Rainbow Connectivity
An edge-colored graph G is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connectivity
of a connected graph G, denoted rc(G), is the smallest number of colors that
are needed in order to make G rainbow connected. In addition to being a natural
combinatorial problem, the rainbow connectivity problem is motivated by
applications in cellular networks. In this paper we give the first proof that
computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to
decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given
edge-colored (with an unbounded number of colors) graph is rainbow connected.
On the positive side, we prove that for every > 0, a connected graph
with minimum degree at least has bounded rainbow connectivity,
where the bound depends only on , and the corresponding coloring can
be constructed in polynomial time. Additional non-trivial upper bounds, as well
as open problems and conjectures are also pre sented
A Unified View of Graph Regularity via Matrix Decompositions
We prove algorithmic weak and \Szemeredi{} regularity lemmas for several
classes of sparse graphs in the literature, for which only weak regularity
lemmas were previously known. These include core-dense graphs, low threshold
rank graphs, and (a version of) upper regular graphs. More precisely, we
define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these
graphs, and then we show that cut pseudorandomness captures all of the above
graph classes as special cases.
The core of our approach is an abstracted matrix decomposition, roughly
following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy
[Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by
Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs,
and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new
PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded
class of input graphs. (It is NP Hard to get PTASes for these graphs in
general.
Solving Stochastic B\"uchi Games on Infinite Arenas with a Finite Attractor
We consider games played on an infinite probabilistic arena where the first
player aims at satisfying generalized B\"uchi objectives almost surely, i.e.,
with probability one. We provide a fixpoint characterization of the winning
sets and associated winning strategies in the case where the arena satisfies
the finite-attractor property. From this we directly deduce the decidability of
these games on probabilistic lossy channel systems.Comment: In Proceedings QAPL 2013, arXiv:1306.241
Hamilton decompositions of regular expanders: applications
In a recent paper, we showed that every sufficiently large regular digraph G
on n vertices whose degree is linear in n and which is a robust outexpander has
a decomposition into edge-disjoint Hamilton cycles. The main consequence of
this theorem is that every regular tournament on n vertices can be decomposed
into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large.
This verified a conjecture of Kelly from 1968. In this paper, we derive a
number of further consequences of our result on robust outexpanders, the main
ones are the following: (i) an undirected analogue of our result on robust
outexpanders; (ii) best possible bounds on the size of an optimal packing of
edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range
of values for d. (iii) a similar result for digraphs of given minimum
semidegree; (iv) an approximate version of a conjecture of Nash-Williams on
Hamilton decompositions of dense regular graphs; (v) the observation that dense
quasi-random graphs are robust outexpanders; (vi) a verification of the `very
dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint
Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the
size of an optimal packing of edge-disjoint Hamilton cycles in a random
tournament.Comment: final version, to appear in J. Combinatorial Theory
Quasi-randomness and algorithmic regularity for graphs with general degree distributions
We deal with two intimately related subjects: quasi-randomness and regular partitions. The purpose of the concept of quasi-randomness is to express how much a given graph “resembles” a random one. Moreover, a regular partition approximates a given graph by a bounded number of quasi-random graphs. Regarding quasi-randomness, we present a new spectral characterization of low discrepancy, which extends to sparse graphs. Concerning regular partitions, we introduce a concept of regularity that takes into account vertex weights, and show that if satisfies a certain boundedness condition, then admits a regular partition. In addition, building on the work of Alon and Naor [Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), Chicago, IL, ACM, New York, 2004, pp. 72–80], we provide an algorithm that computes a regular partition of a given (possibly sparse) graph in polynomial time. As an application, we present a polynomial time approximation scheme for MAX CUT on (sparse) graphs without “dense spots.
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
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