220 research outputs found
The min-max edge q-coloring problem
In this paper we introduce and study a new problem named \emph{min-max edge
-coloring} which is motivated by applications in wireless mesh networks. The
input of the problem consists of an undirected graph and an integer . The
goal is to color the edges of the graph with as many colors as possible such
that: (a) any vertex is incident to at most different colors, and (b) the
maximum size of a color group (i.e. set of edges identically colored) is
minimized. We show the following results: 1. Min-max edge -coloring is
NP-hard, for any . 2. A polynomial time exact algorithm for min-max
edge -coloring on trees. 3. Exact formulas of the optimal solution for
cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial
lower bound of the optimal solution with respect to the average degree of the
graph. 5. An approximation algorithm for planar graphs.Comment: 16 pages, 5 figure
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
New Tools and Connections for Exponential-Time Approximation
In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of
1.
r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r)))
time,
2.
r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r)))
time,
3.
(2−1/r)
for minimum vertex cover in O∗(exp(n/rΩ(r)))
time, and
4.
(k−1/r)
for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr)))
time.
(Throughout, O~
and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. i
Hitting all Maximal Independent Sets of a Bipartite Graph
We prove that given a bipartite graph G with vertex set V and an integer k,
deciding whether there exists a subset of V of size k hitting all maximal
independent sets of G is complete for the class Sigma_2^P.Comment: v3: minor chang
Online and quasi-online colorings of wedges and intervals
We consider proper online colorings of hypergraphs defined by geometric
regions. We prove that there is an online coloring algorithm that colors
intervals of the real line using colors such that for every
point , contained in at least intervals, not all the intervals
containing have the same color. We also prove the corresponding result
about online coloring a family of wedges (quadrants) in the plane that are the
translates of a given fixed wedge. These results contrast the results of the
first and third author showing that in the quasi-online setting 12 colors are
enough to color wedges (independent of and ). We also consider
quasi-online coloring of intervals. In all cases we present efficient coloring
algorithms
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