735 research outputs found
Optimality program in segment and string graphs
Planar graphs are known to allow subexponential algorithms running in time
or for most of the paradigmatic
problems, while the brute-force time is very likely to be
asymptotically best on general graphs. Intrigued by an algorithm packing curves
in by Fox and Pach [SODA'11], we investigate which
problems have subexponential algorithms on the intersection graphs of curves
(string graphs) or segments (segment intersection graphs) and which problems
have no such algorithms under the ETH (Exponential Time Hypothesis). Among our
results, we show that, quite surprisingly, 3-Coloring can also be solved in
time on string graphs while an algorithm running
in time for 4-Coloring even on axis-parallel segments (of unbounded
length) would disprove the ETH. For 4-Coloring of unit segments, we show a
weaker ETH lower bound of which exploits the celebrated
Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over
to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent
Dominating Set.Comment: 19 pages, 15 figure
Algorithms for Coloring Quadtrees
We describe simple linear time algorithms for coloring the squares of
balanced and unbalanced quadtrees so that no two adjacent squares are given the
same color. If squares sharing sides are defined as adjacent, we color balanced
quadtrees with three colors, and unbalanced quadtrees with four colors; these
results are both tight, as some quadtrees require this many colors. If squares
sharing corners are defined as adjacent, we color balanced or unbalanced
quadtrees with six colors; for some quadtrees, at least five colors are
required.Comment: 7 pages, 9 figure
A tabu search heuristic for the Equitable Coloring Problem
The Equitable Coloring Problem is a variant of the Graph Coloring Problem
where the sizes of two arbitrary color classes differ in at most one unit. This
additional condition, called equity constraints, arises naturally in several
applications. Due to the hardness of the problem, current exact algorithms can
not solve large-sized instances. Such instances must be addressed only via
heuristic methods. In this paper we present a tabu search heuristic for the
Equitable Coloring Problem. This algorithm is an adaptation of the dynamic
TabuCol version of Galinier and Hao. In order to satisfy equity constraints,
new local search criteria are given. Computational experiments are carried out
in order to find the best combination of parameters involved in the dynamic
tenure of the heuristic. Finally, we show the good performance of our heuristic
over known benchmark instances
An extensive English language bibliography on graph theory and its applications, supplement 1
Graph theory and its applications - bibliography, supplement
Conflict-free star-access in parallel memory systems
We study conflict-free data distribution schemes in parallel memories in multiprocessor system architectures. Given a host graph G, the problem is to map the nodes of G into memory modules such that any instance of a template type T in G can be accessed without memory conflicts. A conflict occurs if two or more nodes of T are mapped to the same memory module. The mapping algorithm should: (i) be fast in terms of data access (possibly mapping each node in constant time); (ii) minimize the required number of memory modules for accessing any instance in G of the given template type; and (iii) guarantee load balancing on the modules. In this paper, we consider conflict-free access to star templates. i.e., to any node of G along with all of its neighbors. Such a template type arises in many classical algorithms like breadth-first search in a graph, message broadcasting in networks, and nearest neighbor based approximation in numerical computation. We consider the star-template access problem on two specific host graphs-tori and hypercubes-that are also popular interconnection network topologies. The proposed conflict-free mappings on these graphs are fast, use an optimal or provably good number of memory modules, and guarantee load balancing. (C) 2006 Elsevier Inc. All rights reserved
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