240 research outputs found
Approximate Graph Coloring by Semidefinite Programming
We consider the problem of coloring k-colorable graphs with the fewest
possible colors. We present a randomized polynomial time algorithm that colors
a 3-colorable graph on vertices with min O(Delta^{1/3} log^{1/2} Delta log
n), O(n^{1/4} log^{1/2} n) colors where Delta is the maximum degree of any
vertex. Besides giving the best known approximation ratio in terms of n, this
marks the first non-trivial approximation result as a function of the maximum
degree Delta. This result can be generalized to k-colorable graphs to obtain a
coloring using min O(Delta^{1-2/k} log^{1/2} Delta log n), O(n^{1-3/(k+1)}
log^{1/2} n) colors. Our results are inspired by the recent work of Goemans and
Williamson who used an algorithm for semidefinite optimization problems, which
generalize linear programs, to obtain improved approximations for the MAX CUT
and MAX 2-SAT problems. An intriguing outcome of our work is a duality
relationship established between the value of the optimum solution to our
semidefinite program and the Lovasz theta-function. We show lower bounds on the
gap between the optimum solution of our semidefinite program and the actual
chromatic number; by duality this also demonstrates interesting new facts about
the theta-function
Covering orthogonal polygons with star polygons: The perfect graph approach
AbstractThis paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a minimum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an O(n8) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n3) algorithm
Approximation Techniques for Average Completion Time Scheduling
We consider the problem of nonpreemptive scheduling to minimize average ( weighted) completion time, allowing for release dates, parallel machines, and precedence constraints. Recent work has led to constant-factor approximations for this problem based on solving a preemptive or linear programming relaxation and then using the solution to get an ordering on the jobs. We introduce several new techniques which generalize this basic paradigm. We use these ideas to obtain
improved approximation algorithms for one-machine scheduling to minimize average completion time with release dates. In the process, we obtain an optimal randomized on-line algorithm for the same problem that beats a lower bound for deterministic on-line algorithms. We consider extensions to the case of parallel machine scheduling, and for this we introduce two new ideas: first, we show that a preemptive one-machine relaxation is a powerful tool for designing parallel machine scheduling algorithms that simultaneously produce good approximations and have small running times; second, we show that a nongreedy “rounding” of the relaxation yields better approximations than a greedy one. We also prove a general theore mrelating the value of one- machine relaxations to that of the schedules obtained for the original m-machine problems. This theorem applies even when there are precedence constraints on the jobs. We apply this result to obtain improved approximation ratios for precedence graphs such as in-trees, out-trees, and series-parallel graphs
Challenges in web search engines
This article presents a high-level discussion of some problems in information retrieval that are unique to web search engines. The goal is to raise awareness and stimulate research in these areas
Width and size of regular resolution proofs
This paper discusses the topic of the minimum width of a regular resolution
refutation of a set of clauses. The main result shows that there are examples
having small regular resolution refutations, for which any regular refutation
must contain a large clause. This forms a contrast with corresponding results
for general resolution refutations.Comment: The article was reformatted using the style file for Logical Methods
in Computer Scienc
On Computational Power of Quantum Read-Once Branching Programs
In this paper we review our current results concerning the computational
power of quantum read-once branching programs. First of all, based on the
circuit presentation of quantum branching programs and our variant of quantum
fingerprinting technique, we show that any Boolean function with linear
polynomial presentation can be computed by a quantum read-once branching
program using a relatively small (usually logarithmic in the size of input)
number of qubits. Then we show that the described class of Boolean functions is
closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226
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