4 research outputs found
The NP-hard problems solving as a graph coloring task using the genetic algorithm
У статті розглядається проблема пофарбування графів за допомогою генетичного алгоритму. Показано, що велика кількість NP-складних задач зводяться до проблеми пофарбування теоретико-графових моделей. На основі 0-1 програмування та лінійно-квадратичної оптимізації розроблено комбінований алгоритм, який дозволяє ефективно пофарбовувати графи зі 6*10306 вершинами у мінімальну кількість кольорів.In this paper the graph coloring problem using the genetic algorithm is investigated. It is shown that a great majority of NP-hard problems can be converged to graph coloring problem. The combined algorithm is elaborated, basing on the 0-1 programming and linear-quadratic optimization. The designed algorithm enables to color graphs up to 6*10306 vertex in the minimal number of colors
Optimization Framework and Graph-Based Approach for Relay-Assisted Bidirectional OFDMA Cellular Networks
This paper considers a relay-assisted bidirectional cellular network where
the base station (BS) communicates with each mobile station (MS) using OFDMA
for both uplink and downlink. The goal is to improve the overall system
performance by exploring the full potential of the network in various
dimensions including user, subcarrier, relay, and bidirectional traffic. In
this work, we first introduce a novel three-time-slot time-division duplexing
(TDD) transmission protocol. This protocol unifies direct transmission, one-way
relaying and network-coded two-way relaying between the BS and each MS. Using
the proposed three-time-slot TDD protocol, we then propose an optimization
framework for resource allocation to achieve the following gains: cooperative
diversity (via relay selection), network coding gain (via bidirectional
transmission mode selection), and multiuser diversity (via subcarrier
assignment). We formulate the problem as a combinatorial optimization problem,
which is NP-complete. To make it more tractable, we adopt a graph-based
approach. We first establish the equivalence between the original problem and a
maximum weighted clique problem in graph theory. A metaheuristic algorithm
based on any colony optimization (ACO) is then employed to find the solution in
polynomial time. Simulation results demonstrate that the proposed protocol
together with the ACO algorithm significantly enhances the system total
throughput.Comment: 27 pages, 8 figures, 2 table
Reductions for the Stable Set Problem
One approach to finding a maximum stable set (MSS) in a graph is to try to reduce the size of the problem by transforming the problem into an equivalent problem on a smaller graph. This paper introduces several new reductions for the MSS problem, extends several well-known reductions to the maximum weight stable set (MWSS) problem, demonstrates how reductions for the generalized stable set problem can be used in conjunction with probing to produce powerful new reductions for both the MSS and MWSS problems, and shows how hypergraphs can be used to expand the capabilities of clique projections. The effectiveness of these new reduction techniques are illustrated on the DIMACS benchmark graphs, planar graphs, and a set of challenging MSS problems arising from Steiner Triple Systems
Independent set problems and odd-hole-preserving graph reductions
Methods are described that implement a branch-and-price decomposition
approach to solve the maximum weight independent set (MWIS) problem. The
approach is first described by Warrier et. al, and herein our contributions to this
research are presented. The decomposition calls for the exact solution of the
MWIS problem on induced subgraphs of the original graph. The focus of our
contribution is the use of chordal graphs as the induced subgraphs in this solution
framework.
Three combinatorial branch-and-bound solvers for the MWIS problem are
described. All use weighted clique covers to generate upper bounds, and all
branch according to the method of Balas and Yu. One extends and speeds up
the method of Babel. A second one modifies a method of Balas and Xue to
produce clique covers that share structural similarities with those produced by
Babel. Each of these improves on its predecessor. A third solver is a hybrid of
the other two. It yields the best known results on some graphs.
The related matter of deciding the imperfection or perfection of a graph
is also addressed. With the advent of the Strong Perfect Graph Theorem, this
problem is reduced to the detection of odd holes and anti-holes or the proof of
their absence. Techniques are provided that, for a given graph, find subgraphs in
polynomial time that contain odd holes whenever they are present in the given graph. These techniques and some basic structural results on such subgraphs
narrow the search for odd holes.
Results are reported for the performance of the three new solvers for the
MWIS problem that demonstrate that the third, hybrid solver outperforms its
clique-cover-based ancestors and, in some cases, the best current open-source
solver. The techniques for narrowing the search for odd holes are shown to
provide a polynomial-time reduction in the size of the input required to decide
the perfection or imperfection of a graph