20,091 research outputs found
A Tutorial on Clique Problems in Communications and Signal Processing
Since its first use by Euler on the problem of the seven bridges of
K\"onigsberg, graph theory has shown excellent abilities in solving and
unveiling the properties of multiple discrete optimization problems. The study
of the structure of some integer programs reveals equivalence with graph theory
problems making a large body of the literature readily available for solving
and characterizing the complexity of these problems. This tutorial presents a
framework for utilizing a particular graph theory problem, known as the clique
problem, for solving communications and signal processing problems. In
particular, the paper aims to illustrate the structural properties of integer
programs that can be formulated as clique problems through multiple examples in
communications and signal processing. To that end, the first part of the
tutorial provides various optimal and heuristic solutions for the maximum
clique, maximum weight clique, and -clique problems. The tutorial, further,
illustrates the use of the clique formulation through numerous contemporary
examples in communications and signal processing, mainly in maximum access for
non-orthogonal multiple access networks, throughput maximization using index
and instantly decodable network coding, collision-free radio frequency
identification networks, and resource allocation in cloud-radio access
networks. Finally, the tutorial sheds light on the recent advances of such
applications, and provides technical insights on ways of dealing with mixed
discrete-continuous optimization problems
Capacitated max-Batching with Interval Graph Compatibilities
We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7, 4, 5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.
Independent sets of maximum weight in apple-free graphs
We present the first polynomial-time algorithm to solve the maximum weight independent set problem for apple-free graphs, which is a common generalization of several important classes where the problem can be solved efficiently, such as claw-free graphs, chordal graphs, and cographs. Our solution is based on a combination of two algorithmic techniques (modular decomposition and decomposition by clique separators) and a deep combinatorial analysis of the structure of apple-free graphs. Our algorithm is robust in the sense that it does not require the input graph G to be apple-free; the algorithm either finds an independent set of maximum weight in G or reports that G is not apple-free
Polyhedral characteristics of balanced and unbalanced bipartite subgraph problems
We study the polyhedral properties of three problems of constructing an
optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the
first problem we consider a balanced biclique with the same number of vertices
in both parts and arbitrary edge weights. In the other two problems we are
dealing with unbalanced subgraphs of maximum and minimum weight with
nonnegative edges. All three problems are established to be NP-hard. We study
the polytopes and the cone decompositions of these problems and their
1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope
of the balanced complete bipartite subgraph problem. The clique number of
1-skeleton is estimated from below by a superpolynomial function. For both
unbalanced biclique problems we establish the superpolynomial lower bounds on
the clique numbers of the graphs of nonnegative cone decompositions. These
values characterize the time complexity in a broad class of algorithms based on
linear comparisons
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