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 $k$-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

Combinatorial algorithms for the seriation problem

In this thesis we study the seriation problem, a combinatorial problem arising in data analysis, which asks to sequence a set of objects in such a way that similar objects are ordered close to each other. We focus on the combinatorial structure and properties of Robinsonian matrices, a special class of structured matrices which best achieve the seriation goal. Our contribution is both theoretical and practical, with a particular emphasis on algorithms. In Chapter 2 we introduce basic concepts about graphs, permutations and proximity matrices used throughout the thesis. In Chapter 3 we present Robinsonian matrices, discussing their characterizations and recognition algorithms existing in the literature. In Chapter 4 we discuss Lexicographic Breadth-First search (Lex-BFS), a special graph traversal algorithm used in multisweep algorithms for the recognition of several classes of graphs. In Chapter 5 we introduce a new Lex-BFS based algorithm to recognize Robinsonian matrices, which is derived from a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. In Chapter 6 we introduce the novel Similarity-First Search algorithm (SFS), a weighted version of Lex-BFS which we use in a multisweep algorithm for the recognition of Robinsonian matrices. In Chapter 7 we model the seriation problem as an instance of Quadratic Assignment Problem (QAP) and we show that if the data has a Robinsonian structure, then one can find an optimal solution for QAP using a Robinsonian recognition algorithm. In Chapter 8 we discuss how to solve the seriation problem when the data does not have a Robinsonian structure, by finding a Robinsonian approximation of the original data. Finally, in Chapter 9 we discuss some experiments which we have carried out in order to compare the performance of the algorithms introduced in the thesis

Linear-time Constant-ratio Approximation Algorithm and Tight Bounds for the Contiguity of Cographs

International audienceIn this paper we consider a graph parameter called contiguity which aims at encoding a graph by a linear ordering of its vertices. We prove that the contiguity of cographs is unbounded but is always dominated by O(log n), where n is the number of vertices of the graph. And we prove that this bound is tight in the sense that there exists a family of cographs on n vertices whose contiguity is Omega(log n). In addition to these results on the worst-case contiguity of cographs, we design a linear-time constant-ratio approximation algorithm for computing the contiguity of an arbitrary cograph, which constitutes our main result. As a by-product of our proofs, we obtain a min-max theorem, which is worth of interest in itself, stating equality between the rank of a tree and the minimum height of its path partitions

(Nearly-)tight bounds on the contiguity and linearity of cographs

International audienceIn this paper we show that the contiguity and linearity of cographs on n vertices are both O(n). Moreover, we show that this bound is tight for contiguity as there exists a family of cographs on n vertices whose contiguity is Ω(log n). We also provide an Ω(log n / log log n) lower bound on the maximum linearity of cographs on n vertices. As a by-product of our proofs, we obtain a min-max theorem, which is worth of interest in itself, stating equality between the rank of a tree and the minimum height of one of its path partitions

Succinct and Compact Data Structures for Intersection Graphs

This thesis designs space efficient data structures for several classes of intersection graphs, including interval graphs, path graphs and chordal graphs. Our goal is to support navigational operations such as adjacent and neighbourhood and distance operations such as distance efficiently while occupying optimal space, or a constant factor of the optimal space. Using our techniques, we first resolve an open problem with regards to succinctly representing ordinal trees that is able to convert between the index of a node in a depth-first traversal (i.e. pre-order) and in a breadth-first traversal (i.e. level-order) of the tree. Using this, we are able to augment previous succinct data structures for interval graphs with the \GDistance operation. We also study several variations of the data structure problem in interval graphs: beer interval graphs and dynamic interval graphs. In beer interval graphs, we are given that some vertices of the graph are beer nodes (representing beer stores) and we consider only those paths that pass through at least one of these beer nodes. We give data structure results and prove space lower bounds for these graphs. We study dynamic interval graphs under several well known dynamic models such as incremental or offline, and we give data structures for each of these models. Finally we consider path graphs where we improve on previous works by exploiting orthogonal range reporting data structures. For optimal space representations, we improve the run time of the queries, while for non-optimal space representations (but optimal query times), we reduce the space needed