9 research outputs found
ProblĂšme de k-SĂ©parateur
Let G be a vertex-weighted undirected graph. We aim to compute a minimum weight subset of vertices whose removal leads to a graph where the size of each connected component is less than or equal to a given positive number k. If k = 1 we get the classical vertex cover problem. Many formulations are proposed for the problem. The linear relaxations of these formulations are theoretically compared. A polyhedral study is proposed (valid inequalities, facets, separation algorithms). It is shown that the problem can be solved in polynomial time for many special cases including the path, the cycle and the tree cases and also for graphs not containing some special induced sub-graphs. Some (k + 1)-approximation algorithms are also exhibited. Most of the algorithms are implemented and compared. The k-separator problem has many applications. If vertex weights are equal to 1, the size of a minimum k-separator can be used to evaluate the robustness of a graph or a network. Another application consists in partitioning a graph/network into different sub-graphs with respect to different criteria. For example, in the context of social networks, many approaches are proposed to detect communities. By solving a minimum k-separator problem, we get different connected components that may represent communities. The k-separator vertices represent persons making connections between communities. The k-separator problem can then be seen as a special partitioning/clustering graph problemConsidĂ©rons un graphe G = (V,E,w) non orientĂ© dont les sommets sont pondĂ©rĂ©s et un entier k. Le problĂšme Ă Ă©tudier consiste Ă la construction des algorithmes afin de dĂ©terminer le nombre minimum de nĆuds quâil faut enlever au graphe G pour que toutes les composantes connexes restantes contiennent chacune au plus k-sommets. Ce problĂšme nous lâappelons problĂšme de k-SĂ©parateur et on dĂ©signe par k-sĂ©parateur le sous-ensemble recherchĂ©. Il est une gĂ©nĂ©ralisation du Vertex Cover qui correspond au cas k = 1 (nombre minimum de sommets intersectant toutes les arĂȘtes du graphe
Ensemble approach for generalized network dismantling
Finding a set of nodes in a network, whose removal fragments the network
below some target size at minimal cost is called network dismantling problem
and it belongs to the NP-hard computational class. In this paper, we explore
the (generalized) network dismantling problem by exploring the spectral
approximation with the variant of the power-iteration method. In particular, we
explore the network dismantling solution landscape by creating the ensemble of
possible solutions from different initial conditions and a different number of
iterations of the spectral approximation.Comment: 11 Pages, 4 Figures, 4 Table
Treewidth versus clique number. II. Tree-independence number
In 2020, we initiated a systematic study of graph classes in which the
treewidth can only be large due to the presence of a large clique, which we
call -bounded. While -bounded graph
classes are known to enjoy some good algorithmic properties related to clique
and coloring problems, it is an interesting open problem whether
-boundedness also has useful algorithmic implications for
problems related to independent sets.
We provide a partial answer to this question by means of a new min-max graph
invariant related to tree decompositions. We define the independence number of
a tree decomposition of a graph as the maximum independence
number over all subgraphs of induced by some bag of . The
tree-independence number of a graph is then defined as the minimum
independence number over all tree decompositions of . Generalizing a result
on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is
given together with a tree decomposition with bounded independence number, then
the Maximum Weight Independent Packing problem can be solved in polynomial
time.
Applications of our general algorithmic result to specific graph classes will
be given in the third paper of the series [Dallard, Milani\v{c}, and
\v{S}torgel, Treewidth versus clique number. III. Tree-independence number of
graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A
previous version of this arXiv post has been reorganized into two parts; this
is the first of the two parts (the second one is arXiv:2206.15092
Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure
We continue the study of -bounded graph classes, that
is, hereditary graph classes in which the treewidth can only be large due to
the presence of a large clique, with the goal of understanding the extent to
which this property has useful algorithmic implications for the Independent Set
and related problems. In the previous paper of the series [Dallard,
Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. II.
Tree-independence number], we introduced the tree-independence number, a
min-max graph invariant related to tree decompositions. Bounded
tree-independence number implies both -boundedness and
the existence of a polynomial-time algorithm for the Maximum Weight Independent
Set problem, provided that the input graph is given together with a tree
decomposition with bounded independence number.
In this paper, we consider six graph containment relations and for each of
them characterize the graphs for which any graph excluding with respect
to the relation admits a tree decomposition with bounded independence number.
The induced minor relation is of particular interest: we show that excluding
either a minus an edge or the -wheel implies the existence of a tree
decomposition in which every bag is a clique plus at most vertices, while
excluding a complete bipartite graph implies the existence of a tree
decomposition with independence number at most . Our constructive
proofs are obtained using a variety of tools, including -refined tree
decompositions, SPQR trees, and potential maximal cliques. They imply
polynomial-time algorithms for the Independent Set and related problems in an
infinite family of graph classes; in particular, the results apply to the class
of -perfectly orientable graphs, answering a question of Beisegel,
Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.Comment: 46 pages; abstract has been shortened due to arXiv requirements. A
previous arXiv post (arXiv:2111.04543) has been reorganized into two parts;
this is the second of the two part
Models and algorithms for decomposition problems
This thesis deals with the decomposition both as a solution method and as a problem itself. A decomposition approach can be very effective for mathematical problems presenting a specific structure in which the associated matrix of coefficients is sparse and it is diagonalizable in blocks. But, this kind of structure may not be evident from the most natural formulation of the problem. Thus, its coefficient matrix may be preprocessed by solving a structure detection problem in order to understand if a decomposition method can successfully be applied. So, this thesis deals with the k-Vertex Cut problem, that is the problem of finding the minimum subset of nodes whose removal disconnects a graph into at least k components, and it models relevant applications in matrix decomposition for solving systems of equations by parallel computing. The capacitated k-Vertex Separator problem, instead, asks to find a subset of vertices of minimum cardinality the deletion of which disconnects a given graph in at most k shores and the size of each shore must not be larger than a given capacity value. Also this problem is of great importance for matrix decomposition algorithms.
This thesis also addresses the Chance-Constrained Mathematical Program that represents a significant example in which decomposition techniques can be successfully applied. This is a class of stochastic optimization problems in which the feasible region depends on the realization of a random variable and the solution must optimize a given objective function while belonging to the feasible region with a probability that must be above a given value. In this thesis, a decomposition approach for this problem is introduced.
The thesis also addresses the Fractional Knapsack Problem with Penalties, a variant of the knapsack problem in which items can be split at the expense of a penalty depending on the fractional quantity
Some contributions to mathematical programming and combinatorial optimization
Consider the optimization (i.e. maximization or minimization) of a real valued function f defined on a subset of the n-dimensional real vector space, subject to some set of constraints on the variables. The present document gathers and summarizes part of the work I could carry out on such optimization problems when the objective function is either linear or quadratic and when the constraints set on the variables are linear and/or integrality constraints.In the first part, methods are presented to deal with general families of mathematical programs: (mixed-integer) linear programs and unconstrained binary quadratic programs, namely. Chapter 1 is concerned with different types of cutting-plane methods for (mixed-integer) linear programs. In Chapter 2 we present a combinatorial algorithm to solve a particular family of unconstrained binary quadratic programs in polynomial time. General unconstrained quadratic binary programs are then considered in Chapter 3, for which we show how information related to the spectrum of the matrix defining the objective may be used to derive bounds on the optimal objective value.In the second part, we present our contributions w.r.t. several classical combinatorial optimization problems. They namely contain studies on descriptions of polytopes related to the maximum cut problem when the sizes of the shores are fixed (Chapter 4). Given an edge weighted graph, this problem consists in identifying a node subset S of prescribed size, such that the sum of the weights of the edges having exactly one endpoint in S is maximized. Another very classical problem in combinatorial optimization is the one which consists in identifying a minimum cardinality dominating set. Recall that given a graph, a dominating set is a node subset S such that each node not in S has at least one neighbor in S. Polyhedral investigations have been carried out w.r.t. the weighted version of the problem and for different generalizations of the domination concept (Chapter 5). A graph partitioning problem called the k-separator problem is then dealt with from different perspectives (complexity, approximations and polyhedral descriptions) in Chapter 6. Basically it consists in identifying a node subset in a graph whose removal induces a graph having connected components of bounded size. Finally, in Chapter 7, we present several features of an original orientation problem in graphs, including the identification of polynomial time solvable cases, several formulations and connections with the maximum cut problem