It is well known that many of the optimization problems which arise in applications are <I>hard</I>, which usually means that they are NP-hard. Hence much research has been devoted to finding <I>good</I> relaxations for these hard problems. Usually a <I>good</I> relaxation is one which can be solved (either exactly or within a prescribed numerical tolerance) in polynomial-time. Nesterov and Nemirovskii showed that by this criterion, many convex optimization problems are good relaxations. This thesis presents new convex relaxations for two such hard problems, namely the Maximum-Cut (Max-Cut) problem and the VLSI (Very Large Scale Integration of electronic circuits) layout problem. We derive and study the properties of two new strengthened semidefinite programming relaxations for the Max-Cut problem. Our theoretical results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. The first relaxation provides a strengthening of the well-known Goemans-Williamson relaxation, and the second relaxation is a further tightening of the first. We prove that the tighter relaxation automatically enforces the well-known triangle inequalities, and in fact is stronger than the simple addition of these inequalities to the Goemans-Williamson relaxation. We further prove that the tighter relaxation fully characterizes some low dimensional faces of the cut polytope via the rank of its feasible matrices. We also address some practical issues arising in the solution of these relaxations and present numerical results showing the remarkably good bounds computed by the tighter relaxation. For the VLSI layout problem, we derive a new relaxation by extending the <I>target distance</I> concept recently introduced by Etawil and Vannelli. The resulting AR (Attractor-Repeller) model improves on the NLT (Non-linear optimization Layout Technique) method of van Camp et al. by efficiently finding a good initial point for the NLT solver. Because the AR model is not a convex optimization problem, we isolate the source of the non-convexity and thereby derive a convex version of the model. This derivation leads to the definition of certain generalized target distances with an interesting practical interpretation. Finally, we present numerical results that demonstrate the potential of the AR model.

It is well known that many of the optimization problems which arise in applications are <I>hard</I>, which usually means that they are NP-hard. Hence much research has been devoted to finding <I>good</I> relaxations for these hard problems. Usually a <I>good</I> relaxation is one which can be solved (either exactly or within a prescribed numerical tolerance) in polynomial-time. Nesterov and Nemirovskii showed that by this criterion, many convex optimization problems are good relaxations. This thesis presents new convex relaxations for two such hard problems, namely the Maximum-Cut (Max-Cut) problem and the VLSI (Very Large Scale Integration of electronic circuits) layout problem. We derive and study the properties of two new strengthened semidefinite programming relaxations for the Max-Cut problem. Our theoretical results hold for every instance of Max-Cut; in particular, we make no assumptions about the edge weights. The first relaxation provides a strengthening of the well-known Goemans-Williamson relaxation, and the second relaxation is a further tightening of the first. We prove that the tighter relaxation automatically enforces the well-known triangle inequalities, and in fact is stronger than the simple addition of these inequalities to the Goemans-Williamson relaxation. We further prove that the tighter relaxation fully characterizes some low dimensional faces of the cut polytope via the rank of its feasible matrices. We also address some practical issues arising in the solution of these relaxations and present numerical results showing the remarkably good bounds computed by the tighter relaxation. For the VLSI layout problem, we derive a new relaxation by extending the <I>target distance</I> concept recently introduced by Etawil and Vannelli. The resulting AR (Attractor-Repeller) model improves on the NLT (Non-linear optimization Layout Technique) method of van Camp et al. by efficiently finding a good initial point for the NLT solver. Because the AR model is not a convex optimization problem, we isolate the source of the non-convexity and thereby derive a convex version of the model. This derivation leads to the definition of certain generalized target distances with an interesting practical interpretation. Finally, we present numerical results that demonstrate the potential of the AR model.

Cut problems on graphs are a well-known and intensively studied class of optimization problems. In this thesis, we study the maximum cut problem, the maximum bond problem, and the minimum multicut problem through their associated polyhedra, i.e., the cut polytope, the bond polytope, and the multicut dominant, respectively. Continuing the research on the maximum cut problem and the cut polytope, we present a linear description for cut polytopes of K_{3,3}-minor-free graphs as well as an algorithm solving the maximum cut problem on these graphs in the same running time as planar maximum cut. Moreover, we give a complete characterization of simple and simplicial cut polytopes. Considering the maximum cut problem from an algorithmic point of view, we propose an FPT-algorithm for the maximum cut problem parameterized by the crossing number. We start the structural study of the bond polytope by investigating its basic properties and the effect of graph operations on the bond polytope and its facet-defining inequalities. After presenting a linear-time reduction of the maximum bond problem to the maximum bond problem on 3-connected graphs, we discuss valid and facet defining inequalities arising from edges and cycles. These inequalities yield a complete linear description for bond polytopes of 3-connected planar (K_5-e)-minor-free graphs. This polytopal result is mirrored algorithmically by proposing a linear-time algorithm for the maximum bond problem on (K_5-e)-minor-free graphs. Finally, we start the structural study of the multicut dominant. We investigate basic properties, which gives rise to lifting and projection results for the multicut dominant. Then, we study the effect of graph operations on the multicut dominant and its facet-defining inequalities. Moreover, we present facet-defining inequalities supported on stars, trees, and cycles as well as separation algorithms for the former two classes of inequalities.

In this work we study two random procedures in cubic graphs with large girth. The first procedure finds a probability distribution on edge-cuts such that each edge is in a randomly chosen cut with probability at least 0.88672. As corollaries, we derive lower bounds for the size of maximum cut in cubic graphs with large girth and in random cubic graphs, and also an upper bound for the fractional cut covering number in cubic graphs with large girth. The second procedure finds a probability distribution on independent sets such that each vertex is in an independent set with probability at least 0.4352. This implies lower bounds for the size of maximum independent set in cubic graphs with large girth and in random cubic graphs, as well as an upper bound for the fractional chromatic number in cubic graphs with large girth.

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs-a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs. The original bubble model was used by Boyaci, Ekim, and Shalom for proving the polynomiality of the MaxCut problem on unit interval graphs. However, we found a significant mistake in the proof which seems to be hardly repairable. Moreover, we demonstrate advantages of such model by providing a subexponential-time algorithm solving the MaxCut problem on mixed unit interval graphs using our extended version of the bubble model. In addition, it gives us a polynomial-time...

Le déséquilibre d'un sommet dans un graphe orienté est la valeur absolue de la différence entre son degré sortant et son degré entrant. Nous étudions le problème de trouver une orientation des arêtes du graphe telle que l'image du vecteur dont les composantes sont les déséquilibres des sommets par une fonction objectif f est maximisée. Le premier cas considéré est le problème de maximiser le minimum des déséquilibres sur toutes les orientations possibles. Nous caractérisons les graphes dont la valeur objective optimale est nulle. Ensuite nous donnons plusieurs résultats concernant la complexité du problème. Enfin, nous introduisons différentes formulations du problème et présentons quelques résultats numériques. Par la suite, nous montrons que le cas f=1/2 | |·| |₁ mène au célèbre problème de la coupe de cardinalité maximale. Nous introduisons de nouvelles formulations ainsi qu'un nouveau majorant qui domine celui de Goemans et Williamson. Des résultats théoriques et numériques concernant la performance des approches sont présentés. Pour finir, dans le but de renforcer certaines des formulations des problèmes étudiés, nous étudions une famille de polyèdres spécifique consistant en l'enveloppe convexe des matrices d'affectation 0/1 (où chaque colonne contient exactement une composante égale à 1) annexée avec l'indice de leur ligne non-identiquement nulle la plus basse. Nous donnons une description complète de ce polytope ainsi que certaines de ses variantes qui apparaissent naturellement dans le contexte de divers problèmes d'optimisation combinatoire. Nous montrons également que résoudre un programme linéaire sur un tel polytope peut s'effectuer en temps polynomial / The imbalance of a vertex in a directed graph is the absolute value of the difference between its outdegree and indegree. In this thesis we study the problem of orienting the edges of a graph in such a way that the image of the vector which components are the imbalances of the vertices of the graph under an objective function f is maximized. The first case considered is the problem of maximizing the minimum imbalance of all the vertices over all the possible orientations of the input graph. We first characterize graphs for which the optimal objective value is zero. Next we give several results concerning the computational complexity of the problem. Finally, we deal with several mixed integer programming formulations for this problem and present some numerical experiments. Next, we show that the case for f=1/2 | |·| |₁ leads to the famous unweighted maximum cut problem. We introduce some new formulations along with a new bound shown to be tighter than Michel Goemans & David Williamson's. Theoretical and computational results regarding bounds quality and performance are also reported. Finally, in order to strengthen some formulations of the studied problems, we study a specific class of polytopes. Consider the polytope consisting in the convex hull of the 0/1 assignment matrices where each column contains exactly one coefficient equal to 1 appended with their index of the lowest row that is not identically equal to the zero row. We give a full description of this polytope and some of its variants which naturally appear in the context of several combinatorial optimization problems. We also show that linear optimization over those polytopes can be done in polynomial time