The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Applications of Powerset Operators, Especially to Matroids

Let $\mathcal{V}$ denote a vector space over an arbitrary field with an inner product. For any collection $\mathcal{S}$ of vectors from $\mathcal{V}$ the collection of all vectors orthogonal to each vector in $\mathcal{S}$ is a subspace, denoted as $\mathcal{S}^{\perp_v}$ and called the \textit{orthogonal complement} of $\mathcal{S}$. One of the fundamental theorems of vector space theory states that, $(\mathcal{S}^{\perp_v})^{\perp_v}$ is the subspace \textit{spanned} by $\mathcal{S}$. Thus the ``spanning\u27\u27 operator on the subsets of a vector space is the square of the ``orthogonal complement\u27\u27 operator. In matroid theory, the orthogonal complement of a matroid $M$ is also well-defined and similarly results in another matroid. Although this new matroid is more commonly referred to as the `dual matroid\u27, denoted as $M^*$, and typically formed using a very different approach. There is an interesting relation between the circuits of a matroid $M$ and the cocircuits of $M$ (the circuits of its dual matroid $M^*$) which aligns much more closely to the orthogonal complement of a vector space. We expand on this relation to define a powerset operator: $(\phantom{S})^*$. Given $\mathcal{S} \subseteq \mathcal{P}(E)$, we denote $\mathcal{S}^*$ to be the minimal sets of $\{X \subseteq E \colon X \text{ is nonempty, } |X \cap A| \neq 1 \text{ for each } A \in \mathcal{S}\}$. We call this powerset operator the \textbf{circuit duality operator}. Unlike the vector space orthogonal complement operator, this circuit duality operator may not behave as nicely when applied to collections that do not correspond to a matroid. This thesis is an investigation into the development of tools and additional operators to help understand the collections of sets that result in a matroid under one or more applications of the circuit duality operator

Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments

Polyhedral techniques in combinatorial optimization

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience