From matchings to independent sets

In 1965, Jack Edmonds proposed his celebrated polynomial-time algorithm to find a maximum matching in a graph. It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard. What makes this problem easy in the class of line graphs and what other restrictions can lead to an efficient solution of the problem? In the present paper, we analyze these and related questions. We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations. Both techniques have been used in the solution of Edmonds to the maximum matching problem, i.e. in the solution to the maximum independent set problem in the class of line graphs. We survey various results that exploit these techniques beyond the line graphs

Constant-Factor Approximation Algorithms for the Parity-Constrained Facility Location Problem

Facility location is a prominent optimization problem that has inspired a large quantity of both theoretical and practical studies in combinatorial optimization. Although the problem has been investigated under various settings reflecting typical structures within the optimization problems of practical interest, little is known on how the problem behaves in conjunction with parity constraints. This shortfall of understanding was rather discouraging when we consider the central role of parity in the field of combinatorics. In this paper, we present the first constant-factor approximation algorithm for the facility location problem with parity constraints. We are given as the input a metric on a set of facilities and clients, the opening cost of each facility, and the parity requirement - odd, even, or unconstrained - of every facility in this problem. The objective is to open a subset of facilities and assign every client to an open facility so as to minimize the sum of the total opening costs and the assignment distances, but subject to the condition that the number of clients assigned to each open facility must have the same parity as its requirement. Although the unconstrained facility location problem as a relaxation for this parity-constrained generalization has unbounded gap, we demonstrate that it yields a structured solution whose parity violation can be corrected at small cost. This correction is prescribed by a T-join on an auxiliary graph constructed by the algorithm. This auxiliary graph does not satisfy the triangle inequality, but we show that a carefully chosen set of shortcutting operations leads to a cheap and sparse T-join. Finally, we bound the correction cost by exhibiting a combinatorial multi-step construction of an upper bound

Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-Shortest Induced Paths

For vertices $u$ and $v$ of an $n$-vertex graph $G$, a $uv$-trail of $G$ is an induced $uv$-path of $G$ that is not a shortest $uv$-path of $G$. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in $O(n^{18})$ time, to either output a $uv$-trail of $G$ or ensure that $G$ admits no $uv$-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of $n^2\times n^2$ Boolean matrices, leading to a largely improved $O(n^{4.75})$-time algorithm.Comment: 18 pages, 6 figures, a preliminary version appeared in STACS 202

Improved Algorithms for Recognizing Perfect Graphs and Finding Shortest Odd and Even Holes

Various classes of induced subgraphs are involved in the deepest results of graph theory and graph algorithms. A prominent example concerns the {\em perfection} of $G$ that the chromatic number of each induced subgraph $H$ of $G$ equals the clique number of $H$. The seminal Strong Perfect Graph Theorem confirms that the perfection of $G$ can be determined by detecting odd holes in $G$ and its complement. Chudnovsky et al. show in 2005 an $O(n^9)$ algorithm for recognizing perfect graphs, which can be implemented to run in $O(n^{6+\omega})$ time for the exponent $\omega<2.373$ of square-matrix multiplication. We show the following improved algorithms. 1. The tractability of detecting odd holes was open for decades until the major breakthrough of Chudnovsky et al. in 2020. Their $O(n^9)$ algorithm is later implemented by Lai et al. to run in $O(n^8)$ time, leading to the best formerly known algorithm for recognizing perfect graphs. Our first result is an $O(n^7)$ algorithm for detecting odd holes, implying an $O(n^7)$ algorithm for recognizing perfect graphs. 2. Chudnovsky et al. extend in 2021 the $O(n^9)$ algorithms for detecting odd holes (2020) and recognizing perfect graphs (2005) into the first polynomial algorithm for obtaining a shortest odd hole, which runs in $O(n^{14})$ time. We reduce the time for finding a shortest odd hole to $O(n^{13})$. 3. Conforti et al. show in 1997 the first polynomial algorithm for detecting even holes, running in about $O(n^{40})$ time. It then takes a line of intensive efforts in the literature to bring down the complexity to $O(n^{31})$, $O(n^{19})$, $O(n^{11})$, and finally $O(n^9)$. On the other hand, the tractability of finding a shortest even hole has been open for 16 years until the very recent $O(n^{31})$ algorithm of Cheong and Lu in 2022. We improve the time of finding a shortest even hole to $O(n^{23})$.Comment: 29 pages, 5 figure

Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms

An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both. Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results

Propriétés géométriques du nombre chromatique : polyèdres, structures et algorithmes

Computing the chromatic number and finding an optimal coloring of a perfect graph can be done efficiently, whereas it is an NP-hard problem in general. Furthermore, testing perfection can be carried- out in polynomial-time. Perfect graphs are characterized by a minimal structure of their sta- ble set polytope: the non-trivial facets are defined by clique-inequalities only. Conversely, does a similar facet-structure for the stable set polytope imply nice combinatorial and algorithmic properties of the graph ? A graph is h-perfect if its stable set polytope is completely de- scribed by non-negativity, clique and odd-circuit inequalities. Statements analogous to the results on perfection are far from being understood for h-perfection, and negative results are missing. For ex- ample, testing h-perfection and determining the chromatic number of an h-perfect graph are unsolved. Besides, no upper bound is known on the gap between the chromatic and clique numbers of an h-perfect graph. Our first main result states that the operations of t-minors keep h- perfection (this is a non-trivial extension of a result of Gerards and Shepherd on t-perfect graphs). We show that it also keeps the Integer Decomposition Property of the stable set polytope, and use this to answer a question of Shepherd on 3-colorable h-perfect graphs in the negative. The study of minimally h-imperfect graphs with respect to t-minors may yield a combinatorial co-NP characterization of h-perfection. We review the currently known examples of such graphs, study their stable set polytope and state several conjectures on their structure. On the other hand, we show that the (weighted) chromatic number of certain h-perfect graphs can be obtained efficiently by rounding-up its fractional relaxation. This is related to conjectures of Goldberg and Seymour on edge-colorings. Finally, we introduce a new parameter on the complexity of the matching polytope and use it to give an efficient and elementary al- gorithm for testing h-perfection in line-graphs.Le calcul du nombre chromatique et la détermination d'une colo- ration optimale des sommets d'un graphe sont des problèmes NP- difficiles en général. Ils peuvent cependant être résolus en temps po- lynomial dans les graphes parfaits. Par ailleurs, la perfection d'un graphe peut être décidée efficacement. Les graphes parfaits sont caractérisés par la structure de leur poly- tope des stables : les facettes non-triviales sont définies exclusivement par des inégalités de cliques. Réciproquement, une structure similaire des facettes du polytope des stables détermine-t-elle des propriétés combinatoires et algorithmiques intéressantes? Un graphe est h-parfait si les facettes non-triviales de son polytope des stables sont définies par des inégalités de cliques et de circuits impairs. On ne connaît que peu de résultats analogues au cas des graphes parfaits pour la h-perfection, et on ne sait pas si les problèmes sont NP-difficiles. Par exemple, les complexités algorithmiques de la re- connaissance des graphes h-parfaits et du calcul de leur nombre chro- matique sont toujours ouvertes. Par ailleurs, on ne dispose pas de borne sur la différence entre le nombre chromatique et la taille maxi- mum d'une clique d'un graphe h-parfait. Dans cette thèse, nous montrons tout d'abord que les opérations de t-mineurs conservent la h-perfection (ce qui fournit une extension non triviale d'un résultat de Gerards et Shepherd pour la t-perfection). De plus, nous prouvons qu'elles préservent la propriété de décompo- sition entière du polytope des stables. Nous utilisons ce résultat pour répondre négativement à une question de Shepherd sur les graphes h-parfaits 3-colorables. L'étude des graphes minimalement h-imparfaits (relativement aux t-mineurs) est liée à la recherche d'une caractérisation co-NP com- binatoire de la h-perfection. Nous faisons l'inventaire des exemples connus de tels graphes, donnons une description de leur polytope des stables et énonçons plusieurs conjectures à leur propos. D'autre part, nous montrons que le nombre chromatique (pondéré) de certains graphes h-parfaits peut être obtenu efficacement en ar- rondissant sa relaxation fractionnaire à l'entier supérieur. Ce résultat implique notamment un nouveau cas d'une conjecture de Goldberg et Seymour sur la coloration d'arêtes. Enfin, nous présentons un nouveau paramètre de graphe associé aux facettes du polytope des couplages et l'utilisons pour donner un algorithme simple et efficace de reconnaissance des graphes h- parfaits dans la classe des graphes adjoints

Path Parity and Perfection

Two non-adjacent vertices x and y in a graph G form an even pair if every induced path between them has an even number of edges. For a given pair fx; yg in a graph G, we denote by G xy the graph obtained from G by contracting x and y. In 1982, Fonlupt and Uhry proved that if G is perfect then so is G xy . In 1987, Meyniel used this fact to prove that no minimal imperfect graph contains an even pair. In the last eight years, even pairs have become an important tool for proving that certain classes of graphs are perfect and for designing optimization algorithms on special classes of perfect graphs. This paper surveys results of these types. It also discusses numerous related concepts including odd pairs

Even Pairs

Two non-adjacent vertices in a graph form an even pair if every chordless path between them has an even number of edges. The salient fact about even pairs is that contracting an even pair in a graph G yields a graph that has the same clique number and chromatic number as G. It follows that (a) the contraction of two vertices that form an even pair in a perfect graph produces a new perfect graph, and (b) no minimal imperfect graph can contain an even pair. Fact (a) can be exploited to devise simple combinatorial algorithms for coloring many perfect graphs. Fact (b) can be used to define large classes of perfect graphs. We will review the wealth of results that have appeared on these topics and discuss various related concepts such as odd pairs. This is an updated version of [H. Everett, C.M.H. de Figueiredo, C. Linhares-Sales, F. Maffray, O. Porto, B.A. Reed, Path Parity and Perfection, Discrete Mathematics, 165-166 (1997), 223-242.