Bilevel Clique Interdiction and Related Problems

I introduce a formulation of the bilevel clique interdiction problem. Interdiction, a military term, describes the removal of enemy resources. The single level clique interdiction problem describes the attempt of an attacker to interdict a maximum number of cliques. The bilevel form of the problem introduces a defender who attempts to minimize the number of cliques interdicted by the attacker. An algorithm and formulation for the bilevel clique interdiction problem has not previously been investigated. I start by introducing a formulation and a column-generation algorithm to solve the problem of bilevel interdiction of a minimum clique transversal and move forward to the creation of a delayed row-and-column generation algorithm for bilevel clique interdiction. Next, I introduce a formulation and algorithm to solve the bilevel interdiction of a maximum stable set problem. Bilevel interdiction of a maximum stable set is choosing a maximum stable set, but with a defender who is attempting to minimize the maximum stable set that can be chosen by the interdictor. I introduce a deterministic formulation and a delayed column generation algorithm. Additionally, I introduce a stochastic formulation of the problem. I solve this problem using a cross-decomposition method that involves L-shaped cuts into a master problem as well as new ``clique" cuts for the inner problem. Lastly, I define new classes of valid inequalities and facets for the clique transversal polytope. The valid inequalities come from two graph structures who have a closed form for their vertex cover number, which we use as a specific case for finding a minimum clique transversal. The first class of facets are just the maximal clique constraints of the clique transversal polytope. The next class contains an odd hole with distinct cliques on each edge of the hole. Another similar class contains an odd clique with distinct maximal cliques on the edges of one of its spanning cycles. The fourth class contains a clique with distinct maximal cliques on every edge of the initial clique, while the last class is a prism graph with distinct maximal cliques on every edge of the prism

Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

A bipartite graph $G=(U,V,E)$ is convex if the vertices in $V$ can be linearly ordered such that for each vertex $u\in U$, the neighbors of $u$ are consecutive in the ordering of $V$. An induced matching $H$ of $G$ is a matching such that no edge of $E$ connects endpoints of two different edges of $H$. We show that in a convex bipartite graph with $n$ vertices and $m$ weighted edges, an induced matching of maximum total weight can be computed in $O(n+m)$ time. An unweighted convex bipartite graph has a representation of size $O(n)$ that records for each vertex $u\in U$ the first and last neighbor in the ordering of $V$. Given such a compact representation, we compute an induced matching of maximum cardinality in $O(n)$ time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in $O(n)$ time. If no compact representation is given, the cover can be computed in $O(n+m)$ time. All of our algorithms achieve optimal running time for the respective problem and model. Previous algorithms considered only the unweighted case, and the best algorithm for computing a maximum-cardinality induced matching or a minimum chain cover in a convex bipartite graph had a running time of $O(n^2)$

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Counting patterns in strings and graphs

We study problems related to finding and counting patterns in strings and graphs. In the string-regime, we are interested in counting how many substring of a text are at Hamming (or edit) distance at most to a pattern . Among others, we are interested in the fully-compressed setting, where both and are given in a compressed representation. For both distance measures, we give the first algorithm that runs in (almost) linear time in the size of the compressed representations. We obtain the algorithms by new and tight structural insights into the solution structure of the problems. In the graph-regime, we study problems related to counting homomorphisms between graphs. In particular, we study the parameterized complexity of the problem #IndSub(), where we are to count all -vertex induced subgraphs of a graph that satisfy the property . Based on a theory of Lovász, Curticapean et al., we express #IndSub() as a linear combination of graph homomorphism numbers to obtain #W[1]-hardness and almost tight conditional lower bounds for properties that are monotone or that depend only on the number of edges of a graph. Thereby, we prove a conjecture by Jerrum and Meeks. In addition, we investigate the parameterized complexity of the problem #Hom(ℋ → ) for graph classes ℋ and . In particular, we show that for any problem in the class #W[1], there are classes ℋ_ and _ such that is equivalent to #Hom(ℋ_ → _ ).Wir untersuchen Probleme im Zusammenhang mit dem Finden und Zählen von Mustern in Strings und Graphen. Im Stringbereich ist die Aufgabe, alle Teilstrings eines Strings zu bestimmen, die eine Hamming- (oder Editier-)Distanz von höchstens zu einem Pattern haben. Unter anderem sind wir am voll-komprimierten Setting interessiert, in dem sowohl , als auch in komprimierter Form gegeben sind. Für beide Abstandsbegriffe entwickeln wir die ersten Algorithmen mit einer (fast) linearen Laufzeit in der Größe der komprimierten Darstellungen. Die Algorithmen nutzen neue strukturelle Einsichten in die Lösungsstruktur der Probleme. Im Graphenbereich betrachten wir Probleme im Zusammenhang mit dem Zählen von Homomorphismen zwischen Graphen. Im Besonderen betrachten wir das Problem #IndSub(), bei dem alle induzierten Subgraphen mit Knoten zu zählen sind, die die Eigenschaft haben. Basierend auf einer Theorie von Lovász, Curticapean, Dell, and Marx drücken wir #IndSub() als Linearkombination von Homomorphismen-Zahlen aus um #W[1]-Härte und fast scharfe konditionale untere Laufzeitschranken zu erhalten für , die monoton sind oder nur auf der Kantenanzahl der Graphen basieren. Somit beweisen wir eine Vermutung von Jerrum and Meeks. Weiterhin beschäftigen wir uns mit der Komplexität des Problems #Hom(ℋ → ) für Graphklassen ℋ und . Im Besonderen zeigen wir, dass es für jedes Problem in #W[1] Graphklassen ℋ_ und _ gibt, sodass äquivalent zu #Hom(ℋ_ → _ ) ist