Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles

Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed

Topics in Graph Algorithms: Structural Results and Algorithmic Techniques, with Applications

Coping with computational intractability has inspired the development of a variety of algorithmic techniques. The main challenge has usually been the design of polynomial time algorithms for NP-complete problems in a way that guarantees some, often worst-case, satisfactory performance when compared to exact (optimal) solutions. We mainly study some emergent techniques that help to bridge the gap between computational intractability and practicality. We present results that lead to better exact and approximation algorithms and better implementations. The problems considered in this dissertation share much in common structurally, and have applications in several scientific domains, including circuit design, network reliability, and bioinformatics. We begin by considering the relationship between graph coloring and the immersion order, a well-quasi-order defined on the set of finite graphs. We establish several (structural) results and discuss their potential algorithmic consequences. We discuss graph metrics such as treewidth and pathwidth. Treewidth is well studied, mainly because many problems that are NP-hard in general have polynomial time algorithms when restricted to graphs of bounded treewidth. Pathwidth has many applications ranging from circuit layout to natural language processing. We present a linear time algorithm to approximate the pathwidth of planar graphs that have a fixed disk dimension. We consider the face cover problem, which has potential applications in facilities location and logistics. Being fixed-parameter tractable, we develop an algorithm that solves it in time O(5k + n2) where k is the input parameter. This is a notable improvement over the previous best known algorithm, which runs in O(8kn). In addition to the structural and algorithmic results, this text tries to illustrate the practicality of fixed-parameter algorithms. This is achieved by implementing some algorithms for the vertex cover problem, and conducting experiments on real data sets. Our experiments advocate the viewpoint that, for many practical purposes, exact solutions of some NP-complete problems are affordable

Graph Pricing Problem on Bounded Treewidth, Bounded Genus and k-partite graphs

Consider the following problem. A seller has infinite copies of $n$ products represented by nodes in a graph. There are $m$ consumers, each has a budget and wants to buy two products. Consumers are represented by weighted edges. Given the prices of products, each consumer will buy both products she wants, at the given price, if she can afford to. Our objective is to help the seller price the products to maximize her profit. This problem is called {\em graph vertex pricing} ({\sf GVP}) problem and has resisted several recent attempts despite its current simple solution. This motivates the study of this problem on special classes of graphs. In this paper, we study this problem on a large class of graphs such as graphs with bounded treewidth, bounded genus and $k$-partite graphs. We show that there exists an {\sf FPTAS} for {\sf GVP} on graphs with bounded treewidth. This result is also extended to an {\sf FPTAS} for the more general {\em single-minded pricing} problem. On bounded genus graphs we present a {\sf PTAS} and show that {\sf GVP} is {\sf NP}-hard even on planar graphs. We study the Sherali-Adams hierarchy applied to a natural Integer Program formulation that $(1+\epsilon)$-approximates the optimal solution of {\sf GVP}. Sherali-Adams hierarchy has gained much interest recently as a possible approach to develop new approximation algorithms. We show that, when the input graph has bounded treewidth or bounded genus, applying a constant number of rounds of Sherali-Adams hierarchy makes the integrality gap of this natural {\sf LP} arbitrarily small, thus giving a $(1+\epsilon)$-approximate solution to the original {\sf GVP} instance. On $k$-partite graphs, we present a constant-factor approximation algorithm. We further improve the approximation factors for paths, cycles and graphs with degree at most three.Comment: Preprint of the paper to appear in Chicago Journal of Theoretical Computer Scienc

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

On Reconfiguration Problems: Structure and Tractability

Given an n-vertex graph G and two vertices s and t in G, determining whether there exists a path and computing the length of the shortest path between s and t are two of the most fundamental graph problems. In the classical battle of P versus NP or ``easy'' versus ``hard'', both of these problems are on the easy side. That is, they can be solved in poly(n) time, where poly is any polynomial function. But what if our input consisted of a 2^n-vertex graph? Of course, we can no longer assume G to be part of the input, as reading the input alone requires more than poly(n) time. Instead, we are given an oracle encoded using poly(n) bits and that can, in constant or poly(n) time, answer queries of the form ``is u a vertex in G'' or ``is there an edge between u and v?''. Given such an oracle and two vertices of the 2^n-vertex graph, can we still determine if there is a path or compute the length of the shortest path between s and t in poly(n) time? A slightly different, but equally insightful, formulation of the question above is as follows. Given a set S of n objects, consider the graph R(S) which contains one vertex for each set in the power set of S, 2^S, and two vertices are adjacent in R(S) whenever the size of their symmetric difference is equal to one. Clearly, this graph contains 2^n vertices and can be easily encoded in poly(n) bits using the oracle described above. It is not hard to see that there exists a path between any two vertices of R(S). Moreover, computing the length of a shortest path can be accomplished in constant time; it is equal to the size of the symmetric difference of the two underlying sets. If the vertex set of R(S) were instead restricted to a subset of 2^S, both of our problems can become NP-complete or even PSPACE-complete. Therefore, another interesting question is whether we can determine what types of ``restriction'' on the vertex set of R(S) induce such variations in the complexity of the two problems. These two seemingly artificial questions are in fact quite natural and appear in many practical and theoretical problems. In particular, these are exactly the types of questions asked under the reconfiguration framework, the main subject of this thesis. Under the reconfiguration framework, instead of finding a feasible solution to some instance I of a search problem Q, we are interested in structural and algorithmic questions related to the solution space of Q. Naturally, given some adjacency relation A defined over feasible solutions of Q, size of the symmetric difference being one such relation, the solution space can be represented using a graph R_Q(I). R_Q(I) contains one vertex for each feasible solution of Q on instance I and two vertices share an edge whenever their corresponding solutions are adjacent under A. An edge in R_Q(I) corresponds to a reconfiguration step, a walk in R_Q(I) is a sequence of such steps, a reconfiguration sequence, and R_Q(I) is a reconfiguration graph. Studying problems related to reconfiguration graphs has received considerable attention in recent literature, the most popular problem being to determine whether there exists a reconfiguration sequence between two given feasible solutions; for most NP-complete problems, this problem has been shown to be PSPACE-complete. The purpose of our work is to embark on a systematic investigation of the tractability and structural properties of such problems under both classical and parameterized complexity assumptions. Parameterized complexity is another framework which has become an essential tool for researchers in computational complexity during the last two decades or so and one of its main goals is to provide a better explanation of why some hard problems (in a classical sense) can be in fact much easier than others. Hence, we are interested in what separates the tractable instances from the intractable ones and the fixed-parameter tractable instances from the fixed-parameter intractable ones. It is clear from the generic definition of reconfiguration problems that several factors affect their complexity status. Our work aims at providing a finer classification of the complexity of reconfiguration problems with respect to some of these factors, including the definition of the adjacency relation A, structural properties of the input instance I, structural properties of the reconfiguration graph, and the length of a reconfiguration sequence. As most of these factors can be numerically quantified, we believe that the investigation of reconfiguration problems under both parameterized and classical complexity assumptions will help us further understand the boundaries between tractability and intractability. We consider reconfiguration problems related to Satisfiability, Coloring, Dominating Set, Vertex Cover, Independent Set, Feedback Vertex Set, and Odd Cycle Transversal, and provide lower bounds, polynomial-time algorithms, and fixed-parameter tractable algorithms. In doing so, we answer some of the questions left open in recent work and push the known boundaries between tractable and intractable even closer. As a byproduct of our initiating work on parameterized reconfiguration problems, we present a generic adaptation of parameterized complexity techniques which we believe can be used as a starting point for studying almost any such problem

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

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum