New Integrality Gap Results for the Firefighters Problem on Trees

The firefighter problem is NP-hard and admits a $(1-1/e)$ approximation based on rounding the canonical LP. In this paper, we first show a matching integrality gap of $(1-1/e+\epsilon)$ on the canonical LP. This result relies on a powerful combinatorial gadget that can be used to prove integrality gap results for many problem settings. We also consider the canonical LP augmented with simple additional constraints (as suggested by Hartke). We provide several evidences that these constraints improve the integrality gap of the canonical LP: (i) Extreme points of the new LP are integral for some known tractable instances and (ii) A natural family of instances that are bad for the canonical LP admits an improved approximation algorithm via the new LP. We conclude by presenting a $5/6$ integrality gap instance for the new LP.Comment: 22 page

Approximation algorithms for network design and cut problems in bounded-treewidth

This thesis explores two optimization problems, the group Steiner tree and firefighter problems, which are known to be NP-hard even on trees. We study the approximability of these problems on trees and bounded-treewidth graphs. In the group Steiner tree, the input is a graph and sets of vertices called groups; the goal is to choose one representative from each group and connect all the representatives with minimum cost. We show an O(log^2 n)-approximation algorithm for bounded-treewidth graphs, matching the known lower bound for trees, and improving the best possible result using previous techniques. We also show improved approximation results for group Steiner forest, directed Steiner forest, and a fault-tolerant version of group Steiner tree. In the firefighter problem, we are given a graph and a vertex which is burning. At each time step, we can protect one vertex that is not burning; fire then spreads to all unprotected neighbors of burning vertices. The goal is to maximize the number of vertices that the fire does not reach. On trees, a classic (1-1/e)-approximation algorithm is known via LP rounding. We prove that the integrality gap of the LP matches this approximation, and show significant evidence that additional constraints may improve its integrality gap. On bounded-treewidth graphs, we show that it is NP-hard to find a subpolynomial approximation even on graphs of treewidth 5. We complement this result with an O(1)-approximation on outerplanar graphs.Diese Arbeit untersucht zwei Optimierungsprobleme, von welchen wir wissen, dass sie selbst in Bäumen NP-schwer sind. Wir analysieren Approximationen für diese Probleme in Bäumen und Graphen mit begrenzter Baumweite. Im Gruppensteinerbaumproblem, sind ein Graph und Mengen von Knoten (Gruppen) gegeben; das Ziel ist es, einen Knoten von jeder Gruppe mit minimalen Kosten zu verbinden. Wir beschreiben einen O(log^2 n)-Approximationsalgorithmus für Graphen mit beschränkter Baumweite, dies entspricht der zuvor bekannten unteren Schranke für Bäume und ist zudem eine Verbesserung über die bestmöglichen Resultate die auf anderen Techniken beruhen. Darüber hinaus zeigen wir verbesserte Approximationsresultate für andere Gruppensteinerprobleme. Im Feuerwehrproblem sind ein Graph zusammen mit einem brennenden Knoten gegeben. In jedem Zeitschritt können wir einen Knoten der noch nicht brennt auswählen und diesen vor dem Feuer beschützen. Das Feuer breitet sich anschließend zu allen Nachbarn aus. Das Ziel ist es die Anzahl der Knoten die vom Feuer unberührt bleiben zu maximieren. In Bäumen existiert ein lang bekannter (1-1/e)-Approximationsalgorithmus der auf LP Rundung basiert. Wir zeigen, dass die Ganzzahligkeitslücke des LP tatsächlich dieser Approximation entspricht, und dass weitere Einschränkungen die Ganzzahligkeitslücke möglicherweise verbessern könnten. Für Graphen mit beschränkter Baumweite zeigen wir, dass es NP-schwer ist, eine sub-polynomielle Approximation zu finden

Efficient algorithms on trees.

Yang, Lin.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 57-61).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Problems and Main Results --- p.2Chapter 1.1.1 --- Firefighting on Trees --- p.2Chapter 1.1.2 --- Maximum k-Vertex Cover on Trees --- p.3Chapter 1.2 --- Background --- p.3Chapter 1.2.1 --- Random Separation --- p.4Chapter 1.2.2 --- Kernelization --- p.5Chapter 1.2.3 --- Infeasibility of Polynomial Kernel --- p.6Chapter 1.3 --- Organization of the Thesis --- p.7Chapter 2 --- Firefighting on Trees --- p.9Chapter 2.1 --- Definitions and Notation --- p.10Chapter 2.2 --- FPT Algorithms --- p.13Chapter 2.2.1 --- Saving k Vertices --- p.14Chapter 2.2.2 --- Saving k Leaves --- p.19Chapter 2.2.3 --- Protecting k Vertices --- p.23Chapter 2.3 --- Approximation --- p.29Chapter 2.3.1 --- A (1 ´ؤ 1/e)-Approximation Algorithm --- p.29Chapter 2.3.2 --- LP-Repsecting Rounding cannot Do Better --- p.33Chapter 3 --- Maximum k-Vertex Cover on Trees --- p.38Chapter 3.1 --- Maximum k Vertex Cover on Trees --- p.39Chapter 3.2 --- k-MVC on Degree Bounded Graphs --- p.45Chapter 3.3 --- k-MVC on Degeneracy Bounded Graphs --- p.46Chapter 3.4 --- Extension to Maximum k Dominating Set --- p.47Chapter 4 --- Conclusion --- p.49Chapter 4.1 --- The Firefighter problem --- p.49Chapter 4.2 --- The Maximum k-Vertex Cover problem --- p.53Acknowledgement --- p.55Bibliography --- p.5

Cops, robbers and firefighters on graphs

This thesis focuses on the game of cops and robbers on graphs, which was introduced independently by Quilliot in 1978 and by Nowakowski and Winkler in 1983, and one of its variants, the firefighter problem. In the game of cops and robbers, the cops start by choosing their starting positions on vertices of a graph, then the robber chooses his starting point. Then, they move each in turn along the edges of the graph. The basic objective is to determine whether the cops have a strategy which allows them to catch the robber. Looped vertices allow the cops and the robber to pass their turn. The first chapter explores the effect of loops on the cop number and the capture time. It provides examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play. In the second chapter, we investigate the links between this game and algebraic topology. We extend the game of cops and robbers on graphs by considering the case where the cops chase the image of the robber by a graph homomorphism. We prove that the cop number associated with a graph homomorphism is a homotopic invariant. Homotopies between graph homomorphisms or homotopy equivalences between graphs allow us to compare their cop numbers and also their capture times. Finally, using homotopic invariants such as homology groups, we investigate structural properties of copwin graphs. Finally, in the third chapter, we explore the Firefighter problem, introduced by Hartnell in 1995, where a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. While focusing on the case of trees, we also consider a variant game called Fractional Firefighter in which the amount of protection allocated to a vertex lies between 0 and 1. While most of the work in this area deals with a constant amount of firefighters available at each turn, we consider three research questions which arise when including the sequence of firefighters as part of the instance. We first introduce an online version of both Firefighter and Fractional Firefighter, in which the number of firefighters available at each turn is revealed over time. We show that a greedy algorithm on finite trees is 1/2-competitive for both online versions, which generalises a result previously known for special cases of Firefighter. We also show that the optimal competitive ratio of online Firefighter ranges between 1/2 and the inverse of the golden ratio. Next, given two firefighter sequences, we discuss sufficient conditions for the existence of an infinite tree that separates them, in the sense that the fire can be contained with one sequence but not with the other. To this aim, we study a new purely numerical game called targeting game. Finally, we give sufficient conditions for the fire to be contained on infinite trees, expressed as the asymptotic comparison of the number of firefighters and the size of the tree levels

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Legislative hope and utopia

Legislation, inspired by utopian ideals of rationality and justice, has long been experienced as a source of hope but frequently fails in meeting expectations. Current legislation is mainly focused on realising short-term policy goals without offering hope. This contribution aims to investigate how legislation relates to hope and what role utopian thought plays with respect to that and evaluates current legislative policy on its hope-inspiring properties. To this end, this contribution analyses the features of utopias and investigates how these find expression in legislation. It then evaluates to what extent utopianism may inspire hope or may lead to disappointment or even despair. Conceptions of time, knowledge and identity seem of relevance, connected to substantive reasoning and the constitutive function of legislation. The author's contention is that legislative hope hinges on a balance between effectiveness, room for substantive reasoning and the quality of the political aspect of the legislative process