69 research outputs found
Fine-grained Lower Bounds on Cops and Robbers
Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniel\u27s conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time.
In recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game
On the Cop Number of String Graphs
Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The cop number of a graph is the minimum number of cops required to capture the robber. We show that the cop number of a string graph is at most 13, improving upon a result of Gaven?iak et al. [Eur. J. of Comb. 72, 45-69 (2018)]. Using similar techniques, we also show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al. [Austr. J. Comb. 76(2), 248-265 (2020)]
Parameterized Analysis of the Cops and Robber Game
Pursuit-evasion games have been intensively studied for several decades due to their numerous applications in artificial intelligence, robot motion planning, database theory, distributed computing, and algorithmic theory. Cops and Robber (CnR) is one of the most well-known pursuit-evasion games played on graphs, where multiple cops pursue a single robber. The aim is to compute the cop number of a graph, k, which is the minimum number of cops that ensures the capture of the robber.
From the viewpoint of parameterized complexity, CnR is W[2]-hard parameterized by k [Fomin et al., TCS, 2010]. Thus, we study structural parameters of the input graph. We begin with the vertex cover number (vcn). First, we establish that k ? vcn/3+1. Second, we prove that CnR parameterized by vcn is FPT by designing an exponential kernel. We complement this result by showing that it is unlikely for CnR parameterized by vcn to admit a polynomial compression. We extend our exponential kernels to the parameters cluster vertex deletion number and deletion to stars number, and design a linear vertex kernel for neighborhood diversity. Additionally, we extend all of our results to several well-studied variations of CnR
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width
The authentic punk: an ethnography of DIY music ethics
This thesis examines how select participants came to be involved in DiY punk culture,
what they do in it, and how, if they do, they exit from the culture. Underpinning this
will be an ethnographic examination of how the ethics of punk informs their views of
remaining authentic and what they consider to be a sell out and betrayal of these
values. I illustrate how such ethics have evolved and how they inform the daily
practice of two chosen DiY punk communities in Leeds and Bradford. I show how
these communities reciprocally relate to each other. I ask such questions as what do
the participants get out of what is often experienced as hard work and toil, particularly
where it is fraught with a series of dilemmas bound up in politics, ethics, identity and
integrity. I offer a grounded theory of how and what ways those involved in DiY
punk authenticate themselves in their actions. This will demonstrate how and, more
importantly, why DiY punks distinguish their ethical version of punk over and above
what are taken as less favourable forms of punk. What happens if previous
passionately held DiY beliefs are surrendered? Severe consequences follow should a participant sell out. I present an account of these and suggest that what they involve
is not the clear-cut question that is sometimes assumed, either sincerely or selfrighteously
Dagstuhl News January - December 2011
"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic
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