Dimitrios M. Thilikos
National and Kapodistrian University of Athens
Network
Latest external collaboration on country level. Dive into details by clicking on the dots.
Publication
Featured researches published by Dimitrios M. Thilikos.
Theoretical Computer Science | 2008
Fedor V. Fomin; Dimitrios M. Thilikos
Graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers. In this annotated bibliography, we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field.
Journal of the ACM | 2005
Erik D. Demaine; Fedor V. Fomin; Mohammad Taghi Hajiaghayi; Dimitrios M. Thilikos
We introduce a new framework for designing fixed-parameter algorithms with subexponential running time---2O(&kradic;) nO(1). Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, disk dimension, and many others restricted to bounded-genus graphs (phrased as bipartite-graph problem). Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes, as special cases, all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, single-crossing-minor-free graphs, and/or map graphs; we extend these results to apply to bounded-genus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a bounded-genus graph that excludes some planar graph H as a minor. This bound depends linearly on the size ¦V(H)¦ of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graph-minors work of Robertson and Seymour.Building on these results, we develop subexponential fixed-parameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. In particular, this general category of graphs includes planar graphs, bounded-genus graphs, single-crossing-minor-free graphs, and any class of graphs that is closed under taking minors. Specifically, the running time is 2O(&kracic;) nh, where h is a constant depending only on H, which is polynomial for k = O(log2 n). We introduce a general approach for developing algorithms on H-minor-free graphs, based on structural results about H-minor-free graphs at the heart of Robertson and Seymours graph-minors work. We believe this approach opens the way to further development on problems in H-minor-free graphs.
ACM Transactions on Algorithms | 2005
Erik D. Demaine; Fedor V. Fomin; Mohammad Taghi Hajiaghayi; Dimitrios M. Thilikos
The <i>(<i>k</i>, <i>r</i>)-center problem</i> asks whether an input graph <i>G</i> has ≤<i>k</i> vertices (called <i>centers</i>) such that every vertex of <i>G</i> is within distance ≤<i>r</i> from some center. In this article, we prove that the (<i>k</i>, <i>r</i>)-center problem, parameterized by <i>k</i> and <i>R</i>, is fixed-parameter tractable (FPT) on planar graphs, i.e., it admits an algorithm of complexity <i>f</i>(<i>k</i>, <i>r</i>)<i>n</i><sup><i>O</i>(1)</sup> where the function <i>f</i> is independent of <i>n</i>. In particular, we show that <i>f</i>(<i>k,r</i>) = 2<sup><i>O</i>(<i>r</i> log <i>r</i>) &ksqrt;</sup>, where the exponent of the exponential term grows sublinearly in the number of centers. Moreover, we prove that the same type of FPT algorithms can be designed for the more general class of <i>map graphs</i> introduced by Chen, Grigni, and Papadimitriou. Our results combine dynamic-programming algorithms for graphs of small branchwidth and a graph-theoretic result bounding this parameter in terms of <i>k</i> and <i>r</i>. Finally, a byproduct of our algorithm is the existence of a PTAS for the <i>r</i>-domination problem in both planar graphs and map graphs.Our approach builds on the seminal results of Robertson and Seymour on Graph Minors, and as a result is much more powerful than the previous machinery of Alber et al. for exponential speedup on planar graphs. To demonstrate the versatility of our results, we show how our algorithms can be extended to general parameters that are “large” on grids. In addition, our use of branchwidth instead of the usual treewidth allows us to obtain much faster algorithms, and requires more complicated dynamic programming than the standard leaf/introduce/forget/join structure of nice tree decompositions. Our results are also unique in that they apply to classes of graphs that are not minor-closed, namely, constant powers of planar graphs and map graphs.
SIAM Journal on Computing | 2006
Fedor V. Fomin; Dimitrios M. Thilikos
Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance. The main purpose of this paper is to show how very deep min-max and duality theorems from Graph Minors can be used to obtain essential speed-up to many known algorithms on different domination problems.
Algorithmica | 2008
Michael R. Fellows; Christian Knauer; Naomi Nishimura; Prabhakar Ragde; Frances A. Rosamond; Ulrike Stege; Dimitrios M. Thilikos; Sue Whitesides
Abstract We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2O(k)), an improvement over previous algorithms for some of these problems running in time O(n+kO(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent.
workshop on graph theoretic concepts in computer science | 2003
Lali Barrière; Pierre Fraigniaud; Nicola Santoro; Dimitrios M. Thilikos
This paper is concerned with the graph searching game: we are given a graph containing a fugitive (or lost) entity or item; the goal is to clear the edges of the graph, using searchers; an edge is clear if it cannot contain the searched entity, contaminated otherwise. The search numbers(G) of a graph G is the smallest number of searchers required to “clear” G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed (i.e., searchers can not jump but only move along the edges). The difficulty of the “connected” version and of the “monotone internal” version of the graph searching problem comes from the fact that none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of graph searching. We prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, s(G) = is(G) = ms(G) ≤ mis(G) ≤ cs(G) = ics(G) ≤ mcs(G) = mics(G). The first two inequalities can be strict. Motivated by the fact that connected graph searching and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is exactly one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T) and cs(T) ≤ 2 s(T) − 2, using that ics(T)=mcs(T). This implies that there are only two different search numbers, and these search numbers differ by a factor of 2 at most.
international colloquium on automata languages and programming | 1997
Hans L. Bodlaender; Dimitrios M. Thilikos
Let G k be the class of graphs with branchwidth at most k. In this paper we prove that one can construct, for any k, a linear time algorithm that checks if a graph belongs to G k and, if so, outputs a branch decomposition of minimum width. Moreover, we find the obstruction set for G k and, for the same class, we give a safe and complete set of reduction rules. Our results lead to a practical linear time algorithm that checks if a graph has branchwidth ≤3 and, if so, outputs a branch decomposition of minimum width.
international colloquium on automata languages and programming | 2007
Frederic Dorn; Fedor V. Fomin; Dimitrios M. Thilikos
We present a series of techniques for the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch- (or tree-) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of sub-exponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2O(√k) ˙ nO(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter.
Journal of Discrete Algorithms | 2009
Hannes Moser; Dimitrios M. Thilikos
The r-Regular Induced Subgraph problem asks, given a graph G and a non-negative integer k, whether G contains an r-regular induced subgraph of size at least k, that is, an induced subgraph in which every vertex has degree exactly r. In this paper we examine its parameterization k-Sizer-Regular Induced Subgraph with k as parameter and prove that it is W[1]-hard. We also examine the parameterized complexity of the dual parameterized problem, namely, the k-Almostr-Regular Graph problem, which asks for a given graph G and a non-negative integer k whether G can be made r-regular by deleting at most k vertices. For this problem, we prove the existence of a problem kernel of size O(kr(r+k)^2).
Discrete Applied Mathematics | 1997
Hans L. Bodlaender; Dimitrios M. Thilikos
A graph G is K-chordal, if it does not contain chordless cycles of length larger than k. The chordality lc of a graph G is the minimum k for which G is k-chordal. The degeneracy or the width of a graph is the maximum min-degree of any of its subgraphs. Our results are the following: 1. (1) The problem of treewidth remains NP-complete when restricted to graphs with small maximum degree. 2. (2) An upper bound is given for the treewidth of a graph as a function of its maximum degree and chordality. A consequence of this result is that when maximum degree and chordality are fixed constants, then there is a linear algorithm for treewidth and a polynomial algorithm for pathwidth. 3. (3) For any constant s ⩾ 1, it is shown that any (s + 2)-chordal graph with bounded width contains an 12-separator of size O(n(s − 1)s), computable in O(n3 −(1s)) time. Our results extent the many applications of the separator theorems in [1, 32, 33] to the class of K-chordal graphs. Several natural classes of graphs have small chordality. Weakly chordal graphs and cocomparability graphs are 4-chordal. We investigate the complexity of treewidth and pathwidth on these classes when an additional degree restriction is used. We present an application of our separator theorem on approximating the maximum independent set on K-chordal graphs with small width.