Fedor V. Fomin

An annotated bibliography on guaranteed graph searching

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.

Exact exponential algorithms

Discovering surprises in the face of intractability.

Parameterized and Exact Computation

We consider the problem of extracting a maximum-size reflect ed network in a linear program. This problem has been studied before and a st ate-of-the-art SGA heuristic with two variations have been proposed. In this paper we apply a new approach to evaluate the quality o f SGA. In particular, we solve majority of the instances in the testbe d to optimality using a new fixed-parameter algorithm, i.e., an algorithm whose run time is polynomial in the input size but exponential in terms of an additional para meter associated with the given problem. This analysis allows us to conclude that the the existing SGA heuristic, in fact, produces solutions of a very high quality and often reaches t optimal objective values. However, SGA contain two components which leave som e space for improvement: building of a spanning tree and searching for an i ndependent set in a graph. In the hope of obtaining even better heuristic, we tri ed to replace both of these components with some equivalent algorithms. We tried to use a fixed-parameter algorithm instead of a greed y one for searching of an independent set. But even the exact solution of this subproblem improved the whole heuristic insignificantly. Hence, the crucial par t of SGA is building of a spanning tree. We tried three different algorithms, and it a ppears that the DepthFirst search is clearly superior to the other ones in buildin g of the spanning tree for SGA. Thereby, by application of fixed-parameter algorithms, we m anaged to check that the existing SGA heuristic is of a high quality and selec ted the component which required an improvement. This allowed us to intensify the research in a proper direction which yielded a superior variation of SGA. This variation significantly improves the results of the basic SGA solving most of the instances in our experiments to optimality in a short time. A preliminary version of this paper will appear in the Procee dings of the 4th International Workshop on Parameterized and Exact Computation (IWPEC’09). Department of Computer Science, Royal Holloway, Universit y of London, Egham, Surrey TW20 0EX, England, UK,gutin@cs.rhul.ac.uk Department of Computer Science, Royal Holloway, Universit y of London, Egham, Surrey TW20 0EX, England, UK,daniel.karapetyan@gmail.com Department of Computer Science, University College Cork, I reland,i.razgon@cs.ucc.ieThe Workshop on Parameterized and Exact Computation (IWPEC) is an - ternational workshop series that covers research in all aspects of parameterized and exact algorithms and complexity, and especially encourages the study of parameterized and exact computations for real-world applications and algori- mic engineering. The goal of the workshop is to present recent research results, including signi?cant work-in-progress,and to identify and explore directions for future research. IWPEC2009wasthefourthworkshopintheseries,heldinCopenhagen,D- mark, during September 10-11, 2009. The workshop was part of ALGO 2009, which also hosted the 17th European Symposium on Algorithms (ESA 2009), the 9th Workshop on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2009), and the 7th Workshop on Appr- imation and Online Algorithms (WAOA 2009). Three previous meetings of the IWPEC series were held in Bergen, Norway, 2004, Zu rich, Switzerland, 2006, and Victoria, Canada, 2008. At IWPEC 2009, we had two plenary speakers,Noga Alon (Tel Aviv Univ- sity, Israel) and Hans Bodlaender (Utrecht University, The Netherlands), giving 50-minutetalkseach.ProfessorAlonspokeon ColorCoding,BalancedHashing andApproximateCounting, andProfessorBodlaenderon Kernelization:New Upper and Lower Bound Techniques. Their respective abstracts accompanying the talks are included in these proceedings. InresponsetotheCallforPapers,52papersweresubmitted.Eachsubmission was reviewed by at least three reviewers (most by at least four). The reviewers were either Program Committee members or invited external reviewers. The ProgramCommittee held electronic meetings using the EasyChair system, went throughthoroughdiscussions,andselected25ofthesubmissionsforpresentation at the workshop and inclusion in this LNCS volume

Subexponential parameterized algorithms on bounded-genus graphs and H -minor-free graphs

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.

A measure & conquer approach for the analysis of exact algorithms

For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call “Measure & Conquer””, as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis. In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples show that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

Efficient exact algorithms on planar graphs: exploiting sphere cut branch decompositions

Divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with new techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an

Measure and conquer: domination – a case study

O(2^{6.903\sqrt{n}}n^{3/2}+n^{3})

Improved algorithms for feedback vertex set problems

time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time

Fixed-parameter algorithms for ( k , r )-center in planar graphs and map graphs

O(2^{10.8224\sqrt{n}}n^{3/2}+n^{3})

Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up

. Our approach can be used to design parameterized algorithms as well. For example we introduce the first