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Model checking existential logic on partially ordered sets
 In LICS’14
, 2014
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Characterisations of Nowhere Dense Graphs
, 2013
"... Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) ..."
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Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) model theory. Therefore, the concept of nowhere density seems to capture a natural property of graph classes generalising for example classes of graphs which exclude a fixed minor, have bounded degree or bounded local treewidth. In this paper we give a selfcontained introduction to the concept of nowhere dense classes of graphs focussing on the various ways in which they can be characterised. We also briefly sketch algorithmic applications these characterisations have found in the literature.
Complexity of counting subgraphs: Only the boundedness of the vertexcover number counts
"... Abstract—For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertexcover number (equivalently, the size of the maximum matching in C is bounded), the ..."
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Abstract—For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertexcover number (equivalently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomialtime solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertexcover number, then #Sub(C) is #W[1]hard parameterized by the size of H and hence not polynomialtime solvable and not even fixedparameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting kmatchings in bipartite graphs is #W[1]hard. Recently, Curticapean [ICALP 2013] proved the #W[1]hardness of counting kmatchings in general graphs; our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hypothesis (ETH), there is no f(k)no(k / log k) time algorithm for counting kmatchings in bipartite graphs for any computable function f. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]hard, as well as a similar almosttight ETHbased lower bound on the exponent. I.
Approximation Algorithms for PolynomialExpansion and LowDensity Graphs∗
, 2015
"... We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ε)approximation algorith ..."
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We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient (1 + ε)approximation algorithms for polynomial expansion graphs, for Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTAS’s for these problems are known for subclasses of this graph family. These results have immediate interesting applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow). We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth. 1.
FO Model Checking on Posets of Bounded Width
"... Over the past two decades the main focus of research into firstorder (FO) model checking algorithms have been sparse relational structures—culminating in the FPTalgorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs [STOC’14], with dense structures start ..."
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Over the past two decades the main focus of research into firstorder (FO) model checking algorithms have been sparse relational structures—culminating in the FPTalgorithm by Grohe, Kreutzer and Siebertz for FO model checking of nowhere dense classes of graphs [STOC’14], with dense structures starting to attract attention only recently. Bova, Ganian and Szeider [LICS’14] initiated the study of the complexity of FO model checking on partially ordered sets (posets). Bova, Ganian and Szeider showed that model checking existential FO logic is fixedparameter tractable (FPT) on posets of bounded width, where the width of a poset is the size of the largest antichain in the poset. The existence of an FPT algorithm for general FO model checking on posets of bounded width, however, remained open. We resolve this question in the positive by giving an algorithm that takes as its input an nelement poset P of width w and an FO logic formula ϕ, and determines whether ϕ holds on P in time f(ϕ,w) · n2. 1
Reconfiguration on sparse graphs
"... Abstract. A vertexsubset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertexsubset problem asks, given two feasible solutions Ss and St of size k, whether it is possible to transform Ss into St by a sequence of verte ..."
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Abstract. A vertexsubset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertexsubset problem asks, given two feasible solutions Ss and St of size k, whether it is possible to transform Ss into St by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertexsubset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACEcomplete on graphs of bounded bandwidth and W[1]hard parameterized by k on general graphs. We show that ISR is fixedparameter tractable parameterized by k when the input graph is of bounded degeneracy or nowheredense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixedparameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixedparameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowheredense graphs. 1
Enumerating Answers to FirstOrder Queries over Databases of Low Degree
"... A class of relational databases has low degree if for all δ, all but finitely many databases in the class have degree at most nδ, where n is the size of the database. Typical examples are databases of bounded degree or of degree bounded by logn. It is known that over a class of databases having low ..."
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A class of relational databases has low degree if for all δ, all but finitely many databases in the class have degree at most nδ, where n is the size of the database. Typical examples are databases of bounded degree or of degree bounded by logn. It is known that over a class of databases having low degree, firstorder boolean queries can be checked in pseudolinear time, i.e. in time bounded by n1+ε, for all ε. We generalise this result by considering query evaluation. We show that counting the number of answers to a query can be done in pseudolinear time and that enumerating the answers to a query can be done with constant delay after a pseudolinear time preprocessing.
1Distributed Alarming in the OnDuty and OffDuty Models
"... Abstract—Decentralized monitoring and alarming systems can be an attractive alternative to centralized architectures. Distributed sensor nodes (e.g., in the smart grid’s distribution network) are closer to an observed event than a global and remote observer or controller. This improves the visibilit ..."
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Abstract—Decentralized monitoring and alarming systems can be an attractive alternative to centralized architectures. Distributed sensor nodes (e.g., in the smart grid’s distribution network) are closer to an observed event than a global and remote observer or controller. This improves the visibility and response time of the system. Moreover, in a distributed system, local problems may also be handled locally and without overloading the communication network. This article studies alarming from a distributed computing perspective and for two fundamentally different scenarios: onduty and offduty. We model the alarming system as a sensor network consisting of a set of distributed nodes performing local measurements to sense events. In order to avoid false alarms, the sensor nodes cooperate and only escalate an event (i.e., raise an alarm) if the number of sensor nodes sensing an event exceeds a certain threshold. In the onduty scenario, nodes not affected by the event can actively help in the communication process, while in the offduty scenario nonevent nodes are inactive. This article presents and analyzes algorithms that minimize the reaction time of the monitoring system while avoiding unnecessary message transmissions. We investigate time and message complexity tradeoffs in different settings, and also shed light on the optimality of our algorithms by deriving cost lower bounds for distributed alarming systems. I.
Kernelization and Sparseness: the case of Dominating Set∗
, 2014
"... The search for linear kernels for the Dominating Set problem on classes of graphs of a topological nature has been one of the leading trends in kernelization in recent years. Following the fundamental work of Alber et al. [2] that established a linear kernel for the problem on planar graphs, linear ..."
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The search for linear kernels for the Dominating Set problem on classes of graphs of a topological nature has been one of the leading trends in kernelization in recent years. Following the fundamental work of Alber et al. [2] that established a linear kernel for the problem on planar graphs, linear kernels have been given for boundedgenus graphs [4], apexminorfree graphs [15], Hminorfree graphs [16], and Htopologicalminorfree graphs [17]. These generalizations are based on bidimensionality and powerful decomposition theorems for Hminorfree graphs and Htopologicalminorfree graphs of Robertson and Seymour [28] and of Grohe and Marx [22]. In this work we investigate a new approach to kernelization algorithms for Dominating Set on sparse graph classes. The approach is based on the theory of bounded expansion and nowhere dense graph classes, developed in the recent years by Nešetřil and Ossona de Mendez, among others. More precisely, we prove that Dominating Set admits a linear kernel on any hereditary graph class of bounded expansion and an almost linear kernel on any hereditary nowhere dense graph class. Since the class of Htopologicalminorfree graphs has bounded expansion, our results strongly generalize all the above mentioned works on kernelization of Dominating Set.