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D.: Independent set, induced matching, and pricing: connections and tight (subexponential time) approximation hardnesses
 CoRR abs/1308.2617
, 2013
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FixedParameter and Approximation Algorithms: A New Look
"... Abstract. A FixedParameter Tractable (FPT) ρapproximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPTalgorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k e ..."
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Abstract. A FixedParameter Tractable (FPT) ρapproximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPTalgorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For wellknown intractable problems such as the W[1]hard Clique and W[2]hard Set Cover problems, the natural question is whether we can get any FPTapproximation. It is widely believed that both Clique and SetCover admit no FPT ρapproximation algorithm, for any increasing function ρ. However, to the best of our knowledge, there has been no progress towards proving this conjecture. Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) [18] and the Projection Games Conjecture (PGC) [27], we make the first progress towards proving this conjecture by showing that – Under the ETH and PGC, there exist constants F1, F2> 0 such that the Set Cover problem does not admit a FPT approximation algorithm with ratio k F1 k in 2 F2 · poly(N, M) time, where N is the size of the universe and M is the number of sets. – Unless NP ⊆ SUBEXP, for every 1> δ> 0 there exists a constant F (δ)> 0 such that Clique has no FPT cost approximation with ratio k 1−δ in 2 kF · poly(n) time, where n is the number of vertices in the graph. In the second part of the paper we consider various W[1]hard problems
Parameterized algorithms for boxicity
 In: Proceedings of ISAAC 2010, LNCS 6506
"... In this paper we initiate an algorithmic study of Boxicity, a combinatorially well studied graph invariant, from the viewpoint of parameterized algorithms. The boxicity of an arbitrary graph G with the vertex set V (G) and the edge set E(G), denoted by box(G), is the minimum number of interval grap ..."
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In this paper we initiate an algorithmic study of Boxicity, a combinatorially well studied graph invariant, from the viewpoint of parameterized algorithms. The boxicity of an arbitrary graph G with the vertex set V (G) and the edge set E(G), denoted by box(G), is the minimum number of interval graphs on the same set of vertices such that the intersection of the edge sets of the interval graphs is E(G). In the Boxicity problem we are given a graph G together with a positive integer k, and asked whether the box(G) is at most k. The problem is notoriously hard and it is known to be NPcomplete even to determine whether the boxicity of a graph is at most two. This rules out any possibility of having an algorithm with running time V (G)O(f(k)), where f is an arbitrary function depending on k alone. Hence we look for other structural parameters like “vertex cover number ” and “max leaf number ” and see their effect on the problem complexity.
Parameterized Inapproximability of Target Set Selection and Generalizations
"... In this paper, we consider the Target Set Selection problem: given a graph and a threshold value thr(v) for each vertex v of the graph, find a minimum size vertexsubset to “activate” s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex v is activated d ..."
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In this paper, we consider the Target Set Selection problem: given a graph and a threshold value thr(v) for each vertex v of the graph, find a minimum size vertexsubset to “activate” s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex v is activated during the propagation process if at least thr(v) of its neighbors are activated. This problem models several practical issues like faults in distributed networks or wordtomouth recommendations in social networks. We show that for any functions f and ρ this problem cannot be approximated within a factor of ρ(k) in f(k) · nO(1) time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results.
The Constant Inapproximability of the Parameterized Dominating Set Problem
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DIRECTED GRAPHS: FIXEDPARAMETER TRACTABILITY & BEYOND
, 2014
"... Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The pa ..."
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Most interesting optimization problems on graphs are NPhard, implying that (unless P = NP) there is no polynomial time algorithm that solves all the instances of an NPhard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analysis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized complexity is to design efficient algorithms for NPhard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixedparameter tractable (FPT) if instances of size n and parameter k can be solved in f (k) · nO(1) time, where f is a computable function which does not depend on n. A parameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f (k) ·nO(g(k)) time, where f and g are both computable functions. In this thesis we focus on the parameterized complexity of transversal and connectivity problems on directed graphs. This research direction has been hitherto relatively