1,083 research outputs found

    Inapproximability of Maximum Biclique Problems, Minimum kk-Cut and Densest At-Least-kk-Subgraph from the Small Set Expansion Hypothesis

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    The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly states that it is NP-hard to distinguish between a graph with a small subset of vertices whose edge expansion is almost zero and one in which all small subsets of vertices have expansion almost one. In this work, we prove inapproximability results for the following graph problems based on this hypothesis: - Maximum Edge Biclique (MEB): given a bipartite graph GG, find a complete bipartite subgraph of GG with maximum number of edges. - Maximum Balanced Biclique (MBB): given a bipartite graph GG, find a balanced complete bipartite subgraph of GG with maximum number of vertices. - Minimum kk-Cut: given a weighted graph GG, find a set of edges with minimum total weight whose removal partitions GG into kk connected components. - Densest At-Least-kk-Subgraph (DALkkS): given a weighted graph GG, find a set SS of at least kk vertices such that the induced subgraph on SS has maximum density (the ratio between the total weight of edges and the number of vertices). We show that, assuming SSEH and NP ⊈\nsubseteq BPP, no polynomial time algorithm gives n1−Δn^{1 - \varepsilon}-approximation for MEB or MBB for every constant Δ>0\varepsilon > 0. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum kk-Cut and DALkkS to within (2−Δ)(2 - \varepsilon) factor of the optimum for every constant Δ>0\varepsilon > 0. The ratios in our results are essentially tight since trivial algorithms give nn-approximation to both MEB and MBB and efficient 22-approximation algorithms are known for Minimum kk-Cut [SV95] and DALkkS [And07, KS09]. Our first result is proved by combining a technique developed by Raghavendra et al. [RST12] to avoid locality of gadget reductions with a generalization of Bansal and Khot's long code test [BK09] whereas our second result is shown via elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis

    Synchronization Problems in Automata without Non-trivial Cycles

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    We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We investigate the complexity of finding a synchronizing set of states of maximum size. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete.Comment: Extended and corrected version, including arXiv:1608.00889. Conference version was published at CIAA 2017, LNCS vol. 10329, pages 188-200, 201

    Parameterized (in)approximability of subset problems

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    We discuss approximability and inapproximability in FPT-time for a large class of subset problems where a feasible solution SS is a subset of the input data and the value of SS is ∣S∣|S|. The class handled encompasses many well-known graph, set, or satisfiability problems such as Dominating Set, Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first time, we introduce the notion of intersective approximability that generalizes the one of safe approximability and show strong parameterized inapproximability results for many of the subset problems handled. Then, we study approximability of these problems with respect to the dual parameter n−kn-k where nn is the size of the instance and kk the standard parameter. More precisely, we show that under such a parameterization, many of these problems, while W[⋅\cdot]-hard, admit parameterized approximation schemata.Comment: 7 page

    From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

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    We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting OPT\text{OPT} be the optimum and NN be the size of the input, is there an algorithm that runs in t(OPT)poly(N)t(\text{OPT})\text{poly}(N) time and outputs a solution of size f(OPT)f(\text{OPT}), for any functions tt and ff that are independent of NN (for Clique, we want f(OPT)=ω(1)f(\text{OPT})=\omega(1))? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(OPT)o(\text{OPT})-FPT-approximation algorithm for Clique and no f(OPT)f(\text{OPT})-FPT-approximation algorithm for DomSet, for any function ff (e.g., this holds even if ff is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no 2o(n)2^{o(n)}-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1−ϔ)(1 - \epsilon)-satisfiable for some constant Ï”>0\epsilon > 0. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out ko(1)k^{o(1)}-FPT-approximation algorithm for Densest kk-Subgraph although this ratio does not yet match the trivial O(k)O(k)-approximation algorithm.Comment: 43 pages. To appear in FOCS'1

    QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

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    A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails

    Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension

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    Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD's paradigmatic problem: combinatorial auctions. We present a new generalization of the VC dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer-Shelah Lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximability results for deterministic truthful mechanisms for combinatorial auctions to date and establish new separation gaps between truthful and non-truthful algorithms

    Syntactic Separation of Subset Satisfiability Problems

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    Variants of the Exponential Time Hypothesis (ETH) have been used to derive lower bounds on the time complexity for certain problems, so that the hardness results match long-standing algorithmic results. In this paper, we consider a syntactically defined class of problems, and give conditions for when problems in this class require strongly exponential time to approximate to within a factor of (1-epsilon) for some constant epsilon > 0, assuming the Gap Exponential Time Hypothesis (Gap-ETH), versus when they admit a PTAS. Our class includes a rich set of problems from additive combinatorics, computational geometry, and graph theory. Our hardness results also match the best known algorithmic results for these problems

    A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

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    Given a kk-uniform hyper-graph, the Ekk-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that Ekk-Vertex-Cover is NP-hard to approximate within factor (k−1−ϔ)(k-1-\epsilon) for any k≄3k \geq 3 and any Ï”>0\epsilon>0. The result is essentially tight as this problem can be easily approximated within factor kk. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of ss-wise tt-intersecting families of subsets
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