6,091 research outputs found

    Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

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    We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: - For any r larger than some constant, any r-approximation algorithm for the maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph pricing problem, where n is the number of vertices in an input graph. This almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.Comment: The full version of FOCS 201

    Gap Amplification for Small-Set Expansion via Random Walks

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    In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness ϵ\epsilon and soundness 12\frac{1}{2} is at least as difficult as Small-Set Expansion with completeness ϵ\epsilon and soundness f(ϵ)f(\epsilon), for any function f(ϵ)f(\epsilon) which grows faster than ϵ\sqrt{\epsilon}. We achieve this amplification via random walks -- our gadget is the graph with adjacency matrix corresponding to a random walk on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same

    Lower bounds for adaptive linearity tests

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    Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c - the probability it accepts linear functions, its soundness s - the probability it accepts functions far from linear, and its query complexity q - the number of queries it makes. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to generalize this lower bound also to adaptive linearity tests. We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in (Samorodnitsky and Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions..

    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−εn^{1 - \varepsilon}-approximation for MEB or MBB for every constant ε>0\varepsilon > 0. Moreover, assuming SSEH, we show that it is NP-hard to approximate Minimum kk-Cut and DALkkS to within (2−ε)(2 - \varepsilon) factor of the optimum for every constant ε>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

    Towards a navigational logic for graphical structures

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    One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions. In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.Peer ReviewedPostprint (author's final draft

    Directed Security Policies: A Stateful Network Implementation

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    Large systems are commonly internetworked. A security policy describes the communication relationship between the networked entities. The security policy defines rules, for example that A can connect to B, which results in a directed graph. However, this policy is often implemented in the network, for example by firewalls, such that A can establish a connection to B and all packets belonging to established connections are allowed. This stateful implementation is usually required for the network's functionality, but it introduces the backflow from B to A, which might contradict the security policy. We derive compliance criteria for a policy and its stateful implementation. In particular, we provide a criterion to verify the lack of side effects in linear time. Algorithms to automatically construct a stateful implementation of security policy rules are presented, which narrows the gap between formalization and real-world implementation. The solution scales to large networks, which is confirmed by a large real-world case study. Its correctness is guaranteed by the Isabelle/HOL theorem prover.Comment: In Proceedings ESSS 2014, arXiv:1405.055

    Testing product states, quantum Merlin-Arthur games and tensor optimisation

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    We give a test that can distinguish efficiently between product states of n quantum systems and states which are far from product. If applied to a state psi whose maximum overlap with a product state is 1-epsilon, the test passes with probability 1-Theta(epsilon), regardless of n or the local dimensions of the individual systems. The test uses two copies of psi. We prove correctness of this test as a special case of a more general result regarding stability of maximum output purity of the depolarising channel. A key application of the test is to quantum Merlin-Arthur games with multiple Merlins, where we obtain several structural results that had been previously conjectured, including the fact that efficient soundness amplification is possible and that two Merlins can simulate many Merlins: QMA(k)=QMA(2) for k>=2. Building on a previous result of Aaronson et al, this implies that there is an efficient quantum algorithm to verify 3-SAT with constant soundness, given two unentangled proofs of O(sqrt(n) polylog(n)) qubits. We also show how QMA(2) with log-sized proofs is equivalent to a large number of problems, some related to quantum information (such as testing separability of mixed states) as well as problems without any apparent connection to quantum mechanics (such as computing injective tensor norms of 3-index tensors). As a consequence, we obtain many hardness-of-approximation results, as well as potential algorithmic applications of methods for approximating QMA(2) acceptance probabilities. Finally, our test can also be used to construct an efficient test for determining whether a unitary operator is a tensor product, which is a generalisation of classical linearity testing.Comment: 44 pages, 1 figure, 7 appendices; v6: added references, rearranged sections, added discussion of connections to classical CS. Final version to appear in J of the AC
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