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  • 1.
    Austrin, Per
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
    Mossel, Elchanan
    Approximation resistant predicates from pairwise independence2009Ingår i: Computational Complexity, ISSN 1016-3328, E-ISSN 1420-8954, Vol. 18, nr 2, s. 249-271Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q](k) whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that For general k >= 3 and q <= 2, the Max k-CSPq problem is UG-hard to approximate within O(kq(2))/q(k) + epsilon. For the special case of q = 2, i.e., boolean variables, we can sharpen this bound to (k + O(k(0.525)))/2(k) + epsilon, improving upon the best previous bound of 2k/2(k) + epsilon (Samorodnitsky and Trevisan, STOC'06) by essentially a factor 2. Finally, again for q = 2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the O(k(0.525)) term can be replaced by the constant 4.

  • 2.
    Chattopadhyay, Arkadev
    et al.
    Tata Institute of Fundamental Research, Mumbai, Mumbai, India.
    Koucký, Michal
    Charles University, Prague Praha, Czech Republic.
    Loff, Bruno
    INESC-TEC & U. Porto, Porto, Portugal.
    Mukhopadhyay, Sagnik
    KTH, Skolan för elektroteknik och datavetenskap (EECS), Teoretisk datalogi, TCS.
    Simulation Theorems via Pseudo-random Properties2019Ingår i: Computational Complexity, ISSN 1016-3328, E-ISSN 1420-8954, Vol. 28, nr 4, s. 617-659Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999), to any gadget which satisfies a certainhitting property. We prove that inner product and gap-Hammingsatisfy this property, and as a corollary, we obtain a deterministic simulationtheorem for these gadgets, where the gadget’s input size is logarithmicin the input size of the outer function. This yields the firstdeterministic simulation theorem with a logarithmic gadget size, answeringan open question posed by Göös, Pitassi & Watson (in: Proceedingsof the 56th FOCS, 2015). Our result also implies the previous results for the indexing gadget, withbetter parameters than was previously known. Moreover, a simulationtheorem with logarithmic-sized gadget implies a quadratic separationin the deterministic communication complexity and the logarithm ofthe 1-partition number, no matter how high the 1-partition number iswith respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson (2015).

  • 3.
    Håstad, Johan
    KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
    EVERY 2-CSP ALLOWS NONTRIVIAL APPROXIMATION2008Ingår i: Computational Complexity, ISSN 1016-3328, E-ISSN 1420-8954, Vol. 17, nr 4, s. 549-566Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We use semidefinite programming to prove that any constraint satisfaction problem in two variables over any domain allows an efficient approximation algorithm that does better than picking a random assignment. Specifically we consider the case when each variable can take values in [d] and that each constraint rejects t out of the d(2) possible input pairs. Then, for some universal constant c, we can, in probabilistic polynomial time, find an assignment whose objective value is, in expectation, within a factor 1 - t/d(2) + ct/d(4) log d of optimal, improving on the trivial bound of 1 - t/d(2).

  • 4.
    Håstad, Johan
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
    On the Approximation Resistance of a Random Predicate2009Ingår i: Computational Complexity, ISSN 1016-3328, E-ISSN 1420-8954, Vol. 18, nr 3, s. 413-434Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A predicate is called approximation resistant if it is NP-hard to approximate the corresponding constraint satisfaction problem significantly better than what is achieved by the naive algorithm that picks an assignment uniformly at random. In this paper we study predicates of Boolean inputs where the width of the predicate is allowed to grow. Samorodnitsky and Trevisan proved that, assuming the Unique Games Conjecture, there is a family of very sparse predicates that are approximation resistant. We prove that, under the same conjecture, any predicate implied by their predicate remains approximation resistant and that, with high probability, this condition applies to a randomly chosen predicate.

  • 5.
    Håstad, Johan
    KTH, Skolan för datavetenskap och kommunikation (CSC), Teoretisk datalogi, TCS.
    Special issue "conference on computational complexity 2009" guest editor's foreword2010Ingår i: Computational Complexity, ISSN 1016-3328, E-ISSN 1420-8954, Vol. 19, nr 2, s. 151-152Artikel i tidskrift (Refereegranskat)
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