88,540 research outputs found

    Property Testing of LP-Type Problems

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    Given query access to a set of constraints S, we wish to quickly check if some objective function ? subject to these constraints is at most a given value k. We approach this problem using the framework of property testing where our goal is to distinguish the case ?(S) ? k from the case that at least an ? fraction of the constraints in S need to be removed for ?(S) ? k to hold. We restrict our attention to the case where (S,?) are LP-Type problems which is a rich family of combinatorial optimization problems with an inherent geometric structure. By utilizing a simple sampling procedure which has been used previously to study these problems, we are able to create property testers for any LP-Type problem whose query complexities are independent of the number of constraints. To the best of our knowledge, this is the first work that connects the area of LP-Type problems and property testing in a systematic way. Among our results are property testers for a variety of LP-Type problems that are new and also problems that have been studied previously such as a tight upper bound on the query complexity of testing clusterability with one cluster considered by Alon, Dar, Parnas, and Ron (FOCS 2000). We also supply a corresponding tight lower bound for this problem and other LP-Type problems using geometric constructions

    A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models

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    This note uses a simple example to show how moment inequality models used in the empirical economics literature lead to general minimax relative efficiency comparisons. The main point is that such models involve inference on a low dimensional parameter, which leads naturally to a definition of "distance" that, in full generality, would be arbitrary in minimax testing problems. This definition of distance is justified by the fact that it leads to a duality between minimaxity of confidence intervals and tests, which does not hold for other definitions of distance. Thus, the use of moment inequalities for inference in a low dimensional parametric model places additional structure on the testing problem, which leads to stronger conclusions regarding minimax relative efficiency than would otherwise be possible

    Existence, iteration procedures and directional differentiability for parabolic QVIs

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    We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.Comment: 41 page

    Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP

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    Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it. In this paper, we show that similar results hold for constant-time approximation algorithms in the bounded-degree model. Specifically, we present the followings: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution to a basic LP relaxation of the CSP. (ii) Using the oracle, we give a constant-time rounding scheme that achieves an approximation ratio coincident with the integrality gap of the basic LP. (iii) Finally, we give a generic conversion from integrality gaps of basic LPs to hardness results. All of those results are \textit{unconditional}. Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all. A CSP instance is called ϵ\epsilon-far from satisfiability if we must remove at least an ϵ\epsilon-fraction of constraints to make it satisfiable. A CSP is called testable if there is a constant-time algorithm that distinguishes satisfiable instances from ϵ\epsilon-far instances with probability at least 2/32/3. Using the results above, we also derive, under a technical assumption, an equivalent condition under which a CSP is testable in the bounded-degree model
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