Analytical approach to parallel repetition

Dinur, Irit; Steurer, David

doi:10.1145/2591796.2591884

Cited by 333 publications

(331 citation statements)

References 29 publications

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“…In fact, if the communication radius R c is the same as the sensing radius R s , MIN-CSC reduces to MIN-CDS. In general graphs, MIN-CDS inherits the inapproximability of set cover, so it is NP-hard to approximation MIN-CDS within a factor of ρ ln n for any ρ < 1 [14,10]. Improving upon Klein et al [25], Guha et al [18] obtained a 1.35 ln n-approximation, which is the best result known for general graphs.…”

Section: Theoremmentioning

confidence: 94%

“…It is well known that a simple greedy algorithm can achieve an approximation factor of ln n for HitSet and the factor is essentially optimal [14,10]. In this paper, we use a geometric version of HitSet in which the set of given elements are points in R 2 and the subsets are induced by given disks (i.e., each S ∈ S is the subset of points that can be covered by a given disk).…”

Section: Preliminariesmentioning

confidence: 99%

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Approximation Algorithms for the Connected Sensor Cover Problem

Huang

Shi

2015

Lecture Notes in Computer Science

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Abstract. We study the minimum connected sensor cover problem (MIN-CSC) and the budgeted connected sensor cover (Budgeted-CSC) problem, both motivated by important applications in wireless sensor networks. In both problems, we are given a set of sensors and a set of target points in the Euclidean plane. In MIN-CSC, our goal is to find a set of sensors of minimum cardinality, such that all target points are covered, and all sensors can communicate with each other (i.e., the communication graph is connected). We obtain a constant factor approximation algorithm, assuming that the ratio between the sensor radius and communication radius is bounded. In Budgeted-CSC problem, our goal is to choose a set of B sensors, such that the number of targets covered by the chosen sensors is maximized and the communication graph is connected. We also obtain a constant approximation under the same assumption.

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Section: Theoremmentioning

confidence: 94%

Section: Preliminariesmentioning

confidence: 99%

Approximation Algorithms for the Connected Sensor Cover Problem

Huang

Shi

2015

Lecture Notes in Computer Science

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“…In this section, for every ε > 0, we show the (1 − ε) ln n-approximation hardness by the approximationgap preserving reduction from the MinSC. Note that, very recently, Dinur and Steurer [9] show that it is NP-hard to approximate MinSC to within a factor of (1 − ε) ln n, by relaxing the previous assumption in [10].…”

Section: Inapproximability Of Minsdcmentioning

confidence: 99%

Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes

Eto

Kawahara

Miyano

et al. 2018

IEICE Trans. Inf. & Syst.

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SUMMARYIn this paper, we study a variant of the Minimum Dominating Set problem. Given an unweighted undirected graph G = (V, E) of n = |V| vertices, the goal of the Minimum Single Dominating Cycle problem (MinSDC) is to find a single shortest cycle which dominates all vertices, i.e., a cycle C such that for the set V(C) of vertices in C and the set [24]. In this paper we consider the (in)approximability of MinSDC if input graphs are restricted to some special classes of graphs. We first show that MinSDC is still NPhard to approximate even when restricted to planar, bipartite, chordal, or r-regular (r ≥ 3). Then, we show the (ln n + 1)-approximability and the (1 − ε) ln n-inapproximability of MinSDC on split graphs under P NP. Furthermore, we explicitly design a linear-time algorithm to solve MinSDC for graphs with bounded treewidth and estimate the hidden constant factor of its running time-bound.

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“…A recent work of Dinur and Steurer [DS13] introduces a new approach to the parallel repetition question, focused on the case of projection games. A projection game is one in which the referee's acceptance criterion has a special form: for any pair of questions (u, v), any answer b from the second player determines at most one valid answer a = π uv (b) for the first player.…”

Section: Introductionmentioning

confidence: 99%

“…The approach of [DS13] is based on the introduction of a relaxation of the game value, denoted VAL + . This relaxation can be defined for any game (we give the definition in Section 1.2 below), and it is perfectly multiplicative.…”

Section: Introductionmentioning

confidence: 99%

A parallel repetition theorem for entangled projection games

2015

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We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1 − ε < 1, the value of the k-fold repetition of G goes to zero as O((1 − ε c ) k ), for some universal constant c ≥ 1. If furthermore the constraint graph of G is expanding we obtain the optimal c = 1. Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games.Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call "vector quantum strategy" to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, the algorithm relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our "quantum correlated sampling lemma" generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two non-communicating parties, given classical descriptions of bipartite states |ψ , |ϕ respectively such that |ψ ≈ |ϕ , are able to locally generate a joint entangled state |Ψ ≈ |ψ ≈ |ϕ using an initial entangled state that is independent of their inputs.

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Analytical approach to parallel repetition

Cited by 333 publications

References 29 publications

Approximation Algorithms for the Connected Sensor Cover Problem

Approximation Algorithms for the Connected Sensor Cover Problem

Complexity of the Minimum Single Dominating Cycle Problem for Graph Classes

A parallel repetition theorem for entangled projection games

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