A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP

Abstract. All-Optical Label Switching (AOLS) is a new technology that performs packet forwarding without any Optical-Electrical-Optical (OEO) conversions. In this paper, we study the problem of routing a set of requests in AOLS networks using GMPLS technology, with the aim of minimizing the number of labels required to ensure the forwarding. We first formalize the problem by associating to each routing strategy a logical hypergraph whose hyperarcs are dipaths of the physical graph, called tunnels in GMPLS terminology. Such a hypergraph is called a hypergraph layout, to which we assign a cost function given by its physical length plus the total number of hops traveled by the traffic. Minimizing the cost of the design of an AOLS network can then be expressed as finding a minimum cost hypergraph layout. We prove hardness results for the problem, namely for general directed networks we prove that it is NPhard to find a C log n-approximation, where C is a a positive constant and n is the number of nodes of the network. For symmetric directed networks, we prove that the problem is APX-hard. These hardness results hold even is the traffic instance is a partial broadcast. On the other hand, we provide an O(log n)-approximation algorithm to the problem for a general symmetric network. Finally, we focus on the case where the physical network is a path, providing a polynomial-time dynamic programming algorithm for a bounded number of sources, thus extending the algorithm given in [2] for a single source.

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“…Since, unless P = NP, approximating Minimum Set Cover within a factor C log n for some constant C > 0 is NP-hard [12], our result follows.…”

Section: General Casementioning

confidence: 69%

“…Raz and Safra [12] proved that Minimum Set Cover is not approximable within a factor C log n, for some constant C > 0, unless P = NP. To a Set Cover instance with sets S 1 , S 2 , .…”

Section: General Casementioning

confidence: 99%

Designing Hypergraph Layouts to GMPLS Routing Strategies

Bermond

Coudert

Mouliérac

et al. 2010

Structural Information and Communication Complexity

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“…He also observed, that for directed graphs, even for rooted {0, 1}-requirements, CA is at least as hard as the SetCover problem. Combined with the result of [22] this implies an Ω(log n)-approximation threshold for this simple variant (namely, the problem cannot be approximated within c ln n for some universal constant c < 1, unless P=NP). By extending the construction from [7], a similar threshold was shown in [21] for the undirected rooted CA with root s and S = V − {s}, but for {0, k}-requirements with k = Θ(n).…”

Section: The Problem and Our Resultsmentioning

confidence: 89%

Tight Approximation Algorithm for Connectivity Augmentation Problems

Kortsarz

Nutov

2006

Automata, Languages and Programming

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The S-connectivity λ S G (u, v) of (u, v) in a graph G is the maximum number of uv-paths that no two of them have an edge or a node in S − {u, v} in common. The corresponding Connectivity Augmentation (CA) problem is: given a graph G0 = (V, E0), S ⊆ V , and requirements r(u, v) on V ×V , find a minimum size set F of new edges (any edge is allowed) so thatExtensively studied particular choices of S are the edge-CA (when S = ∅) and the node-CA (when S = V ). A. Frank gave a polynomial algorithm for undirected edge-CA and observed that the directed case even with rooted {0, 1}-requirements is at least as hard as the Set-Cover problem (in rooted requirements there is s ∈ V − S so that if r(u, v) > 0 then: u = s for directed graphs, and u = s or v = s for undirected graphs). Both directed and undirected node-CA have approximation threshold Ω(2 log 1−ε n ). The only polylogarithmic approximation ratio known for CA was for rooted requirements -O(log n · log rmax) = O(log 2 n), where rmax = maxu,v∈V r (u, v). No nontrivial approximation algorithms were known for directed CA even for r(u, v) ∈ {0, 1}, nor for undirected CA with S arbitrary. We give an approximation algorithm for the general case that matches the known approximation thresholds. For both directed and undirected CA with arbitrary requirements our approximation ratio is: O(log n) for S = V arbitrary, and O(rmax · log n) for S = V .

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“…, c k } of subsets of S, find a minimumcardinality subset SC of C such that for every 1 ≤ i ≤ n, s i ∈ c j for some c j ∈ SC. It is known that there exists a constant c > 0 such that approximating Minimum Set-Cover to within a factor of c ln n is NP-hard [18]. Given an instance (S = {s 1 , .…”

Section: Expression Op T 3 (Eq (3)) Is Np-hard To Compute Even When mentioning

confidence: 99%

Commitment Under Uncertainty: Two-Stage Stochastic Matching Problems

Katriel

Kenyon-Mathieu

Upfal

Automata, Languages and Programming

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Abstract. We define and study two versions of the bipartite matching problem in the framework of two-stage stochastic optimization with recourse. In one version the uncertainty is in the second stage costs of the edges, in the other version the uncertainty is in the set of vertices that needs to be matched. We prove lower bounds, and analyze efficient strategies for both cases. These problems model real-life stochastic integral planning problems such as commodity trading, reservation systems and scheduling under uncertainty.

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A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP

Cited by 672 publications

References 23 publications

Designing Hypergraph Layouts to GMPLS Routing Strategies

Designing Hypergraph Layouts to GMPLS Routing Strategies

Tight Approximation Algorithm for Connectivity Augmentation Problems

Commitment Under Uncertainty: Two-Stage Stochastic Matching Problems

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