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88
Packet routing and jobshop scheduling in O(congestion+dilation) steps
 Combinatorica
, 1994
"... In this paper, we prove that there exists a schedule for routing any set of packets with edgesimple paths, on any network, in O(c+d) steps, where c is the congestion of the paths in the network, and d is the length of the longest path. The result has applications to packet routing in parallel machi ..."
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Cited by 119 (9 self)
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In this paper, we prove that there exists a schedule for routing any set of packets with edgesimple paths, on any network, in O(c+d) steps, where c is the congestion of the paths in the network, and d is the length of the longest path. The result has applications to packet routing in parallel machines, network emulations, and jobshop scheduling.
A constructive proof of the general Lovász Local Lemma
, 2009
"... The Lovász Local Lemma [EL75] is a powerful tool to nonconstructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Si ..."
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Cited by 83 (2 self)
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The Lovász Local Lemma [EL75] is a powerful tool to nonconstructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08a]. In [Mos08b], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma.
Approximating the Domatic Number
"... A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and ..."
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Cited by 76 (7 self)
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A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and \Delta the maximum degree.We show that every graph has a domatic partition with (1o(1))(ffi + 1) / ln n dominatingsets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) ln n approximation algorithm for domatic number, since the domaticnumber is always at most ffi + 1. We also show this to be essentially best possible. Namely,extending the approximation hardness of set cover by combining multiprover protocols with zeroknowledge techniques, we show that for every ffl> 0, a (1 ffl) ln napproximation impliesthat N P ` DT IM E(nO(log log n)). This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better.We also show that every graph has a domatic partition with (1o(1))(ffi + 1) / ln \Delta dominating sets, where the &quot; o(1) &quot; term goes to zero as \Delta increases. This can be turned intoan efficient algorithm that produces a domatic partition of \Omega ( ffi / ln \Delta) sets.
A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
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Cited by 75 (10 self)
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The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for converting some of these existence proofs into efficient algorithmic procedures, at the cost of loosing a little in the estimates. His method does not seem to be parallelizable. Here we modify his technique and achieve an algorithmic version that can be parallelized, thus obtaining deterministic NC 1 algorithms for several interesting algorithmic problems.
On Hierarchical Routing in Doubling Metrics
, 2005
"... We study the problem of routing in doubling metrics, and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with small amount of routing information stored at each vertex). We say that a metric (X, d) has doubling dimension dim(X) at most α ..."
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Cited by 70 (9 self)
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We study the problem of routing in doubling metrics, and show how to perform hierarchical routing in such metrics with small stretch and compact routing tables (i.e., with small amount of routing information stored at each vertex). We say that a metric (X, d) has doubling dimension dim(X) at most α if every set of diameter D can be covered by 2 α sets of diameter D/2. (A doubling metric is one whose doubling dimension dim(X) is a constant.) We show how to perform (1 + τ)stretch routing on metrics for any 0 < τ ≤ 1 with routing tables of size at most (α/τ) O(α) log 2 ∆ bits with only (α/τ) O(α) log ∆ entries, where ∆ is the diameter of the graph; hence the number of routing table entries is just τ −O(1) log ∆ for doubling metrics. These results extend and improve on those of Talwar (2004). We also give better constructions of sparse spanners for doubling metrics than those obtained from the routing tables above; for τ> 0, we give algorithms to construct (1 + τ)stretch spanners for a metric (X, d) with maximum degree at most (2 + 1/τ) O(dim(X)) , matching the results of Das et al. for Euclidean metrics.
The repulsive lattice gas, the independentset polynomial, and the Lovász local lemma
, 2004
"... We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independent ..."
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Cited by 44 (7 self)
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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independentset polynomial for G is nonvanishing in the polydisc of radii {px}. Furthermore, we show that the usual proof of the Lovász local lemma — which provides a sufficient condition for this to occur — corresponds to a simple inductive argument for the nonvanishing of the independentset polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37, 38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for “soft” dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternatingsign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.
Distributed Packet Switching in Arbitrary Networks
 In Proceedings of the 28th Annual ACM Symposium on Theory of Computing
, 1996
"... In a seminal paper Leighton, Maggs, and Rao consider the packet scheduling problem when a single packet has to traverse each path. They show that there exists a schedule where each packet reaches its destination in O(C + D) steps, where C is the congestion and D is the dilation. The proof relies o ..."
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Cited by 42 (2 self)
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In a seminal paper Leighton, Maggs, and Rao consider the packet scheduling problem when a single packet has to traverse each path. They show that there exists a schedule where each packet reaches its destination in O(C + D) steps, where C is the congestion and D is the dilation. The proof relies on the Lov'asz Local Lemma, and hence is not algorithmic. In a followup paper Leighton and Maggs use an algorithmic version of the Local Lemma due to Beck to give centralized algorithms for the problem. Leighton, Maggs, and Rao also give a distributed randomized algorithm where all packets reach their destinations with high probability in O(C +D log n) steps. In this paper we develop techniques to guarantee the high probability of delivering packets without resorting to the Lov'asz Local Lemma. We improve the distributed algorithm for problems with relatively high dilation to O(C) + (log n) O(log n) D + poly(log n). We extend the techniques to handle the case of infinite streams of ...
Hardness of approximate hypergraph coloring
 SICOMP: SIAM Journal on Computing
"... We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCP). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the ..."
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Cited by 38 (4 self)
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We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCP). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to “satisfy ” the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular, (hyper)graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple NotallEqual check on the four bits it reads. This enables us to prove that for any constant c, it is NPhard to color a 2colorable 4uniform hypergraph using just c colors, and also yields a superconstant inapproximability result under a stronger hardness
A constructive proof of the Lovász local lemma
 In STOC ’09: Proceedings of the 41st annual ACM Symposium on Theory of Computing
, 2009
"... The Lovász Local Lemma [EL75] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a kCNF formula in which each clause has common variables with at most 2 k− ..."
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Cited by 31 (2 self)
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The Lovász Local Lemma [EL75] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a kCNF formula in which each clause has common variables with at most 2 k−2 other clauses is always satisfiable. All hitherto known proofs of the Local Lemma are nonconstructive and do thus not provide a recipe as to how a satisfying assignment to such a formula can be efficiently found. In his breakthrough paper [Bec91], Beck demonstrated that if the neighbourhood of each clause be restricted to O(2 k/48), a polynomial time algorithm for the search problem exists. Alon simplified and randomized his procedure and improved the bound to O(2 k/8) [Alo91]. Srinivasan presented in [Sri08] a variant that achieves a bound of essentially O(2 k/4). In [Mos08], we improved this to O(2 k/2). In the present paper, we give a randomized algorithm that finds a satisfying assignment to every kCNF formula in which each clause has a neighbourhood of at most the asymptotic optimum of 2 k−5 −1 other clauses and that runs in expected time polynomial in the size of the formula, irrespective of k. If k is considered a constant, we can also give a deterministic variant. In contrast to all previous approaches, our analysis does not anymore invoke the standard nonconstructive versions of the Local Lemma and can therefore be considered an alternative, constructive proof of it. Key Words and Phrases. Lovász Local Lemma, derandomization, bounded occurrence SAT instances, hypergraph colouring.
Path Coloring on the Mesh
 In Proc. of the 37th Annual IEEE Symposium on Foundations of Computer Science
, 1996
"... In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Rag ..."
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Cited by 29 (0 self)
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In the minimum path coloring problem, we are given a list of pairs of vertices of a graph. We are asked to connect each pair by a colored path. Paths of the same color must be edge disjoint. Our objective is to minimize the number of colors used. This problem was raised by Aggarwal et al [1] and Raghavan and Upfal [22] as a model for routing in alloptical networks. It is also related to questions in circuit routing. In this paper, we improve the O(ln N ) approximation result of Kleinberg and Tardos [14] for path coloring on the N \Theta N mesh. We give an O(1) approximation algorithm to the number of colors needed, and a poly(ln ln N ) approximation algorithm to the choice of paths and colors. To the best of our knowledge, these are the first sublogarithmic bounds for any network other than trees, rings, or trees of rings. Our results are based on developing new techniques for randomized rounding. These techniques iteratively improve a fractional solution until it approaches integral...