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18
Beating the random ordering is hard: Inapproximability of maximum acyclic subgraph
 in FOCS, 2008
"... We prove that approximating the Max Acyclic Subgraph problem within a factor better than 1/2 is UniqueGames hard. Specifically, for every constant ε> 0 the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1 − ε) of its edges, if one can efficiently ..."
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Cited by 41 (9 self)
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We prove that approximating the Max Acyclic Subgraph problem within a factor better than 1/2 is UniqueGames hard. Specifically, for every constant ε> 0 the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1 − ε) of its edges, if one can efficiently find an acyclic subgraph of G with more than (1/2+ε) of its edges, then the UGC is false. Note that it is trivial to find an acyclic subgraph with 1/2 the edges, by taking either the forward or backward edges in an arbitrary ordering of the vertices of G. The existence of a ρapproximation algorithm for ρ> 1/2 has been a basic open problem for a while. Our result is the first tight inapproximability result for an ordering problem. The starting point of our reduction is a directed acyclic subgraph (DAG) in which every cut is nearlybalanced in the sense that the number of forward and backward edges crossing the cut are nearly equal; such DAGs were constructed in [3]. Using this, we are able to study Max Acyclic Subgraph, which is a constraint satisfaction problem (CSP) over an unbounded domain, by relating it to a proxy CSP over a bounded domain. The latter is then amenable to powerful techniques based on the invariance principle [13, 19]. Our results also give a superconstant factor inapproximability result for the Min Feedback Arc Set problem. Using our reductions, we also obtain SDP integrality gaps for both the problems. 1
Improved Algorithms via Approximations of Probability Distributions
 Journal of Computer and System Sciences
, 1997
"... We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC ..."
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Cited by 28 (4 self)
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We present two techniques for approximating probability distributions. The first is a simple method for constructing the smallbias probability spaces introduced by Naor & Naor. We show how to efficiently combine this construction with the method of conditional probabilities to yield improved NC algorithms for many problems such as set discrepancy, finding large cuts in graphs, finding large acyclic subgraphs etc. The second is a construction of small probability spaces approximating general independent distributions, which is of smaller size than the constructions of Even, Goldreich, Luby, Nisan & Velickovi'c. Such approximations are useful, e.g., for the derandomization of certain randomized algorithms. Keywords. Derandomization, parallel algorithms, discrepancy, graph coloring, small sample spaces, explicit constructions. 1 Introduction Derandomization, the development of general tools to derive efficient deterministic algorithms from their randomized counterparts, has blossomed ...
Testing Properties of Directed Graphs: Acyclicity and Connectivity
, 2002
"... This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs { acyclicity. Because the choice of representation aects the choice of algorithm, the two main representations of graphs are studied. For the adja ..."
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Cited by 15 (1 self)
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This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs { acyclicity. Because the choice of representation aects the choice of algorithm, the two main representations of graphs are studied. For the adjacencymatrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of ), where is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2=3, every graph whose adjacency matrix should be modi ed in at least fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an jVj ) lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. Along with
Testing Acyclicity of Directed Graphs in Sublinear Time
 In Proceedings of ICALP
, 2000
"... This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs  acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the a ..."
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Cited by 13 (5 self)
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This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs  acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacencymatrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of ~ O(1=ffl 2 ), where ffl is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2=3, every graph whose adjacency matrix should be modified in at least ffl fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an \Omega\Gamma jVj 1=3 ) lower bound is proved on the number of queries and the time required for testing, where V...
BEATING THE RANDOM ORDERING IS HARD: EVERY ORDERING CSP IS APPROXIMATION RESISTANT
, 2011
"... We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if ρ is the expected fraction of constraints satisfied by a ra ..."
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Cited by 13 (3 self)
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We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant. In other words, we show that if ρ is the expected fraction of constraints satisfied by a random ordering, then obtaining a ρ ′ approximation for any ρ ′>ρis UGhard. For the simplest OCSP, the Maximum Acyclic Subgraph (MAS) problem, this implies that obtaining a ρapproximation for any constant ρ>1/2 is UGhard. Specifically, for every constant ε>0the following holds: given a directed graph G that has an acyclic subgraph consisting of a fraction (1−ε) of its edges, it is UGhard to find one with more than (1/2 +ε) of its edges. Note that it is trivial to find an acyclic subgraph with 1/2 the edges by taking either the forward or backward edges in an arbitrary ordering of the vertices of G. The MAS problem has been well studied, and beating the random ordering for MAS has been a basic open problem. An OCSP of arity k is specified by a subset Π ⊆ Sk of permutations on {1, 2,...,k}. An instance of such an OCSP is a set V and a collection of constraints, each of which is an ordered ktuple of V. The objective is to find a global linear ordering of V while maximizing the number of constraints ordered as in Π. A random ordering of V is expected to satisfy a ρ = Π
Partial order reduction for verification of timed systems
, 1999
"... conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of SRC, NSF, DARPA, or the United States Government. ..."
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Cited by 12 (0 self)
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conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of SRC, NSF, DARPA, or the United States Government.
Balanced VertexOrderings of Graphs
, 2002
"... We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains N ..."
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Cited by 9 (4 self)
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We consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NPhard, and remains NPhard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertexordering, obtaining optimal orderings for directed acyclic graphs and graphs with maximum degree three. Finally we