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52
Hardness as randomness: A survey of universal derandomization
 in Proceedings of the International Congress of Mathematicians
, 2002
"... We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that P = BPP, it also indicates that this may be a difficult question to resolve. ..."
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Cited by 11 (5 self)
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We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that P = BPP, it also indicates that this may be a difficult question to resolve. In fact, proving that probalistic algorithms have nontrivial deterministic simulations is basically equivalent to proving circuit lower bounds, either in the algebraic or Boolean models.
Algebraic Algorithms for Matching and Matroid Problems
 SIAM JOURNAL ON COMPUTING
, 2009
"... We present new algebraic approaches for two wellknown combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algori ..."
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Cited by 11 (0 self)
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We present new algebraic approaches for two wellknown combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions.
Readonce Polynomial Identity Testing
"... An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readon ..."
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Cited by 11 (4 self)
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An arithmetic readonce formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of readonce formulas. the following are some of the results that we obtain. 1. Given k ROFs in n variables, over a field F, we give a deterministic (non blackbox) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n O(k2). 2. We give an n O(d+k2) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k2) for the sum of k depth d ROFs. If F  is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs. 3. We give a hitting set of size exp ( Õ( √ n + k 2)) for the sum of k ROFs (without depth restrictions). In particular this implies a subexponential time deterministic algorithm for
Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
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Cited by 11 (3 self)
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we
NC algorithms for comparability graphs, interval graphs, and unique perfect matching
 Proc. 5th Conf. Found. Software Technology and Theor. Comput. Sci., volume 206 of Lect. Notes in Comput. Sci
, 1985
"... Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval rep ..."
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Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for nding a maximum matching in an incomparability graph. 1
Algebraic structures and algorithms for matching and matroid problems
"... We present new algebraic approaches for several wellknown combinatorial problems, including nonbipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For nonbipartite matching, we obtain a simple, pu ..."
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Cited by 10 (2 self)
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We present new algebraic approaches for several wellknown combinatorial problems, including nonbipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n ω) where n is the number of vertices and ω is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time O(nr ω−1) for matroids with n elements and rank r that satisfy some natural conditions. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal.
A new NCalgorithm for finding a perfect matching in bipartite planar and small genus graphs (Extended Abstract)
, 2000
"... It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a Ptime algorithm for the case of graphs of small genus.) H ..."
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Cited by 8 (2 self)
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It has been known for a long time now that the problem of counting the number of perfect matchings in a planar graph is in NC. This result is based on the notion of a pfaffian orientation of a graph. (Recently, Galluccio and Loebl [7] gave a Ptime algorithm for the case of graphs of small genus.) However, it is not known if the corresponding search problem, that of finding one perfect matching in a planar graph, is in NC. This situation is intriguing as it seems to contradict our intuition that search should be easier than counting. For the case of planar bipartite graphs, Miller and Naor [22] showed that a perfect matching can indeed be found using an NC algorithm. We present a very different NCalgorithm for this problem. Unlike the Miller...
Overlay protection against link failures using network coding
 in in the proceedings of the Conference on Information Sciences and Systems (CISS
, 2008
"... Abstract — This paper introduces a network codingbased protection scheme against single and multiple link failures. The proposed strategy makes sure that in a connection, each node receives two copies of the same data unit: one copy on the working circuit, and a second copy that can be extracted fr ..."
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Cited by 8 (4 self)
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Abstract — This paper introduces a network codingbased protection scheme against single and multiple link failures. The proposed strategy makes sure that in a connection, each node receives two copies of the same data unit: one copy on the working circuit, and a second copy that can be extracted from linear combinations of data units transmitted on a shared protection path. This guarantees instantaneous recovery of data units upon the failure of a working circuit. The strategy can be implemented at an overlay layer, which makes its deployment simple and scalable. The proposed strategy is an extension of the scheme presented in [1]. The new scheme is simpler, less expensive, and does not require the synchronization required by the original scheme. The sharing of the protection circuit by a number of connections is the key to the reduction of the cost of protection. A preliminary comparison of the cost of the proposed scheme to the 1+1 protection strategy is conducted, and establishes the benefits of our strategy. I.
Black box polynomial identity testing of generalized depth3 arithmetic circuits with bounded top fanin
 in IEEE Conference on Computational Complexity
"... In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in ..."
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Cited by 7 (2 self)
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In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the blackbox polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f(¯x) = ∑ k i=1 hi(¯x) · gi(¯x), where each hi is a polynomial that depends on only ρ linear functions, and each gi is a product of linear functions (when hi = 1, for each i, then we get the class of depth3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When maxi(deg(hi · gi)) = d we say that f is computable by a ΣΠΣ(k, d, ρ) circuit. We obtain the following results. 1. A deterministic blackbox identity testing algorithm for ΣΠΣ(k, d, ρ) circuits that runs in quasipolynomial time (for ρ = polylog(n + d)). In particular this gives the first blackbox quasipolynomial time PIT algorithm for depth3 circuits with k multiplication gates. 2. A deterministic blackbox identity testing algorithm for readk ΣΠΣ circuits (depth3 circuits where each variable appears at most k times) that runs in time n 2O(k2). In particular