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HardnessRandomness Tradeoffs for Bounded Depth Arithmetic Circuits
"... In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of sma ..."
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In this paper we show that lower bounds for bounded depth arithmetic circuits imply derandomization of polynomial identity testing for bounded depth arithmetic circuits. More formally, if there exists an explicit polynomial f(x1,..., xm) that cannot be computed by a depth d arithmetic circuit of small size then there exists an efficient deterministic algorithm to test whether a given depth d − 8 circuit is identically zero or not (assuming the individual degrees of the tested circuit are not too high). In particular, if we are guaranteed that the tested circuit computes a multilinear polynomial then we can perform the identity test efficiently. To the best of our knowledge this is the first hardnessrandomness tradeoff for bounded depth arithmetic circuits. The above results are obtained using the the arithmetic NisanWigderson generator of [KI04] together with a new theorem on bounded depth circuits, which is the main technical contribution of our work. This theorem deals with polynomial equations of the form P (x1,..., xn, y) ≡ 0 and shows that if P has a circuit of depth d and size s and if the polynomial f(x1,..., xn) satisfies P (x1,..., xn, f(x1,..., xn)) ≡ 0 then f has a circuit of depth d + 3 and size O(s · r + m r), where m is the total degree of f and r is the degree of y in P.
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 interva ..."
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Cited by 12 (0 self)
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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 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
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 11 (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.
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.
The ideal membership problem and polynomial identity testing
, 2007
"... Abstract. Given a monomial ideal I = 〈m1, m2, · · · , mk 〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I. We study this problem and show the following results. (a) If the ideal I = 〈m1, m2, · · · , mk 〉 for a constant k then ..."
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Cited by 10 (1 self)
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Abstract. Given a monomial ideal I = 〈m1, m2, · · · , mk 〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I. We study this problem and show the following results. (a) If the ideal I = 〈m1, m2, · · · , mk 〉 for a constant k then there is a randomized polynomialtime membership algorithm to test if f ∈ I. This result holds even for f given by a blackbox, when f is of small degree. (b) When I = 〈m1, m2, · · · , mk 〉 for a constant k and f is computed by a ΣΠΣ circuit with output gate of bounded fanin we can test whether f ∈ I in deterministic polynomial time. This generalizes the KayalSaxena result [KS07] of deterministic polynomialtime identity testing for ΣΠΣ circuits with bounded fanin output gate. (c) When k is not constant the problem is coNPhard. However, the problem is upper bounded by coAM PP over the field of rationals, and by coNP ModpP over finite fields. (d) Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits. For ideals I = 〈f1, · · · , fℓ 〉 where each fi ∈ F[x1, · · · , xk] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above. 1
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 9 (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 9 (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.