A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally non-trivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show applications of this lemma to parallel computation and randomized reductions.
A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally non-trivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show applications of this lemma to parallel computation and randomized reductions.
In [26], henceforth referred to as Part I, we suggested an approach to the P vs. N P and related lower bound problems in complexity theory through geometric invariant theory. In particular, it reduces the arithmetic (characteristic zero) version of the N P ⊆ P conjecture to the problem of showing that a variety associated with the complexity class N P cannot be embedded in a variety associated with the complexity class P . We shall call these class varieties associated with the complexity classes P and N P . This paper develops this approach further, reducing these lower bound problemswhich are all nonexistence problems-to some existence problems: specifically to proving existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions.The first result is a generalization of the Borel-Weil theorem to a class of orbit * The work of the first author was supported by NSF grant CCR 9800042 and, in part, by Guggenheim Fellowship.† Visiting faculty member 1 closures, which include class varieties. The second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class N C. These results indicate that the fundamental lower bound problems in complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory.The results here were announced in [25]. We are grateful to Madhav Nori for his guidence and encouragement during the course of this work, and to C. S. Seshadri and Burt Totaro for helpful discussions. Main resultsWe shall now state the results precisely. For the sake of completeness, we recall in Section 3 the main results from part I. The rest of this paper is self contained. All groups in this paper are algebraic and the base field is C. In Part I, we reduced arithmetic (characteristic zero) implications of the lower bound problems in complexity theory, such as P vs. N P , and N C vs. P #P , to instances of the following (Section 3):Problem 2.1 (The orbit closure problem)The goal is to show that this is not the case in the problems under consideration.The f 's and g's here depend on the complexity classes in the lower bound problem under consideration. In the context of the P vs. N P problem, the point g will correspond to a judiciously chosen P -complete problem, and f to a judiciously chosen N P -complete problem. We call ∆ V [g] and ∆ V [f ] the class varieties associated with the complexity classes P and N P (this terminology was not used in part I). The orbit closure problem in this context is to show that the class variety associated with N P cannot be embedded in a class variety associated with P . We have oversimplified the story here. There is not just one class variety associated with a given complexity class, but a sequence of 2 class varieties depending on the parameters of the lower bound problem under consideration. In the context of the P vs. N P proble...
Abstract. We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood-Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood-Richardson coefficients, unless P = NP. We also show that there exists a #P -formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that, despite the hardness of the positivity problem, there may well exist a positive combinatorial formula for the Kronecker coefficients. Finding such a formula is a major open problem in representation theory and algebraic combinatorics. Finally, we consider the existence of the partition triples (λ, µ, π) such that the Kronecker coefficient k λ µ,π = 0 but the Kronecker coefficient k lλ lµ,lπ > 0 for some integer l > 1. Such "holes" are of great interest as they witness the failure of the saturation property for the Kronecker coefficients, which is still poorly understood. Using insight from computational complexity theory, we turn our hardness proof into a positive result: We show that not only do there exist many such triples, but they can also be found efficiently. Specifically, we show that, for any 0 < ≤ 1, there exists 0 < a < 1 such that, for all m, there exist Ω(2 m a ) partition triples (λ, µ, µ) in the Kronecker cone such that: (a) the Kronecker coefficient k λ µ,µ is zero, (b) the height of µ is m, (c) the height of λ is ≤ m , and (d) |λ| = |µ| ≤ m 3 . The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.
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