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[Computational Complexity] Favorite Theorems: The Permanent

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  • Lance
    June Edition The permanent of a matrix has a simple form Perm(A)= a 1&sigma(1) a 2&sigma(2) a n&sigma(n) where A is an nn matrix, a ij is the element in the
    Message 1 of 2 , Jul 24 6:21 AM
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      June Edition

      The permanent of a matrix has a simple form

      Perm(A)=Σσa1&sigma(1)a2&sigma(2)···an&sigma(n)

      where A is an n×n matrix, aij is the element in the ith row and jth column and the sum is taken over all permutations of {1,…,n}.

      Very similar to the determinant except without the negations and a 0-1 permanent counts the number of perfect matchings on a bipartite graph. But while we can compute the determinant quickly and easily check if a graph has a perfect matching the permanent turns out to have an apparently much higher complexity.

      Leslie Valiant, The Complexity of Computing the Permanent, TCS 1979.

      Valiant shows that computing the permanent, even for 0-1 matrices, is #P-complete, i.e., as hard as counting the number of solutions for NP problems.

      While an important hardness result in its own right, Valiant's theorem becomes even more important with later work. Toda's theorem implies the permanent is hard for the entire polynomial-time hierarchy. The permanent also has two very important properties:

      1. The permanent on n×n matrices is easily reducible to solving the permanent on (n-1)×(n-1) matrices.
      2. The permanent is a multilinear polynomial on the entries of A.
      Those two facts, combined with Valiant's and Toda's theorems, helped the permanent play a critical role in the development of interactive proofs and in some derandomization results, most notably Impagliazzo-Wigderson and Impagliazzo-Kabanets.

      --
      Posted by Lance to Computational Complexity at 7/24/2006 08:15:00 AM

    • Lance
      June Edition The permanent of a matrix has a simple form Perm(A)= a 1&sigma(1) a 2&sigma(2) a n&sigma(n) where A is an nn matrix, a ij is the element in the
      Message 2 of 2 , Jul 24 6:22 AM
      • 0 Attachment
        June Edition

        The permanent of a matrix has a simple form

        Perm(A)=Σσa1&sigma(1)a2&sigma(2)···an&sigma(n)

        where A is an n×n matrix, aij is the element in the ith row and jth column and the sum is taken over all permutations of {1,…,n}.

        Very similar to the determinant except without the negations and a 0-1 permanent counts the number of perfect matchings on a bipartite graph. But while we can compute the determinant quickly and easily check if a graph has a perfect matching the permanent turns out to have an apparently much higher complexity.

        Leslie Valiant, The Complexity of Computing the Permanent, TCS 1979.

        Valiant shows that computing the permanent, even for 0-1 matrices, is #P-complete, i.e., as hard as counting the number of solutions for NP problems.

        While an important hardness result in its own right, Valiant's theorem becomes even more important with later work. Toda's theorem implies the permanent is hard for the entire polynomial-time hierarchy. The permanent also has two very important properties:

        1. The permanent on n×n matrices is easily reducible to solving the permanent on (n-1)×(n-1) matrices.
        2. The permanent is a multilinear polynomial on the entries of A.
        Those two facts, combined with Valiant's and Toda's theorems, helped the permanent play a critical role in the development of interactive proofs and in some derandomization results, most notably Impagliazzo-Wigderson and Impagliazzo-Kabanets.

        --
        Posted by Lance to Computational Complexity at 7/24/2006 08:15:00 AM

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