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On Tractable Exponential Sums

  • Jin-Yi Cai
  • Xi Chen
  • Richard Lipton
  • Pinyan Lu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6213)

Abstract

We consider the problem of evaluating certain exponential sums. These sums take the form
$$\sum_{x_1,x_2, \ldots, x_n \in \mathbb{Z}_N } e^{\frac{2 \pi i}{N} {f(x_1,x_2, \ldots, x_n)} },$$
where each x i is summed over a ring ℤ N , and f(x 1, x 2,..., x n ) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and logN when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree ≥ 3 we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem — a complete classification of each problem to be either computable in polynomial time or #P-hard — for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks; for the hardness result we prove group-theoretic necessary conditions for tractability.

Keywords

Partition Function Polynomial Time Prime Power Quadratic Polynomial Basic Building Block 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Jin-Yi Cai
    • 1
  • Xi Chen
    • 2
  • Richard Lipton
    • 3
  • Pinyan Lu
    • 4
  1. 1.University of Wisconsin-MadisonUSA
  2. 2.University of Southern CaliforniaUSA
  3. 3.Georgia Institute of TechnologyUSA
  4. 4.Microsoft Research AsiaUSA

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