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Establishing the complexity of {\em Bounded Distance Decoding} for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length $$N$$ and dimension $K=O(N)$, we show that it is NP-hard to decode more than $$ N-K- c\frac{\log N}{\log\log N}$$ errors (with $c>0$ an absolute constant). Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount $$> N-K- c\log{N}$$ (with $c>0$$ an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a {\em Polynomial Reconstruction} problem. In this view, our results show that it is NP-hard to decide whether there exists a degree $$K$$ polynomial passing through $$K+ c\frac{\log N}{\log\log N}$$ points from a given set of points $$(a_1, b_1), (a_2, b_2)\ldots, (a_N, b_N)$$. Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree $$K$$ polynomial passing through $$K+c\log{N}$$ many points. These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called {\em Moments Subset Sum}, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.
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A Faster Solution to Smale's 17th Problem I: Real Binomial Systems
Suppose $$F:=(f_1,\ldots,f_n)$$ is a system of random $$n$$-variate polynomials with $$f_i$$ having degree $$\leq\!d_i$$ and the coefficient of $$x^{a_1}_1\cdots x^{a_n}_n$$ in $$f_i$$ being an independent complex Gaussian of mean $$0$$ and variance $$\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$$. Recent progress on Smale's 17$$\thth$$ Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, B\"{u}rgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of $$F$$ using just $$N^{O(1)}$$ arithmetic operations on average, where $$N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$$ ($$=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$$) is the maximum possible total number of monomial terms for such an $$F$$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when $$F$$ is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $$O(n^3\log^2(n\max_i d_i))$$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in $$n\log \max_i d_i$$ when $$F$$ has more terms.
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- Award ID(s):
- 1900881
- PAR ID:
- 10142360
- Date Published:
- Journal Name:
- ISSAC (International Symposium on Symbolic and Algebraic Computation) 2019
- Page Range / eLocation ID:
- 323 to 330
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3326229.3326267
