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• W2167831266 abstract "Systems of polynomial equations over an algebraically-closed field K can be used to concisely represent combinatorial decision problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, Hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K . If the system of polynomial equations has no solution, then Hilbert's Nullstellensatz yields a certificate that the underlying combinatorial problem is infeasible. We investigate an algorithm aimed at proving combinatorial infeasibility based on the experimentally-observed low degree of Hilbert's Nullstellensatz and large-scale, sparse linear algebra computations over K . We explore the Nullstellensatz Linear Algebra algorithm (Nul LA) from both a computational and a theoretical perspective. From the computational perspective, we compare computations over the rationals to computations over finite fields; we discuss mathematical ideas for optimizing Nul LA ranging from the algebraic to the probabilistic, and we report on experiments proving the non-3-colorability of graphs with almost two thousand vertices and tens of thousands of edges. From a theoretical perspective, we observe that if an NP-complete problem (e.g. graph 3-colorability) is represented as a system of polynomial equations, the resulting infeasibility certificate is a coNP certificate. Thus, if P ≠ NP and NP ≠ coNP, there must exist an infinite family of instances (e.g. an infinite family of graphs) where the minimum-degree of the associated Nullstellensatz certificate grows linearly in the input size and the certificates contain a super-polynomial number of monomials. In the case of graph 3-colorability, we show that the minimum-degree of a Nullstellensatz certificate (associated with a particular encoding) follows the sequence 1,4,7,..., etc.. In the case of the independent set decision problem, we show that a minimum-degree Nullstellensatz certificate (associated with a particular encoding) proving the non-existence of an independent set of size k is equal to the size of the largest independent set in the graph. Moreover, such a Nullstellensatz certificate contains one monomial for each independent set in the graph." @default.
• W2167831266 created "2016-06-24" @default.
• W2167831266 creator A5005906505 @default.
• W2167831266 date "2008-01-01" @default.
• W2167831266 modified "2023-09-26" @default.
• W2167831266 title "Computer algebra, combinatorics, and complexity: hilbert's nullstellensatz and np-complete problems" @default.
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