Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gröbner Bases
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We consider the graph k-colouring problem encoded as a set of polynomial equations in the standard way over 0/1-valued variables. We prove that there are bounded-degree graphs that do not have legal k-colourings but for which the polynomial calculus proof system defined in [Clegg et al '96, Alekhnovich et al '02] requires linear degree, and hence exponential size, to establish this fact. This implies a linear degree lower bound for any algorithms based on Gröbner bases solving graph k-colouring using this encoding. The same bound applies also for the algorithm studied in a sequence of papers [De Loera et al '08,'09,'11,'15] based on Hilbert's Nullstellensatz proofs for a slightly different encoding, thus resolving an open problem mentioned in [De Loera et al '08,'09,'11] and [Li '16]. We obtain our results by combining the polynomial calculus degree lower bound for functional pigeonhole principle (FPHP) formulas over bounded-degree bipartite graphs in [Mikša and Nordström '15] with a reduction from FPHP to k-colouring derivable by polynomial calculus in constant degree.
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