Abstract
We derive upper and lower bounds on the degree $d$ for which the Lovasz $\vartheta$ function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a $k$-coloring in ra...
Highlights
Many constraint satisfaction problems have phase transitions in the random case: as the ratio between the number of constraints and the number of variables increases, there is a critical value at which the probability that a solution exists, in the limit n → ∞, suddenly drops from one to zero
The Lovász θ function, which we review below, gives a lower bound on the chromatic number which can be computed in polynomial time
We first use orthogonal polynomials to derive explicit bounds on θ(G) for arbitrary regular graphs of a given girth – which may be of independent interest – and employ a concentration argument for Gn,d
Summary
Many constraint satisfaction problems have phase transitions in the random case: as the ratio between the number of constraints and the number of variables increases, there is a critical value at which the probability that a solution exists, in the limit n → ∞, suddenly drops from one to zero. Conjectures from statistical physics [40, 25, 26] suggest this exponential difficulty is sometimes unavoidable These conjectures state that polynomial-time algorithms for detection and reconstruction exist if and only if d is above the Kesten-Stigum threshold [34, 35], k−1 2. We define the d-regular block model by choosing a planted partition σ uniformly at random and conditioning Gn,d on the event that σ is good.
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