By Brian Osserman
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Reconﬁguration problems also model questions of evolvability: Can genotype y0 evolve into genotype yt via individual mutations which are each of adequate ﬁtness? Here a genotype is considered feasible if its ﬁtness is above a threshold, and two genotypes are considered adjacent if one is a simple mutation of the other. Finally, reconﬁguration versions of constraint satisfaction problems (the ﬁrst kind studied in the literature ) yield insights into the structure of the solution space, and heuristics, such as survey propagation, whose performance depends crucially on connectivity and other properties of the solution space.
Due to space limitations some proofs in this article are omitted. They can be found in the full version of this paper. 2 Hardness Results Theorem 1. Given an unweighted n-vertex graph with an (∞, 2)-coloring, no polynomial-time algorithm for the MCRP, the MRRP or the MBRP has an approximation ratio of (1 − o(1)) ln ln n unless N P ⊆ DT IM E(nO(log log n) ). Theorem 2. The MCRP on unweighted graphs is N P-hard even if the problem is restricted to trees colored by an initial (∞, 2)-coloring. The Complexity of Minimum Convex Coloring 19 Proof.
Proof. Without loss of generality, let Spp |qq be a staircase with respect to the third quadrant, as Fig. 10 shows. Let S be a staircase polygon in Spp |qq such that CreateStaircasePath(S) is invoked. We will prove L Gpp |qq ∩ S ≤ 2L Gpp |qq ∩ S using induction. The inequality obviously holds in the trivial case S = ∅. Assume the relation holds for smaller staircase polygons in S. For the case L hS1 ≥ L v1S , t1 is connected down and the original problem is reduced to the small one with region Right1 (S).