The best current estimates of the thresholds for the existence of solutions in random constraint satisfaction problems ('CSPs') mostly derive from the first and the second moment method. Yet apart from a very few exceptional cases these methods do not quite yield matching upper and lower bounds. According to deep but non-rigorous arguments from statistical mechanics, this discrepancy is due to a change in the geometry of the set of solutions called condensation that occurs shortly before the actual threshold for the existence of solutions (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). To cope with condensation, physicists have developed a sophisticated but non-rigorous formalism called Survey Propagation (Me-zard, Parisi, Zecchina: Science 2002). This formalism yields precise conjectures on the threshold values of many random CSPs. Here we develop a new Survey Propagation inspired second moment method for the random k-NAESAT problem, which is one of the standard benchmark problems in the theory of random CSPs. This new technique allows us to overcome the barrier posed by condensation rigorously. We prove that the threshold for the existence of solutions in random k-NAESAT is 2k-1ln2-(ln 2 2+1 4)+ek, where |ek| ≤ 2-(1-ok(1))k, thereby verifying the statistical mechanics conjecture for this problem.
Rigorous work. The @math -NAESAT problem is well-known to be NP-complete in the worst case for any @math . In fact, the NP-complete problem of @math -coloring a @math -uniform hypergraph (with @math ) simply is the special case of @math -NAESAT without negations. The results in @cite_18 are actually phrased in terms of hypergraph @math -coloring but carry over to @math -NAESAT directly.
@cite_18
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It is shown that two sorts of problems belong to the NP-complete class. First, it is proven that for a given @k-colorable graph and a given @k-coloring of that graph, determining whether the graph is or is not uniquely @k-colorable is NP-complete. Second, a result by Garey, Johnson, and Stockmeyer is extended with a proof that the coloring of four-regular planar graphs is NP-complete. Graph coloring problems, in which one would like to color the vertices of a given graph with a small number of colors so that no two adjacent vertices receive the same color, arise in many applications, including various scheduling and partitioning problems. In this paper the complexity and performance of algorithms which construct such colorings are investigated. For a graph G , let k( G ) denote the minimum possible number of colors required to color G and, for any graph coloring algorithm A , let A ( G ) denote the number of colors used by A when applied to G . Since the graph coloring problem is known to be “NP-complete,” it is considered unlikely that any efficient algorithm can guarantee A ( G ) = k( G ) for all input graphs. In this paper it is proved that even coming close to khgr;( G ) with a fast algorithm is hard. Specifically, it is shown that if for some constant r d there exists a polynomial-time algorithm A which guarantees A ( G ) ≤ r ·k( G ) + d , then there also exists a polynomial-time algorithm A which guarantees A ( G ) = k( G ). The problem of coloring a graph with the minimum number of colorsis well known to be NP-hard, even restricted to k -colorable graphs for constant k ≥ 3. This paper explores theapproximation problem of coloring k -colorable graphs with as fewadditional colors as possible in polynomial time, with special focus onthe case of k = 3. The previous best upper bound on the number of colors needed forcoloring 3-colorable n -vertex graphsin polynomial time was o n log n colors by Berger and Rompel, improving a bound of o n colors by Wigderson. This paper presents an algorithmto color any 3-colorable graph with o n3 8 polylog n colors, thus breaking an“ O((n 1 2-o(1) ) barrier”. The algorithm given here is based on examiningsecond-order neighborhoods of vertices, rather than just immediateneighborhoods of vertices as in previous approaches. We extend ourresults to improve the worst-case bounds for coloring k -colorable graphs for constant k > 3 as well. We study deterministic, distributed algorithms for two weak variants of the standard graph coloring problem. We consider defective colorings, i.e., colorings where nodes of a color class may induce a graph of maximum degree d for some parameter d>0. We also look at colorings where a minimum number of multi-chromatic edges is required. For an integer k>0, we call a coloring k-partially proper if every node v has at least min k,deg(v) neighbors with a different color. We show that for all d∈ 1,...,Δ , it is possible to compute a O(Δ2 d2)-coloring with defect d in time O(log*n) where Δ is the largest degree of the network graph. Similarly, for all k∈ 1,...,Δ , a k-partially proper O(k2)-coloring can be computed in O(log*n) rounds. As an application of our weak defective coloring algorithm, we obtain a faster deterministic algorithm for the standard vertex coloring problem on graphs with moderate degrees. We show that in time O(Δ+log*n), a (Δ+1)-coloring can be computed, a task for which the best previous algorithm required time O(Δ*log(Δ) + log*n). The same result holds for the problem of computing a maximal independent set.