We introduce a new notion of resilience for constraint satisfaction problems, with the goal of more precisely determining the boundary between NP-hardness and the existence of efficient algorithms for resilient instances. In particular, we study @math -resiliently @math -colorable graphs, which are those @math -colorable graphs that remain @math -colorable even after the addition of any @math new edges. We prove lower bounds on the NP-hardness of coloring resiliently colorable graphs, and provide an algorithm that colors sufficiently resilient graphs. We also analyze the corresponding notion of resilience for @math -SAT. This notion of resilience suggests an array of open questions for graph coloring and other combinatorial problems.
With little progress on coloring general graphs, research has naturally turned to approximation. In approximating the chromatic number of a general graph, the first results were of Garey and Johnson, giving a performance guarantee of @math colors @cite_17 and proving that it is NP-hard to approximate chromatic number to within a constant factor less than two @cite_0 . Further work improved this bound by logarithmic factors @cite_18 @cite_16 . In terms of lower bounds, Zuckerman @cite_3 derandomized the PCP-based results of H stad @cite_7 to prove the best known approximability lower-bound to date, @math .
@cite_18 @cite_7 @cite_3 @cite_0 @cite_16 @cite_17
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The paper considers the computational hardness of approximating the chromatic number of a graph. The authors first give a simple proof that approximating the chromatic number of a graph to within a constant power (of the value itself) in NP-hard. They then consider the hardness of coloring a 3-colorable graph with as few as possible colors. They show that determining whether a graph is 3-colorable or any legal coloring of it requires at least 5 colors is NP-hard. Therefore, coloring a 3-colorable graph with 4 colors is NP-hard. > We deterministically compute a @D+1 coloring and a maximal independent set(MIS) in time O(@D^1^ ^2^+^@Q^(^1^ ^h^)+log^*n) for @D"1"+"[email protected][email protected]^1^+^i^ ^h, where @D"j is defined as the maximal number of nodes within distance j for a node and @D:[email protected]"1. Our greedy coloring and MIS algorithms improve the state-of-the-art algorithms running in O(@D+log^*n) for a large class of graphs, i.e., graphs of (moderate) neighborhood growth with h>=36. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. Our algorithm runs in time O([email protected]+log^*n) for @[email protected][email protected](log^1^+^1^ ^l^o^g^^^*^nn) and @[email protected]?O(@D log^1^+^1^ ^l^o^g^^^*^nn). For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest @D+1 coloring algorithm running in time O([email protected]+logn). The algorithm works without knowledge of @g and uses less than @D colors, i.e., (1-1 O(@g))@D with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number @g into account. We also improve on the state of the art deterministic computation of (2,c)-ruling sets. We give a new proof showing that it is NP-hard to color a 3-colorable graph using just 4 colors. This result is already known , [S. Khanna, N. Linial, and S. Safra, Combinatorica, 20 (2000), pp. 393--415], but our proof is novel because it does not rely on the PCP theorem, while the known one does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor @math hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [M. Bellare, O. Goldreich, and M. Sudan, SIAM J. Comput., 27 (1998), pp. 805--915]. Another aspect in which our proof is novel is in its use of the PCP theorem to show that 4-coloring of 3-colorable graphs remains NP-hard even on bounded-degree graphs (this hardness result does not seem to follow from the earlier reduction of Khanna, Linial, and Safra). We point out that such graphs can always be colored using O(1) colors by a simple greedy algorithm, while the best known algorithm for coloring (general) 3-colorable graphs requires @math colors. Our proof technique also shows that there is an @math such that it is NP-hard to legally 4-color even a @math fraction of the edges of a 3-colorable graph. We deterministically compute a Δ + 1 coloring in time O(Δ5c+2 ċ (Δ5)2 c (Δ1)e + (Δ1)e + log* n) and O(Δ5c+2 ċ (Δ5)1 c Δe + Δe + (Δ5)d log Δ5 log n) for arbitrary constants d, e and arbitrary constant integer c, where Δi is defined as the maximal number of nodes within distance i for a node and Δ := Δ1. Our greedy algorithm improves the state-of-the-art Δ + 1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. If Δ ∈ Ω(log1+1 log* n n) and χ ∈ O(Δ log1+1 log* n n) then our algorithm executes in time O(log χ + log* n) with high probability. For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest Δ + 1 coloring algorithm running in time O(log Δ + √log n). The algorithm works without knowledge of χ and uses less than Δ colors, i.e., (1 - 1 O(χ))Δ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account.