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.
On the other hand there are asymptotic and concrete lower bounds. Khot @cite_9 proved that for sufficiently large @math it is NP-hard to color a @math -colorable graph with fewer than @math colors; this was improved by Huang to @math @cite_25 . It is also known that for every constant @math there exists a sufficiently large @math such that coloring a @math -colorable graph with @math colors is NP-hard @cite_20 . In the non-asymptotic case, Khanna, Linial, and Safra @cite_2 used the PCP theorem to prove it is NP-hard to 4-color a 3-colorable graph, and more generally to color a @math colorable graph with at most @math colors. Guruswami and Khanna give an explicit reduction for @math @cite_4 . Assuming a variant of Khot's 2-to-1 conjecture, prove that distinguishing between chromatic number @math and @math is hard for constants @math @cite_20 . This is the best conditional lower bound we give in , but it does not to our knowledge imply Theorem .
@cite_4 @cite_9 @cite_2 @cite_25 @cite_20
1969999211 2963885173 1492972816 2008724760
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. In this paper we establish a substantially improved lower bound on the k-color ability threshold of the random graph G(n, m) with n vertices and m edges. The new lower bound is ≈ 1.39 less than the 2k ln (k)-ln (k) first-moment upper bound (and approximately 0.39 less than the 2k ln (k) - ln(k) - 1 physics conjecture). By comparison, the best previous bounds left a gap of about 2+ln(k), unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. Furthermore, we prove that, in a precise sense, our lower bound marks the so-called condensation phase transition predicted on the basis of physics arguments [: PNAS 2007]. Our proof technique is a novel approach to the second moment method, inspired by physics conjectures on the geometry of the set of k-colorings of the random graph. We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than (2^ (K^ 1 3 ) ) colors. This improves the previous result of K versus (K^ 1 25 K ) in Khot [1]. In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal.