We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems. 3-SAT is equivalent to (2,3)-CSP while the other problems above are special cases of (3,2)-CSP; there is also a natural duality transformation from (a,b)-CSP to (b,a)-CSP. We give a fast algorithm for (3,2)-CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas.
There has also been some related work on approximate or heuristic 3-coloring algorithms. Blum and Karger @cite_7 show that any 3-chromatic graph can be colored with @math colors in polynomial time. Alon and Kahale @cite_25 describe a technique for coloring random 3-chromatic graphs in expected polynomial time, and Petford and Welsh @cite_9 present a randomized algorithm for 3-coloring graphs which also works well empirically on random graphs although they prove no bounds on its running time. Finally, Vlasie @cite_16 has described a class of instances which are (unlike random 3-chromatic graphs) difficult to color.
@cite_9 @cite_16 @cite_25 @cite_7
2068179205 2161779091 2079035346 1529433769
Let G3n,p,3 be a random 3-colorable graph on a set of 3n vertices generated as follows. First, split the vertices arbitrarily into three equal color classes, and then choose every pair of vertices of distinct color classes, randomly and independently, to be edges with probability p. We describe a polynomial-time algorithm that finds a proper 3-coloring of G3n,p,3 with high probability, whenever p @math c n, where c is a sufficiently large absolute constant. This settles a problem of Blum and Spencer, who asked if an algorithm can be designed that works almost surely for p @math polylog(n) n [J. Algorithms, 19 (1995), pp. 204--234]. The algorithm can be extended to produce optimal k-colorings of random k-colorable graphs in a similar model as well as in various related models. Implementation results show that the algorithm performs very well in practice even for moderate values of c. 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 show how the results of Karger, Motwani, and Sudan (1994) and Blum (1994) can be combined in a natural manner to yield a polynomial-time algorithm for O(n314)-coloring any n-node 3-colorable graph. This improves on the previous best bound of O(n14) colors (, 1994). 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. The following three problems concerning random graphs can be solved in (log n)O(1) expected time using linearly many processors: (1) finding the lexicographically first maximal independent set, (2) coloring the vertices using a number of colors that is almost surely within twice the chromatic number, and (3) finding a Hamiltonian circuit. The author presents improved inapproximability results for three problems: the problem of finding the maximum clique size in a graph, the problem of finding the chromatic number of a graph, and the problem of coloring a graph with a small chromatic number with a small number of colors. J. Hastad's (1996) result shows that the maximum clique size in a graph with n vertices is inapproximable in polynomial time within a factor n sup 1- spl epsi or arbitrarily small constant spl epsi >0 unless NP=ZPP. We aim at getting the best subconstant value of spl epsi in Hastad's result. We prove that clique size is inapproximable within a factor n 2((log n)) sup 1-y corresponding to spl epsi =1 (log n) sup spl gamma for some constant spl gamma >0 unless NP spl sube ZPTIME(2((log n)) sup O(1) ). This improves the previous best inapproximability factor of n 2 sup O(log n spl radic log log n) (corresponding to spl epsi =O(1 spl radic log log n)) due to L. Engebretsen and J. Holmerin (2000). A similar result is obtained for the problem of approximating chromatic number of a graph. We also present a new hardness result for approximate graph coloring. We show that for all sufficiently large constants k, it is NP-hard to color a k-colorable graph with k sup 1 25 (log k) colors. This improves a result of M. Furer (1995) that for arbitrarily small constant spl epsi >0, for sufficiently large constants k, it is hard to color a k-colorable graph with k sup 3 2- spl epsi colors. 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. 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. 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. How to color 3 colorable graphs with few colors is a problem of longstanding interest. The best polynomial-time algorithm uses n0.2072 colors. There are no indications that coloring using say O(log n) colors is hard. It has been suggested that SDP hierarchies could be used to design algorithms that use ne colors for arbitrarily small e > 0. We explore this possibility in this paper and find some cause for optimism. While the case of general graphs is till open, we can analyse the Lasserre relaxation for two interesting families of graphs. For graphs with low threshold rank (a class of graphs identified in the recent paper of Arora, Barak and Steurer on the unique games problem), Lasserre relaxations can be used to find an independent set of size Ω(n) (i.e., progress towards a coloring with O(log n) colors) in nO(D) time, where D is the threshold rank of the graph. This algorithm is inspired by recent work of Barak, Raghavendra, and Steurer on using Lasserre Hierarchy for unique games. The algorithm can also be used to show that known integrality gap instances for SDP relaxations like strict vector chromatic number cannot survive a few rounds of Lasserre lifting, which also seems reason for optimism. For distance transitive graphs of diameter Δ, we can show how to color them using O(log n) colors in n2O(Δ) time. This family is interesting because the family of graphs of diameter O(1 e) is easily seen to be complete for coloring with ne colors. The distance-transitive property implies that the graph "looks" the same in all neighborhoods. The full version of this paper can be found at: http: www.cs.princeton.edu rongge LasserreColoring.pdf . For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast @math -coloring of trees requires random bits: Building on the recent lower bounds of , we prove that the randomized complexity of @math -coloring a tree with maximum degree @math is @math , whereas its deterministic complexity is @math for any @math . This also establishes a large separation between the deterministic complexity of @math -coloring and @math -coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in @math rounds can be transformed to run in @math rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires @math time deterministically. 3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size @math is at least its deterministic complexity on instances of size @math . This shows that a deterministic @math lower bound for any problem implies a randomized @math lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model. 1. Fundamental Concepts. What Is a Graph? Paths, Cycles, and Trails. Vertex Degrees and Counting. Directed Graphs. 2. Trees and Distance. Basic Properties. Spanning Trees and Enumeration. Optimization and Trees. 3. Matchings and Factors. Matchings and Covers. Algorithms and Applications. Matchings in General Graphs. 4. Connectivity and Paths. Cuts and Connectivity. k-connected Graphs. Network Flow Problems. 5. Coloring of Graphs. Vertex Colorings and Upper Bounds. Structure of k-chromatic Graphs. Enumerative Aspects. 6. Planar Graphs. Embeddings and Euler's Formula. Characterization of Planar Graphs. Parameters of Planarity. 7. Edges and Cycles. Line Graphs and Edge-Coloring. Hamiltonian Cycles. Planarity, Coloring, and Cycles. 8. Additional Topics (Optional). Perfect Graphs. Matroids. Ramsey Theory. More Extremal Problems. Random Graphs. Eigenvalues of Graphs. Appendix A: Mathematical Background. Appendix B: Optimization and Complexity. Appendix C: Hints for Selected Exercises. Appendix D: Glossary of Terms. Appendix E: Supplemental Reading. Appendix F: References. Indices. Vertex coloring is a central concept in graph theory and an important symmetry-breaking primitive in distributed computing. Whereas degree-Δ graphs may require palettes of Δ + 1 colors in the worst case, it is well known that the chromatic number of many natural graph classes can be much smaller. In this paper we give new distributed algorithms to find ( Δ k ) -coloring in graphs of girth 4 (triangle-free graphs), girth 5, and trees. The parameter k can be at most ( 1 4 - o ( 1 ) ) ln ? Δ in triangle-free graphs and at most ( 1 - o ( 1 ) ) ln ? Δ in girth-5 graphs and trees, where o ( 1 ) is a function of Δ. Specifically, for Δ sufficiently large we can find such a coloring in O ( k + log * ? n ) time. Moreover, for any Δ we can compute such colorings in roughly logarithmic time for triangle-free and girth-5 graphs, and in O ( log ? Δ + log Δ ? log ? n ) time on trees. As a byproduct, our algorithm shows that the chromatic number of triangle-free graphs is at most ( 4 + o ( 1 ) ) Δ ln ? Δ , which improves on Jamall's recent bound of ( 67 + o ( 1 ) ) Δ ln ? Δ . Finally, we show that ( Δ + 1 ) -coloring for triangle-free graphs can be obtained in sublogarithmic time for any Δ. In this paper we study the acyclic 3-colorability of some subclasses of planar graphs. First, we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Then, we show that every planar graph has a subdivision with one vertex per edge that is acyclically 3-colorable and provide a linear-time coloring algorithm. Finally, we characterize the series-parallel graphs for which every 3-coloring is acyclic and provide a linear-time recognition algorithm for such graphs. We study the maximum differential graph coloring problem, in which the goal is to find a vertex labeling for a given undirected graph that maximizes the label difference along the edges. This problem has its origin in map coloring, where not all countries are necessarily contiguous. We define the differential chromatic number and establish the equivalence of the maximum differential coloring problem to that of k-Hamiltonian path. As computing the maximum differential coloring is NP-Complete, we describe an exact backtracking algorithm and a spectral-based heuristic. We also discuss lower bounds and upper bounds for the differential chromatic number for several classes of graphs. We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on q, we find with a one-step replica-symmetry breaking approximation the precise value of the critical average connectivity @math Moreover, we show that below @math there exists a clustering phase @math in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region, i.e., when @math We consider worst case time bounds for several NP-complete problems, based on a constraint satisfaction (CSP) formulation of these problems: (a, b)-CSP instances consist of a set of variables, each with up to a possible values, and constraints disallowing certain b-tuples of variable values; a problem is solved by assigning values to all variables satisfying all constraints, or by showing that no such assignment exist. 3-SAT is equivalent to (2, 3)-CSP while 3-coloring and various related problems are special cases of (3, 2)-CSP; there is also a natural duality transformation from (a, b)-CSP to (b, a)-CSP. We show that n-variable (3, 2)-CSP instances can be solved in time O(1.3645n), that satisfying assignments to (d, 2)-CSP instances can be found in randomized expected time O((0.4518d)n); that 3-coloring of n-vertex graphs can be solved in time O(1.3289n); that 3-list-coloring of n-vertex graphs can be solved in time O(1.3645n); that 3-edge-coloring of n-vertex graphs can be solved in time O(2n 2), and that 3-satisfiability of a formula with t 3-clauses can be solved in time O(nO(1) + 1.3645t). Given a 3-colorable graph G together with two proper vertex 3-colorings α and β of G, consider the following question: is it possible to transform α into β by recoloring vertices of G one at a time, making sure that all intermediate colorings are proper 3-colorings? We prove that this question is answerable in polynomial time. We do so by characterizing the instances G, α, β where the transformation is possible; the proof of this characterization is via an algorithm that either finds a sequence of recolorings between α and β, or exhibits a structure which proves that no such sequence exists. In the case that a sequence of recolorings does exist, the algorithm uses O(|V(G)|2) recoloring steps and in many cases returns a shortest sequence of recolorings. We also exhibit a class of instances G, α, β that require Ω(|V(G)|2) recoloring steps. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 69-82, 2011 © 2011 Wiley Periodicals, Inc. The problem of coloring a graph with the minimum number of colors is well known to be NP-hard, even restricted to k-colorable graphs for constant k ≥ 3. On the other hand, it is known that random k-colorable graphs are easy to k-color. The algorithms for coloring random k-colorable graphs require fairly high edge densities, however. In this paper we present algorithms that color randomly generated k-colorable graphs for much lower edge densities than previous approaches. In addition, to study a wider variety of graph distributions, we also present a model of graphs generated by the semi-random source of Santha and Vazirani (M. Santha and U. V. Vazirani, J. Comput. System Sci.33 (1986), 75-87) that provides a smooth transition between the worst-case and random models. In this model, the graph is generated by a "noisy adversary"-an adversary whose decisions (whether or not to insert a particular edge) have some small (random) probability of being reversed. We show that even for quite low noise rates, semi-random k-colorable graphs can be optimally colored with high probability.