DSATUR. Tsai-Chen Du. December 2, 2013

Transcription

1 DSATUR Tsai-Chen Du December 2, 2013 Abstract The Graph Coloring Problem (GCP) is a well-known NP-complete problem that has been studied extensively. Heuristics have been widely used for the GCP. The well-known Greedy method is the simplest algorithm which takes an ordering of nodes of a graph and colors these with the smallest color satisfying the constraints that no adjacent nodes are assigned same colors. However, the Greedy method performs poorly in practice. DSATUR uses a heuristic which changes the ordering of nodes and then uses the Greedy method to color these nodes. 1 Application Vertex coloring arises in a variety of scheduling and clustering applications. Compiler optimization is the canonical application for coloring, where we seek to schedule the use of a finite number of registers. In a program fragment to be optimized, each variable has a range of times during which its value must be kept intact, in particular, after it is initialized and before its final use. Any two variables whose life spans intersect cannot be placed in the same register. Construct a graph where there is a variable associated with eachvertex and add an edge between any two vertices whose variable life spans intersect. A coloring of the vertices of this graph assigns the variables to classes such that two variables with the same color do not clash and so can be assigned to the same register. No conflicts can occur if each vertex is colored with a distinct color. However, our goal is to find a coloring using the minimum number of colors, because computers have a limited number of registers. The smallest number of colors sufficient to vertex color a graph is known as its chr omatic number. 1

2 Several special cases of interest arise in practice: * Can I color the graph using only two colors? - An important special case is testing whether a graph is bipartite, meaning it can be coloredusing two different colors. Such a coloring of the vertices of a bipartite graph means that the graph can be drawn with the red vertices on theleft and the blue vertices on the right such that all edges go from left to right. Bipartite graphs are fairly simple, yet they arise naturally in such applications as mapping workers to possible jobs. * Testing whether a graph is bipartite is easy. Color the first vertex blue, and then do a depth-first search of the graph.whenever we discover a new, uncolored vertex, color it opposite that of its parent, since the same color would cause a clash. If we ever find an edge where both vertices have been colored identically, then the graph cannot be bipartite. Otherwise, this coloring will be a 2-coloring, and it is constructed in O(n+m) time. * Is the graph planar, or are all vertices of low degree? - The famous 4-color theorem states that every planar graph can be vertex coloredusing at most 4 distinct colors. Efficient algorithms for finding a 4-coloring are known, although it is NP-complete to decide whether a given planar graph is 3-colorable. * There is a very simple algorithm that finds a vertex coloring of any planar graph using at most 6 colors. In any planar graph, there exists a vertex of degree at most five. Delete this vertex and recursively color the graph. This vertex has at most five neighbors, which means that it can always be colored using one of the six colors that does not appear as a neighbor. This works because deleting a vertex from a planar graph leaves a planar graph, so we always must have a lowdegree vertex to delete. * Is this an edge coloring problem? - Certain vertex coloring problems can be modeled as edge coloring, where we seek to color the edges of a graph G such that no two edges with a vertex in common are colored the same. The payoff is that there is an efficient algorithm that always returns a near-optimal edge coloring. * Computing the chromatic number of a graph is NP-complete, so if you need an exact solution you must resort to backtracking, which can be surprisingly effective in coloring certain random graphs. It remains hard to compute a provably good approximation to the optimal coloring, so expect no guarantees. 2

3 2 Introduction DSATUR dynamically chooses the vertex to color next, by picking the first vertex, in the given input ordering, that maximizes a given score. DSATUR is an exact method for finding an optimal coloring for a graph by implicitly enumerating all possible colorings. It is a sequential algorithm in which nodes are chosen based on the degree of saturation: the number of dierent colors used for its neighbours in the current solution. Harder nodes (higher degree of saturation) are chosen first. Easiest node: We first select the easiest node (lowest degree of saturation) to be colored in each partition, and then we apply standard criteria to pick the hardest one from that set. Hardest partition: We rst pick the hardest partition to be colored according to its degree of saturation, size and uncolored nodes; and then we pick the easiest node (lowest degree) from that partition. v 2 v 3 v 0 v 1 v 4 v 5 Figure 1: Greedy graph. 3 History The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps. While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecture, noting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Guthrie s brother passed on the question to his mathematics teacher Augustus de Morgan at University College, who mentioned it in a letter to William Hamilton in Arthur Cayley raised the problem at a meeting of the London Mathematical Society in The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London 3

4 Mathematical Society.[1] In 1890, Heawood pointed out that Kempe s argument was wrong. However, in that paper he proved the five color theorem, saying that every planar map can be colored with no more than five colors, using ideas of Kempe. In the following century, a vast amount of work and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in 1976 by Kenneth Appel and Wolfgang Haken. The proof went back to the ideas of Heawood and Kempe and largely disregarded the intervening developments.[2]the proof of the four color theorem is also noteworthy for being the first major computer-aided proof. In 1912, George David Birkhoff introduced the chromatic polynomial to study the coloring problems, which was generalised to the Tutte polynomial by Tutte, important structures in algebraic graph theory. Kempe had already drawn attention to the general, non-planar case in 1879,[3] and many results on generalisations of planar graph coloring to surfaces of higher order followed in the early 20th century. In 1960, Claude Berge formulated another conjecture about graph coloring, the strong perfect graph conjecture, originally motivated by an informationtheoretic concept called the zero-error capacity of a graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem in 2002 by Chudnovsky, Robertson, Seymour, Thomas Graph coloring has been studied as an algorithmic problem since the early 1970s: the chromatic number problem is one of Karp s 21 NP-complete problems from 1972, and at approximately the same time various exponential-time algorithms were developed based on backtracking and on the deletion-contraction recurrence of Zykov (1949). One of the major applications of graph coloring, register allocation in compilers, was introduced in Four Color Problem seems to have been mentioned for the first time in writing in an De Morgan to W.R. Hamilton. Nobody thought at that time that it was the beginning of a new theory. The first proof was given by Kempe in 1879.It stood for more than 10 years until Heawood in 1890 found a mistake. The chromatic number problem is one of Karp.s 21 NP-complete problems from 1972,and at approximately the same time various exponential-time algorithms weredeveloped based on backtracking and on the deletion-contraction recurrence of Zykov (1949) 4

5 4 Greedy algorithm DSATUR procedure greedy algorithm(v ) Give an initial ordering of vertices as V = v1, v2...vn; Find a largest clique V of G, assign each vertex in V a distinct color class; V = V V ; while V NULL do Find a vertex v in V, which is adjacent to the largest number of distinctly colored vertices, assign v to the lowest indexed color class that contains no vertices adjancent to v ; if(no existing color class to assignment to) create a new color class for v ; Move v out of V ; end while Return color classes end procedure 5 Conclusions Greedy method is the simplest which takes an ordering of nodes of a graph and colors these with the smallest color satisfying the constraints that no adjacent nodes are assigned same colors. However, the Greedy method performs poorly in practice. DSATUR uses a heuristic which changes the ordering of nodes and then uses the Greedy method to color these nodes. References [1] Barenboim, L.; Elkin, M. (2009), Distributed ( + 1)- coloring in linear (in ) time, Proceedings of the 41st Symposium on Theory of Computing, of_computing pp , doi: / , ISBN [2] Panconesi, A.; Srinivasan, A. (1996), On the complexity of distributed network decomposition, Journal of Algorithms 20 [3] Schneider, J. (2010), A new technique for distributed symmetry breaking, pdf Proceedings of the Symposium on Principles of Distributed Computing [4] W. Klotz, Graph Coloring Algorithms, de/arbeitsgruppen/diskrete-optimierung/publications/2002/gca. pdf 5

6 [5] P. S. Segundo, A new DSATUR-based algorithm for exact vertex coloring, [6] E.H.Norman, Heuristic for Graph Coloring, Japan s emergence as a modern state 1940: International Secretariat, Institute of Pacific Relations, http: //shah.freeshell.org/graphcoloring/ [7] A. Handrizal and A. N. Agdalla, Comparison between Vertex Merge Algorithm and Dsatur Algorithm, Malaysia:Science Publications,2011. Web. 7 Oct