Distance Coloring. Alexa Sharp. Cornell University, Ithaca, NY

Size: px

Start display at page:

Download "Distance Coloring. Alexa Sharp. Cornell University, Ithaca, NY"

Chrystal Simpson
6 years ago
Views:

1 Distance Coloring Alexa Sharp Cornell Universit, Ithaca, NY Abstract. Given a graph G = (V, E), a (d, k)-coloring is a function from the vertices V to colors {1,,..., k} such that an two vertices within distance d of each other are assigned different colors. We determine the complexit of the (d, k)-coloring problem for all d and k, and enumerate some interesting properties of (d, k)-colorable graphs. Our main result is the discover of a dichotom between polnomial and NP-hard instances: for fixed d, the distance coloring problem is polnomial time for k 3d and NP-hard for k > 3d. Ke words: graph coloring, power graphs, complexit threshold 1 Introduction The classic k-coloring problem tries to assign a color from 1 to k to each vertex in a graph such that no two adjacent vertices share the same color [1]. The k- coloring problem, along with man variations and generaliations, is well-studied in both computer science and mathematics [ 6]. Its applications range from frequenc assignment [7] to circuit board testing [8], among others. The distance (d, k)-coloring problem is a generaliation of k-coloring that tries to assign a color from 1 to k to each vertex such that no two vertices within distance d of each other share the same color. Clearl, k-coloring is a special case of (d, k)-coloring with d = 1. Conversel, (d, k)-coloring a graph G is equivalent to k-coloring G d, the dth power graph of G. (The graph G d has the same vertex set as G and an edge between two vertices if and onl if the are within distance d of each other in G.) In this wa (d, k)-coloring is also a special case of k-coloring. Although k-coloring is NP-complete for k 3 and polnomial-time for k [1], the complexit of (d, k)-coloring is not so straightforward. It is NP-complete for large k [9, 10]; other special cases have been studied in depth, such as cubic graphs [11], planar graphs [1], and trees [13, 14]. However, there are man values of d and k for which (d, k)-coloring is polnomial-time; the dichotom between polnomial and NP-hard instances is the subject of this paper. The results are not onl of theoretical interest, but also of practical use in applications with underling structures that fit the power graph model. For example, we ma want to assign k frequencies to a set of radio stations, but require that an two stations within distance d of each other use different frequencies. The main result is significant, as it classifies the complexit of all instances of (d, k)-coloring for d : determining whether a graph is (d, k)-colorable is

2 Alexa Sharp polnomial-time for k 3d, but NP-hard for k > 3d. Moreover, (d, k)- coloring on trees is solvable in polnomial time. This paper presents man opportunities for future work; further improvements on these results, as well as their implications to generalied graph decompositions and complexit are explored in [15]. Preliminaries Given a graph G = (V, E) and integers d 1 and k d + 1, the goal of (d, k) coloring is to find an assignment of k colors to the vertices of G such that no two vertices within distance d of each other share the same color. More precisel, we want a function f : V {1,,..., k} such that d(u, v) d f(u) f(v). We sa G is (d, k)-colorable if such a function exists, and call the function itself a (d, k)-coloring. The standard k-coloring problem is the special case when d = 1. Observe that each connected component of G can be colored independentl; moreover, an component of sie less than or equal to k can be (d, k)-colored b assigning a distinct color to each vertex. Thus in this paper we assume that G is connected and has at least k + 1 vertices. Definition 1. For a vertex v and integer r, let G r v G be the subgraph of radius r around v, that is, the subgraph induced b {w d(w, v) r}. Definition. For a subgraph G G, the diameter of G, denoted diam(g ), is the maximum shortest path distance between an two vertices of G, that is, diam(g ) = max{d(u, v) u, v G }. Definition 3. Given a connected graph G = (V, E), the set V V is a cutset if the subgraph induced b V \ V is no longer connected. Definition 4. Given a graph G of sie at least k +1, a graph G = (V, E ) G is a forbidden (d, k) subgraph if diam(g ) min{d, V (k d) 1}..1 Properties of (d, k)-colorable Graphs Theorem 1. If G contains a forbidden (d, k) subgraph then it is not (d, k)- colorable. Proof. Suppose G = (V, E ) is a subgraph of G with diam(g ) min{d, V (k d) 1}. Let G be a connected graph induced b V and max{0, k +1 V } vertices of V \V ; G has k+1 vertices and diameter at most d, which precludes a (d, k)-coloring. Corollar 1. If there is a vertex v G such that G r v (k d) + r + 1 for an 1 r d then G cannot be (d, k)-colored. Proof. The diameter of G r v is at most r d. But r = (k d + r + 1) (k d) 1 G r v (k d) 1 and Theorem 1 completes the proof.

3 Distance Coloring 3 v (a) v (b) v (c) Fig. 1. Examples of Corollar 1 for (6, 9)-coloring. The graphs of 1(a), 1(b) cannot be (6, 9)-colored because G 3 v = 10; the graph of 1(c) is (6, 9)-colorable because G r v r + 3 for all r 3. Theorem. Given a (d, k)-colorable graph G with k 3d, if Gk d v is a strict subgraph of G for some v G, then G k d v contains a cutset of sie disconnecting v from G \ G k d v. Proof. First observe that an vertex G \ G k d v is connected to v through at least one vertex of each G i v \ G i 1 v for 1 i k d. We will show there is 1 i k d such that G i v \ G i 1 v. Then the removal of G i v \ Gv i 1, of at most two vertices, would disconnect from v. B wa of contradiction, suppose G i v \ G i 1 v 3 for all 1 i k d. Then G 0 v = 1 implies G i v 3i + 1. Hence G k d v (k d) + (k d) + 1, and G is not (d, k)-colorable b Corollar 1, a contradiction. Lemma 1. Let P be a path of length 0 p d, with left and right endpoints v L and v R, respectivel, in a (d, k)-colorable graph G, k 3d. Suppose there exist disjoint subgraphs P L, P R in G \ P such that P L {v L }, P R {v R } are connected and P L, P R d ; let O be the non-p vertices connected to P in the graph G \ (P L P R ). Then O k d 1. Proof. See Appendix for the proof and an illustration of the notation. 3 Negative Complexit Results We know that (1, k)-coloring is NP-hard for k 3; we now use the results of Section to show that for d, (d, k)-coloring is NP-hard for k > 3d. Theorem 3. The (d, k)-coloring problem is NP-hard for d, k > 3d.

4 4 Alexa Sharp Theorem 3 follows via a reduction from (1, k)-coloring (k-col). Given a graph G that we wish to k-color, we construct a graph G such that G is k-colorable if and onl if G is (d, k)-colorable. The building block of this reduction is a triangle gadget G, which is shown in Figure. x d d d x d d d k 3d 1 k 3d 1 (a) (b) Fig.. (a) and (b) show Theorem 3 s G gadget for odd and even d, respectivel. Lemma. If G is (d, k)-colorable then x, and have the same color. Proof. The k 1 vertices in G \ {x,, } are within distance d of each other, and use k 1 distinct colors. The vertices x, and are within distance d of all these k 1 colors, but distance d + 1 from each other. If G is (d, k)-colorable then x, and are colored the same. Reduction from k-col. Given an instance G = (V, E) of k-col, create the graph G = (V, E ) as follows. For each vertex u V, create gadget G u G b concatenating deg(u) k 4 copies of G, overlapping the x and vertices, alwas leaving the vertices open. Ever k 4 th vertex is reserved for use as follows: for each edge e = (u, v) G, create an edge (u e, v e ) G where u e and v e are reserved vertices of G u and G v, respectivel; an example of this reduction is shown in Figure 3. Note that G is polsie, as G = k + and it is copied u V deg(u) k4 times. Lemma 3. If G is (d, k)-colorable then u e, u f G have the same color for all edges e, f incident to u G. Proof. B Lemma we know that G u s x, and vertices must be the same color. The vertices u e and u f are simpl vertices, and the result follows. Lemma 4. If G is (d, k)-colorable then for edge e = (u, v) G, u e, v e G are different colors. Proof. If e = (u, v) G then (u e, v e ) G. Thus d(u e, v e ) d and u e, v e are different colors in G.

5 Distance Coloring 5 v e u t f g w v e u e k 4 1 k 4 1 w f u f t g ug (a) (b) Fig. 3. 3(a) An instance G of k-col. 3(b) A subgraph of the graph G constructed from G in Theorem 3. Lemma 5. If G is (d, k)-colorable then G is k-colorable. Proof. If G has a (d, k)-coloring C then create a k-coloring D of G b setting D(u) = C(u e ) for an e incident to u. Since C(u e ) = C(u f ) for all e, f incident to u b Lemma 3, D is well-defined. Moreover, D(u) D(v) for (u, v) G b Lemma 4, and D uses at most k colors. Lemma 6. If G is k-colorable then G is (d, k)-colorable. The proof of Lemma 6 requires some additional framework. Consider the colors of the vertices of G \{x,, }, labeled as in figure 4(a). If we use the coloring of this G to color an adjacent G as shown in Figures 4(b) or 4(c) then we will have defined a smmetr group based on the colors of G \ {x,, }; the base set is A = {a 1, a,..., a k 1 } and we have two elements σ, π : A A, where σ is the shift operator (a 1 a a k 1 ) and τ is the adjacent transposition operator (a 1 a )(a 3 )(a 4 ) (a k 1 ). Lemma 7. Appling either the shift σ or the transposition τ to the colors of G ields a valid coloring. Proof. The vertices of the same color in adjacent G s are at least distance d + 1 apart. Lemma 8. An adjacent transposition π = (a j a j+1 ) can be generated from k compositions of σ and τ. Proof. Use the shift σ j 1 times, transpose with τ once, then shift again (k 1) (j 1) times. The onl elements transposed are j and j + 1; the remaining elements shift k 1 times and hence are unchanged. Lemma 9. An transposition π = (a i a j ) can be generated from k compositions of adjacent transpositions. Proof. Without loss of generalit, suppose i < j. Then (a i a j ) = (a j 1 a j )(a j a j 1 ) (a i a i+1 )(a i+1 a i+ ) (a j a j 1 )(a j 1 a j ), which uses (j i) 1 k transpositions, as required.

6 6 Alexa Sharp a a 4 x a 3 a 5 +1a d+1 a k 1 a 1 +1 a d+ a d+3 (a) a shift σ a 1 a 4 a 3 x a 3 a 5 a k 1 a 1 +1 a d+1 +1 a a 4 a a k 1 a d+ a d+3 (b) a d+1 a d+ a d+3 a a 4 adjacent transposition τ a 1 a 4 x a 3 a 5 a k 1 a 1 +1 a d+1 +1 a 3 a +1 a d+1 5 a k 1 a +1 a d+ a d+3 (c) a d+ a d+3 Fig. 4. 4(a) The set A for the basis of the permutation group for odd d. The same labeling can be used for even d, ignoring the center vertex. 4(b) The shift operator σ. 4(c) The adjacent transposition operator τ. π p ) can be generated from p composi- Lemma 10. An ccle π = (π 1 π tions of transpositions. Proof. Note that (π 1 π π p ) = (π 1 π p )(π 1 π p 1 ) (π 1 π 3 )(π 1 π ). Lemma 11. An permutation on k elements can be generated from k 4 compositions of the shift σ and the adjacent transposition τ.

7 Distance Coloring 7 Proof. First note that an permutation on A can be generated b at most k ccles of length k. B Lemmas 8-10 we have that k ccles of length k are generated b k transpositions, which are generated b k 3 adjacent transpositions, which are generated b k 4 compositions of σ and τ. Thus an permutation is generated b k 4 compositions of σ and τ. Proof of Lemma 6. Given a k-coloring D of G we show how to (d, k)-color the gadgets of G. First, for each vertex u G, color all of G u s x, and vertices with the color D(u). This is feasible because these vertices are distance d + 1 apart, and their color is different from all neighboring gadgets x,, colors. Next, for each vertex u G, color the G s directl connected to G v for some v u G. Now the remaining uncolored G gadgets in G u are chains of k 4 1 G s between two colored G s. We know from Lemma 11 that there is some chain of at most k 4 compositions of σ and τ that lead from an one coloring of G to the next, and each composition corresponds to a valid coloring of each G. Thus we know there is a sequence of colorings getting us from one G to the one distance k 4 awa, and G can thus be (d, k) colored. Proof of Theorem 3. The polnomial reduction constructs a graph G that is (d, k)-colorable if and onl if G is k-colorable, b Lemmas 5 and 6. Therefore (d, k)-coloring is NP-complete for d and k 3d Positive Complexit Results Thus (d, k)-coloring is NP-hard for d, k > 3d ; we now show that the remaining parameters lead to polnomial-time solutions. 4.1 (d, d + 1)-coloring Theorem 4. The (d, d + 1)-coloring problem is polnomial-time solvable. Proof. A graph is (1, )-colorable if and onl if it is bipartite. For d, if G contains a vertex of degree 3 or greater, then it is not (d, d + 1)-colorable b Corollar 1. Otherwise, G is a path or ccle. If it is a ccle but its length is not a multiple of (d + 1) then it is not (d, d + 1)-colorable. Otherwise G is a path or a ccle whose length is a multiple of (d + 1). In either case, ccle through the colors 1,,..., (d + 1), which ensures that vertices within distance d of each other are colored differentl. 4. (d, k)-coloring for k 3d We describe an algorithm polnomial in G that either produces a (d, k)-coloring of G or declares that no such coloring exists. The algorithm finds a bounded treewidth tree decomposition of G [16], then colors the graph using known coloring algorithms for bounded tree-width graphs [17, 18].

8 8 Alexa Sharp Definition 5. A tree decomposition of G = (V, E) is a triple (T, F, X) consisting of an undirected tree (T, F ) and a map X : T V associating a subset X i V with each i T such that I. V = i T X i; II. for all edges (u, v) E, there exists i T such that u, v X i ; III. if j lies on the path between i and k in (T, F ), then X i X k X j. Definition 6. The width of (T, F, X) is max i { X i 1}. Definition 7. A path decomposition (T, F, X) is a tree decomposition in which the graph (T, F ) is a simple path. We first tr to compute a path decomposition of the graph G. Let P be a simple path of length d + 1 in G, and let s be a center vertex; if no such path exists then diam(g) d and G is not (d, k)-colorable. Otherwise, perform a breadth-first search on G starting from s to get level sets L 0, L 1,..., L m, where L i consists of the set of vertices of distance i from the root s. The level graph H is the graph consisting of vertices V and directed edges from L i to L i+1, ignoring edges between vertices on the same level. We take L j = for j > m. For 0 i max{0, m d}, let X i def = i+d j=i Lemma 1. If G is (d, k)-colorable, then X i 5d. L j. (1) Proof. Label the right side of P as s = s R 0, s R 1,..., s R m R and the left side of P as s = s L 0, s L 1,..., s L m L such that s R i, sl i L i (where m L, m R d because l(p ) d + 1). For 0 i m, let Ti R, T i L be the subgraphs of the level graph rooted at s R i, sl i, respectivel, consisting of all vertices reachable from sr i, sl i b a (directed) path in H. Let T L = H \T1 R, T R = H \T1 L. Note that H = T L T R. See Figure 5. To obtain the result for X i, we bound X i T R and X i T L separatel, showing X i T R, X i T L 5d. The arguments are identical except for notation, so without loss of generalit we argue onl the former. For X i T R, let j = min{m R, i + d} d and l = max{0, i (k d)} (see Figure 6). Then j 0 and l i m R d, and X i T R = X i T R i + X i (T R \ T R i ) () = X i T R j + X i (T R i \ T R j ) + X i (T R \ T R i ) (3) X i T R j + X i (T R i \ T R j ) + X i (T R l \ T R i ), (4) where the last inequalit follows because an vertex v (X i T R ) \ Tl R is at least distance i l + 1 (k d) from its connection point on P, which has d and d vertices of P on either side of it; this forms a forbidden subgraph, therefore no such v exists since G is (d, k)-colorable.

9 Distance Coloring 9 s R 1 s R s R m R T R s R 0 = s = s L 0 T L s L 1 s L s L m L Fig. 5. Notation used in Lemma 1. s R l d s R i s R j s R min{m R,i+d} X i T R Fig. 6. The above inequalities shown pictoriall. The light gra triangle is X i T R j. The dark gra triangles are X i (T R i \ T R j ). The small white triangle is X i (T R l \ T R i ). For X i Tj R, note that there are d vertices of P \ T j R within d of sr j. Either G is not (d, k)-colorable or there are k d d vertices in T j R within distance d of sr j ; since T j R X i is precisel this set of vertices, Tj R X i d. For X i (Ti R \ Tj R), consider the interval of length p = j i d from sr i to s R j. There are at least d vertices in P \ T i R and P Tj R, respectivel, thus b Lemma 1, either G is not (d, k)-colorable or (Ti R \ Tj R ) \ P k d 1, and X i (Ti R \ Tj R) k d 1 + p + 1 k d d. For X i (Tl R \ T i R) consider the interval sr l to sr i of length p = i l d. There are d vertices in P \ T l R and P Ti R, respectivel, and either G is not (d, k)-colorable, or Tl R \ Ti R \ P k d 1, and X i (Tl R \ Ti R) d. Thus X i T R 5d, and X i H = X i 5d, as required. Theorem 5. For fixed constants d, k with k 3d, there is an O(n) algorithm for finding a (d, k)-coloring of G if one exists. Proof. We use the algorithm of Section 4. to either find a path decomposition (T, F, X) of G of width at most 5d or determine that G is not (d, k)-colorable.

10 10 Alexa Sharp Assume the former. Then (T, F, X) is also a path decomposition of the dth power graph G d. The algorithms of [17, 18] can determine the (1, k)-colorabilit of G d in linear time, and a (1, k)-coloring of G d is precisel a (d, k)-coloring of G. 5 Coloring on Trees Although it is alread known that (d, k)-coloring trees is polnomial-time solvable [13], the results of this paper lead to a nice combinatorial method to achieve the same result. This does not require d or k to be fixed. Algorithm to Find a (d, k)-coloring for a Tree T. 1. If d = 1 and k then we (1, )-color T, a bipartite graph.. Otherwise, perform a breadth-first search on T, starting from a leaf node s to get level sets L 0, L 1,..., L m, where L i consists of the set of vertices of distance i from the root s. 3. For i = 0, 1,,..., m, color the vertices of L i in some arbitrar order b assigning v L i the lowest color not et used in G d v. 4. If we need more than k colors then T is not (d, k)-colorable, otherwise return the assigned coloring. Lemma 13. The algorithm returns a coloring of T if and onl if T is (d, k)- colorable. Proof. [ ] Consider a coloring found b the algorithm; it uses at most k colors. Moreover, no two vertices within distance d of each other are assigned the same color. Indeed, consider an two vertices u and v such that d(u, v) d, and suppose u preceded v in the algorithm. Then when v is considered, u s color is excluded from consideration; consequentl, the algorithm will not assign v the color that it used for u. [ ] Suppose T is (d, k)-colorable. Consider one of the vertices v, and suppose there are t vertices alread colored within distance d of v. Then these t vertices along with v are within distance d of each other b properties of the BFS tree, and hence t+1 k necessaril. It follows that this set of t vertices uses t k 1 colors, and so there is a color that can be assigned to v without exceeding the allotted k colors. This work was supported b NSF grant CCF An views and conclusions expressed herein are those of the author and do not necessaril represent the official policies or endorsements of the National Science Foundation or the United States government. Acknowledgments. I would like to thank Dexter Koen for his man useful comments and insights, as well as the anonmous reviewers for their helpful suggestions.

11 Distance Coloring 11 References 1. Gare, M.R., Johnson, D.S.: Computers and Intractabilit; A Guide to the Theor of NP-Completeness. W. H. Freeman & Co. (1990). Bond, J., Murt, U.: Graph Theor with Applications. The MacMillan Press Ltd (1978) 3. Fiorini, S., Wilson, R.J.: Edge-colourings of graphs. In Beineke, L.W., Wilson, R.J., eds.: Selected Topics in Graph Theor. Academic Press, Inc., London (1978) Bollobás, B., Harris, A.J.: List-colourings of graphs. Graphs and Combinatorics 1 (1985) Wilson, B.: Line-distinguishing and harmonious colourings. In Nelson, R., Wilson, R.J., eds.: Graph Colourings. Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Longman house, Burnt Mill, Harlow, Essex, UK (1990) Chetwnd, A.: Total colourings of graphs. In Nelson, R., Wilson, R.J., eds.: Graph Colourings. Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Longman house, Burnt Mill, Harlow, Essex, UK (1990) Gamst, A.: Some lower bounds for a class of frequenc assignment problems. IEEE Trans. Veh. Technol. VT-35 (1986) Gare, M.R., Johnson, D.S., So, H.C.: An application of graph coloring to printed circuit testing. IEEE Transactions on Circuits and Sstems Vol. CAS-3 (1976) Lin, Y.L., Skiena, S.S.: Algorithms for square roots of graphs. SIAM J. Discret. Math. 8 (1995) McCormic, S.T.: Optimal approximation of sparse hessians and its equivalence to a graph coloring problem. Math. Programming 6 (1983) Heggernes, P., Telle, J.A.: Partitioning graphs into generalied dominating sets. Nordic J. of Computing 5 (1998) Agnarsson, G., Halldórsson, M.M.: Coloring powers of planar graphs. In: Proceedings of the 11th annual ACM-SIAM smposium on Discrete algorithms, Philadelphia, PA, USA, Societ for Industrial and Applied Mathematics (000) Bertossi, A.A., Pinotti, M.C., Rii, R.: Channel assignment on strongl-simplicial graphs. In: IPDPS 03: Proceedings of the 17th International Smposium on Parallel and Distributed Processing, Washington, DC, USA, IEEE Computer Societ (003). 14. Agnarsson, G., Greenlow, R., Halldórsson, M.: On powers of chordal graphs and their colorings. Congressus Numerantium 100 (000) Koen, D., Sharp, A.: On distance coloring. Technical Report cul.cis/tr , Cornell Universit (007) 16. Robertson, N., Semour, P.D.: Graph minors. ii. algorithmic aspects of tree-width. J. Algorithms 7 (1986) Andrejak, A.: An algorithm for the tutte polnomials of graphs of bounded treewidth. Discrete Math. 190 (1998) Noble, S.D.: Evaluating the tutte polnomial for graphs of bounded tree-width. Comb. Probab. Comput. 7 (1998) Appendix Lemma 14. Let P be a path of length 0 p d, with left and right endpoints v L and v R, respectivel, in a (d, k)-colorable graph G, k 3d. Suppose there exist

12 1 Alexa Sharp disjoint subgraphs P L, P R in G \ P such that P L {v L}, P R {v R} are connected and P L, P R d ; let O be the non-p vertices connected to P in the graph G\(PL PR). Then O k d 1. Proof. Label P as v L = v 0, v 1,..., v p = v R. Let O 0, O 1,..., O p be a partition of O such that the induced subgraph O i {v i} is connected for all 0 i p. See Figure 7. O 1 O O P L P R v L = v 0 v 1 v v 3 v R = v p Fig. 7. Notation for Lemma 14. Note that O i k d 1 d 1 for all i 0 otherwise d G v i forbidden subgraph and G not (d, k)-colorable b Theorem 1. Let would be a d L = max{ d max{ Oi + i, d }, max i 0 { Oi i}} d i 0 p. (5) Consider the subgraph G consisting of P, the d L d closest vertices of PL to vl and the d p dl d closest vertices of PR to vr; G consists of d +1 vertices within distance d of each other. The vertices of O are within dl + max j 0{ O j + j} of the G P L vertices, where d L + max j 0 max{ Oi } + max{ Oj + j}, i 0 j 0 (6) { Oi i} + max{ Oj + j}} (7) max i 0 j 0 max{ d, max { Oi, Oi + Oj + p}} (8) i 0,j i max{ d, O + p}. (9) Similarl, the vertices of O are within d p dl + max j 0{ O j + p j} of the G P R vertices, where d dl + max j 0 max{ d max{ Oi } i 0 (10) max{ Oj j}, 0} j 0 (11) d. (1) Lastl, the vertices of O are at most distance p + O from each other. Thus we have d O vertices within distance max{ d, O + p} of each other, and hence O k d 1 otherwise there is a forbidden subgraph.

On Distance Coloring

On Distance Coloring A review based on work with Dexter Kozen Alexa Sharp Oberlin College, Oberlin, OH Abstract. An undirected graph G = (V, E) is (d, k)-colorable if there is a vertex coloring using at