JOURNAL OF APPLIED COMPUTER SCIENCE Vol. 22 No. 2 (2014), pp. 19-30 The Backbone Coloring Problem for Small Graphs Robert Janczewski, Krzysztof Turowski Gdansk University of Technology Department of Algorithms and Systems Modelling Narutowicza 11/12, 80-233 Gdańsk, Poland skalar@eti.pg.gda.pl, Krzysztof.Turowski@pg.gda.pl Abstract. In this paper we investigate the values of the backbone chromatic number, derived from a mathematical model for the problem of minimization of bandwidth in radio networks, for small connected graphs and connected backbones (up to 7 vertices). We study the relationship of this parameter with the structure of the graph and compare the results with the solutions obtained using the classical graph coloring algorithms (LF, IS), modified for the backbone coloring problem. Keywords: frequency assignment, graph coloring, backbone chromatic number. 1. Introduction The backbone coloring problem was introduced by Broersma in [1] as a generalization of the classical vertex graph coloring problem. The problem is a model for the frequency assignment problem in which we are given a set of transmitters This project has been partially supported by Narodowe Centrum Nauki under contract DEC- 2011/02/A/ST6/00201.
20 The Backbone Coloring Problem for Small Graphs and receivers and we want to assign them frequencies in such a way that interference is avoided. Transmitters and receivers are modelled by vertices, the interference is modelled by edges of two types (the model takes into account two types of interference) and the frequencies are modelled by integers (colors). Formally, the backbone coloring problem is defined as follows: for a given graph G, its given spanning subgraph H and positive integer λ 2 by a λ-backbone coloring of graph G with backbone H we mean any function c : V(G) N + such that the following conditions hold: c(u) c(v) if uv E(G), c(u) c(v) λ if uv E(H). We want to find optimal λ-backbone coloring of graph G with backbone H, i.e. a λ-backbone coloring c of G with backbone H that minimizes max c(v(g)). The minimal value of max c(v(g)) over all possible λ-backbone colorings c of G with backbone H is denoted by BBC λ (G, H) and is called the λ-backbone chromatic number of G with backbone H. The value of λ-backbone chromatic number is typically bounded by a function of chromatic number of the graph χ(g) and the parameter λ. Basic relationship between these numbers as presented by Broersma in [1]: χ(g) BBC λ (G, H) λ(χ(g) 1) + 1 (1) The bound was improved in [2]: λ(χ(h) 1) + 1 BBC λ (G, H) λ(χ(h) 1) + n χ(h) + 1 (2) where χ(h) is the chromatic number of the backbone and n is the number of vertices of G and H. Moreover, it was proved that the λ-backbone chromatic number is a linear function of λ for large values of this parameter (exceeding the number of the vertices in the graph). The other, tighter bounds were proved for certain graph classes, i.e. planar graphs and split graphs with different backbones. For fixed λ Broersma et al. proved in [3] that the λ-backbone chromatic number for split graphs with matching backbone is bounded from above as follows: ( BBC λ (G, H) 2 2 ) χ(g) + O(1) (3) λ + 1
R. Janczewski, K. Turowski 21 In the case of split graphs with pairwise vertex-disjoint stars backbone the similar bound was proven in the same paper: ( BBC λ (G, H) 2 1 ) χ(g) + O(1) (4) λ The special case λ = 2 was also investigated in terms of relation between the backbone chromatic number, maximum degree of the graph and degeneracy of the backbone. For an arbitrary graph G with maximum degree (G) and its d- degenerated backbone H the following inequality was proved by Miskuf et al. in [4]: BBC 2 (G, H) (G) + d + 1 (5) This bound is important since all frequently studied classes of backbones (matchings, stars, trees) are 1-degenerate. The above inequality could be simply generalized for any value of λ 2: BBC λ (G, H) (λ 1) (G) + d + 1 (6) All above bounds are tight. Most of the works related to the BBC problem concentrate on small values of λ. The computation of the exact value of BBC λ (G, H) is NP-hard in the strong sense (so it cannot be computed in a reasonable time for large graphs unless P = NP) even for complete graphs, we had to limit ourselves to small connected graphs with connected backbones. 2. Tests In [2] we proved that the backbone chromatic number can be expressed for λ n χ(g) + 1 as: BBC λ (G, H) = λ(χ(h) 1) + χ(g; H) (7) where χ(g; H) is a parameter (named the bichromatic number of the graph G with backbone H) independent of λ. This relation suggests the investigation of relation between the λ-backbone chromatic number and the chromatic number of the backbone graph and the parameter λ for small values of λ (not exceeding n). We decided to compute optimal values for all non-isomorphic connected graphs up to 7 vertices with all possible
22 The Backbone Coloring Problem for Small Graphs connected backbones. We restricted the search due to the number of cases (about 8 millions for n 7 and over a billion for n = 8) together with the necessity of using the exact, exponential algorithms in each case. The main purpose of the study was to find answers for the following questions about the linearity of the value of BBC λ (G, H) with respect to the value of λ: for how many graphs does the equation 7 holds also for small values of λ? And if the value of BBC λ (G, H) is not linear for small values, into how many linear parts the relation between the λ-backbone chromatic number and the parameter λ can be divided? we know that BBC λ (G, H) BBC λ 1 (G, H) is true for all G, H and λ 3 but does always the slope rise, i.e. BBC λ (G, H) BBC λ 1 (G, H) BBC λ 1 (G, H) BBC λ 2 (G, H)? Maybe there are examples of G and H, when BBC λ (G, H) BBC λ 1 (G, H) > BBC λ 1 (G, H) BBC λ 2 (G, H) + k even for large values of k? Or does the slope always rise by at most χ(h) 1, when λ increases by 1? Furthermore, we compared the exact values of the λ-backbone chromatic numbers against the approximations given by the equations 1, 2 and 7 and looked for the families of graphs, for which these bounds are tight possibly other than the trivial ones such as (K n, K n ). The other important field of our study are the approximation algorithms, derived from the well-known algorithms for the classical vertex coloring problem. It is known that: Theorem 1 For every λ-backbone coloring instance, there exists an ordering of the vertices of a graph for which the greedy algorithm gives an optimal solution. Therefore, the greedy algorithm, executed over all possible arrangements of vertices, is guaranteed to return the optimal value. However, this is supposedly not the case of another algorithms: first one based on Largest First (LF) ordering (here, during the computation of degrees of the vertices we count each backbone edge λ times normal edges) and the second one based on Independent Set (IS) approach (we sequentially find the uncolored vertices of the graph with the lowest available color). To be precise, we implemented all algorithms in their exponential versions (branching at each possible tie), returning both the minimum and the maximum number of colors used by the chosen method of coloring over all of the orderings of vertices of G. We investigated the quality of LF and IS algorithms in the best and
R. Janczewski, K. Turowski 23 Table 1. The linearity of the value of BBC for arbitrary backbones n linear parts 1 2 3 4 100% 0% 0% 5 98.11% 1.89% 0% 6 89.35% 10.65% 0% 7 83.17% 16.69% 0.14% worst case and we tried to establish the families of graphs that are particularly easy/hard to color for these algorithms. All of the above questions were also investigated in the special case of bipartite and tree backbones. These classes are important subclasses of graphs in backbone coloring problem, since their properties are highly restricted, since χ(h) = 2. 3. Results for arbitrary backbones First, we present the results obtained for the general graphs with no restriction on the structure of their backbones. It turned out that in many cases the linearity of the value of BBC λ (G, H) was violated only in the case λ = 1, so we decided to omit this value and pick λ = 2 as the boundary case. For small graphs, the majority of the tested graphs fit the equation 7, however we can see that the percentage of the graphs with more linear parts increases with the number of vertices. Interestingly, for 0.23% of the graphs on 6 vertices and 0.44% of the graphs on 7 vertices, there is a slope decrease, i.e. the graphs do not satisfy the inequality BBC λ (G, H) BBC λ 1 (G, H) BBC λ 1 (G, H) BBC λ 2 (G, H). One example of such instance is presented in Fig. 1. Similarly, for the 0.23% fraction of the graphs on 6 vertices (for example, the graph in Fig. 1) and 0.32% fraction of the graphs on 7 vertices the value of BBC λ (G, H) BBC λ 1 (G, H) is equal to χ(h) and therefore even for the small graphs the slope may rise by more than χ(h) 1, suggested by the equation 7. It is worth noting that the values in Table 1 and Table 2 are quite similar but not identical. This leads us to the conjecture that for most of the graphs both the maximum slope and the number of the linear parts of the BBC λ (G, H) (parametrized by λ) is given by the similar relation, based on the structure of G and H.
24 The Backbone Coloring Problem for Small Graphs Figure 1. The graph G with backbone H with χ(g) = 6, χ(h) = 3, BBC 2 (G, H) = 7, BBC 3 (G, H) = 7, BBC 4 (G, H) = 10, BBC 5 (G, H) = 12, BBC 6 (G, H) = 14, BBC 7 (G, H) = 16. Table 2. The maximum slope: the difference between the values of BBC λ (G, H) BBC λ 1 (G, H) and BBC λ 1 (G, H) BBC λ 2 (G, H) for arbitrary backbones n slope 0 1 2 4 100% 0% 0% 5 98.11% 1.89% 0% 6 89.59% 10.41% 0% 7 83.48% 15.47% 1.05% Table 3. The maximum difference between the value of BBC λ (G, H) and the upper bound from equation 1 for arbitrary backbones n diff 0 1 2 3 4 5 4 22.22% 0% 0% 28.57% 19.05% 30.16% 5 12.76% 0% 0% 0.38% 19.11% 67.75% 6 9.79% 0% 0% 0.00% 0.43% 89.78% 7 5.45% 0% 0% 0.00% 0.30% 94.25% The number of the graphs, which attain the upper bound is very small and drops fast with the increase of n. Even for graphs on 7 vertices, almost 19 out of every 20 graphs had the values of BBC λ (G, H) at least 5 below the given bound.
R. Janczewski, K. Turowski 25 Table 4. The maximum difference between the value of BBC λ (G, H) and the lower bound from equation 2 for arbitrary backbones n diff 0 1 2 3 4 5 4 41.27% 47.62% 11.11% 0% 0% 0% 5 39.28% 39.91% 19.86% 0.94% 0% 0% 6 32.26% 47.84% 15.10% 4.76% 0.04% 0% 7 29.35% 39.90% 26.95% 2.92% 0.88% 0.00% Table 5. The maximum difference between the value of BBC λ (G, H) and the upper bound from equation 2 for arbitrary backbones n diff 0 1 2 3 4 5 4 22.22% 66.67% 11.11% 0% 0% 0% 5 3.21% 42.80% 52.29% 1.70% 0% 0% 6 0.24% 10.25% 49.11% 39.87% 0.53% 0% 7 0.01% 1.15% 19.39% 51.66% 27.74% 0.05% Moreover, the difference between the chromatic number and the bound from equation 2 exceeded 10 in 12.25% cases for n = 5, in 55.90% cases for n = 6 and in 69.49% cases for n = 7, which lead us to the conclusion that the bound is very weak for general graphs where difference between χ(g) and χ(h) can be arbitrarily large. The lower bound from the same equation was even worse: clearly, it was never exact when λ exceeded n 1 and the error was very significant in most of the cases so we decided to omit these results. On average, the lower bound is far better than the upper bound which clearly follows from the fact that most graphs have rather low value of χ(g; H). During the next stage of our experiments we were concerned with the performance of LF and IS algorithms. The IS algorithm turned out to be far better than LF in the best case, although far worse in the worst case as the LF algorithm solves the problem optimally for almost a quarter of the all instances on 7 vertices. The influence of the values of λ on the fraction of correctly solved cases is only significant for λ = 2. However, we should notice that the algorithm IS in the best case always may solve optimally the classical vertex coloring problem, provided the vertices are correctly preordered and this property doest not hold for λ-backbone coloring
26 The Backbone Coloring Problem for Small Graphs Table 6. The fraction of optimal solutions found by LF algorithm in the best case n λ 2 3 4 5 6 7 4 74.60% 68.25% 68.25% 5 76.62% 68.70% 68.07% 68.07% 6 74.66% 62.77% 60.70% 60.60% 60.60% 7 68.59% 54.03% 51.35% 51.04% 51.03% 51.03% Table 7. The fraction of optimal solutions found by LF algorithm in the worst case n λ 2 3 4 5 6 7 4 63.49% 63.49% 63.49% 5 42.55% 38.78% 38.78% 38.78% 6 33.03% 31.93% 31.82% 31.84% 31.84% 7 23.52% 23.29% 24.12% 24.17% 24.17% 24.17% Table 8. The fraction of optimal solutions found by IS algorithm in the best case n λ 2 3 4 5 6 7 4 100% 100% 100% 5 100% 100% 100% 100% 6 99.98% 99.86% 99.86% 99.86% 99.86% 7 99.95% 99.82% 99.81% 99.81% 99.81% 99.81% Table 9. The fraction of optimal solutions found by IS algorithm in the worst case n λ 2 3 4 5 6 7 4 47.62% 47.62% 47.62% 5 20.49% 19.80% 19.80% 19.80% 6 7.74% 7.76% 7.75% 7.75% 7.75% 7 1.82% 1.98% 1.98% 1.98% 1.98% 1.98% problem even for the graphs on 6 vertices (an example of hard-to-color graphs for LF and IS algorithms are presented in Fig. 2).
R. Janczewski, K. Turowski 27 Figure 2. The hard-to-color graphs for the algorithms LF and IS: BBC 3 (G, H) = 6, but IS and LF need more than 6 colors. Table 10. The linearity of the value of BBC for bipartite backbones n linear parts 1 2 3 4 100% 0% 0% 5 100% 0% 0% 6 92.12% 7.88% 0% 7 88.50% 11.34% 0.16% Table 11. The linearity of the value of BBC for tree backbones n linear parts 1 2 3 4 100% 0% 0% 5 100% 0% 0% 6 90.88% 9.12% 0% 7 89.40% 10.34% 0.27% 4. Results for bipartite and tree backbones We also computed the exact values of BBC λ (G, H) for instances with bipartite and tree backbones. For all graphs on 5 vertices the equality 2 holds, which was not the case of the arbitrary backbones. But apart from that observation, the results suggest that the graphs with bipartite or tree backbones do not tend to have significantly higher
28 The Backbone Coloring Problem for Small Graphs Figure 3. The graph G with bipartite backbone H with χ(g) = 5, χ(h) = 2, BBC 2 (G, H) = 5, BBC 3 (G, H) = 7, BBC 4 (G, H) = 8, BBC 5 (G, H) = 9, BBC 6 (G, H) = 10. Table 12. The maximum slope: the difference between the values of BBC λ (G, H) BBC λ 1 (G, H) and BBC λ 1 (G, H) BBC λ 2 (G, H) for bipartite backbones n slope 0 1 2 4 100% 0% 0% 5 100% 0% 0% 6 93.17% 6.83% 0% 7 91.04% 8.89% 0.07% fraction of linear dependency between BBC λ (G, H) and χ(h) for all values of λ. Interestingly, all graphs on 6 vertices, which do not satisfy the following inequality BBC λ (G, H) BBC λ 1 (G, H) BBC λ 1 (G, H) BBC λ 2 (G, H) have bipartite backbones but this does not hold in general since this is not the case even for graphs on 7 vertices (only 25835 out of 35291 relevant instances have bipartite backbones). However, the percentage of the graphs not satisfying this equality is significantly higher for bipartite graphs: 1.06% for n = 6 and 2.69% for n = 7. These values are quite similar for trees: 0.93% and 3.08%, respectively. The examples of the graphs with such irregularities are presented in Fig. 3 and 4. The λ-backbone chromatic number increases far more regularly for the graphs with bipartite backbones, but nevertheless, even if the backbone is a tree, there are instances for which the difference is greater than 1 (as presented in Fig. 3 and 4).
R. Janczewski, K. Turowski 29 Figure 4. The graph G with tree backbone H with χ(g) = 6, χ(h) = 2, BBC 2 (G, H) = 7, BBC 3 (G, H) = 7, BBC 4 (G, H) = 9, BBC 5 (G, H) = 10, BBC 6 (G, H) = 11. Table 13. The maximum slope: the difference between the values of BBC λ (G, H) BBC λ 1 (G, H) and BBC λ 1 (G, H) BBC λ 2 (G, H) for tree backbones n slope 0 1 2 4 100% 0% 0% 5 100% 0% 0% 6 91.81% 8.19% 0% 7 92.20% 7.63% 0.17% Table 14. The maximum difference between the value of BBC λ (G, H) and the lower bound from equation 2 for bipartite backbones n diff 0 1 2 3 4 5 4 41.27% 63.16% 18.42% 0% 0% 0% 5 4.46% 44.96% 48.10% 2.48% 0% 0% 6 2.38% 35.77% 40.65% 21.03% 0.17% 0% 7 0.43% 25.12% 48.43% 18.40% 7.61% 0.01% Since all instances with the bipartite backbones have χ(h) = 2, the results for the upper bound form equation 2 are just mirrored results from Tables 14 and 15. It turned out that for the instances with bipartite and even tree backbones the
30 The Backbone Coloring Problem for Small Graphs Table 15. The maximum difference between the value of BBC λ (G, H) and the lower bound from equation 2 for tree backbones n diff 0 1 2 3 4 5 4 18.18% 69.70% 12.12% 0% 0% 0% 5 4.37% 50.57% 43.91% 1.15% 0% 0% 6 2.12% 41.57% 41.24% 15.02% 0.06% 0% 7 0.36% 30.46% 51.97% 12.80% 4.40% 0.00% suboptimal results are returned much more frequently by the IS and LF algorithms: 99.38% of the graphs with bipartite backbones and 98.88% of the graphs with tree backbones on 7 vertices may be solved optimally using the algorithm IS still the overwhelming majority, but the failure ratio is significantly higher than in the general case. The LF algorithm performs even worse, solving optimally only 35.52% and 36.02% cases. However, in much simpler case when G is bipartite then IS algorithm solved all cases optimally in the best case (and over 48% in the worst case for the instances on 7 vertices). The other algorithm also performed much better for this class of graphs, solving optimally 99.52% and 96.28% of the instances on 7 vertices in the best and worst case, respectively. References [1] Broersma, H. J., A general framework for coloring problems: old results, new results, and open problems, In: Combinatorial Geometry and Graph Theory, Springer Berlin / Heidelberg, 2003, pp. 65 79. [2] Janczewski, R. and Turowski, K., The backbone coloring problem for bipartite backbones, submitted. [3] Broersma, H. J., Fujisawa, J., Marchal, B., Paulusma, D., Salman, A. N. M., and Yoshimoto, K., λ-backbone colorings along pairwise disjoint stars and matchings, Discrete Mathematics, Vol. 309, No. 18, 2009, pp. 5596 5609. [4] Miskuf, J., Skrekovski, R., and Tancer, M., Backbone colorings of graphs with bounded degree, Discrete Applied Mathematics, Vol. 158, No. 5, 2010, pp. 534 542.