Phase Transition in the Bandwidth Minimization Problem
Phase Transition in the Bandwidth Minimization Problem Nelson Rangel-Valdez and Jose Torres-Jimenez CINVESTAV-Tamaulipas, Information Technology Laboratory Km. 6 Carretera Victoria-Monterrey, 87276 Victoria Tamps., Mexico {nrangel,jtj}@tamps.cinvestav.mx
Abstract. It is known that some NP-Complete problems exhibit sharp phase transitions with respect to some order parameter. Moreover, a correlation between that critical behavior and the hardness of finding a solution exists in some of these problems. This paper shows experimental evidence about the existence of a critical behavior in the computational cost of solving the bandwidth minimization problem for graphs (BMPG). The experimental design involved the density of a graph as order parameter, 200000 random connected graphs of size 16 to 25 nodes, and a branch and bound algorithm taken from the literature. The results reveal a bimodal phase transition in the computational cost of solving the BMPG instances. This behavior was confirmed with the results obtained by metaheuristics that solve a known BMPG benchmark. Keywords: Phase Transition, Bandwidth Minimization Problem, Connected Graphs.
1
Introduction
In general, some computational problems are easier to solve than others. A first approach to understand the hardness of such problems can be obtained using the computational complexity theory [1,2]; a computational problem can be classified as P (can be solved in polynomial time) or N P (there is no known polynomial time algorithm that solves it). Despite the fact that a problem belongs to the class of NP problems, it is possible to find that some particular cases can be easily solved, for example the bandwidth minimization problem for graphs (BMPG) restricted to caterpillars with maximum hair length equal to 2 [3]. In the other side, there are also special cases where the BMPG problem remains NP complete such as in the case of graph caterpillar with maximum hair length equal to 3 [4]. Then, the question: Where the really hard instances are? remains open [5]. The research carried out in [5] presents the transition phenomena as a source of hard typical cases for several computational problems. This work was not pioneering on studying the phase transitions on artificial intelligence (AI). In fact, the application of statistical mechanics to NP-complete optimization problems was presented in [6], and [7] discusses the importance of such phenomena in AI.
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In the last couple of decades there has been an increasing interest in understanding the nature of the hardness of optimization problems. The phase transition phenomena has been studied in some NP-complete problems like Hamiltonian Circuit[5], Traveling Salesman Problem (TSP)[5], Chromatic Number [8], Vertex Covering [9], Independent Set[8], K-Satisfiability (K-SAT) [10] and Constraint Satisfaction Problems [11]. In [12] a work about solving instances that lies in the critical point of the phase transition of K-SAT is presented. The algorithm used was a Tabu Search. In [13] is studied the typical case of complexity of random (2 + p)-SAT formulas, where p varies from 0 to 1. Navarro [14] presents an approach to construct hard instances of the SAT problem; the construction of such instances is based in the location of the phase transition of that hard optimization problem. The location of critical points of phase transitions has been mainly studied for NP-Complete problems. Some works have been performed for NP-Hard problems. Among them are TSP [5], Project Scheduling [15] Chromatic Number and MAX 2-SAT [16,17], and Number Partitioning [18,19]. In this paper we study the existence of a critical behavior in the computational cost of solving instances of BMPG. The evidence was constructed using a set of random connected graphs whose size vary from 16 to 25 nodes. The computational cost of the solution of the instances was determined through the branch and bound algorithm proposed in [20]. The rest of the paper is organized as follows. Section 2 describes the BMPG. Section 3 presents the elements used to identify the phase transition and the experiments performed so that the critical behavior can be identified. Finally, section 4 presents the conclusions of this paper.
2
Bandwidth Minimization Problem
The Bandwidth Minimization Problem for Graphs (BMPG) can be defined as follows. Let G = (V, E) be an undirected graph, where V defines the set of vertices with cardinality n = |V |, and E ⊆ V × V = {{i, j}|i, j ∈ V } is the set of edges with cardinality m = |E|. Given a one-to-one labeling function τ : V → 1, 2, ..., n, the bandwidth β of G for τ is defined according to the Equation (1). βτ = max{|τ (i) − τ (j)| : (i, j) ∈ E}
(1)
∗
Then the BMPG consists in finding a labeling τ for which βτ ∗ (G) is minimum, it is expressed in mathematical terms in the Equation (2). βτ ∗ (G) = min{βτ (G) : τ ∈ T }
(2)
where T is the set of all possible labeling functions. The BMPG has been widely studied since 1960s [21]; and its applications range from sparse systems of equations to applications in electromagnetic industry [22], circuit design [23], hypertext reordering [24] and others.
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The NP-Completeness of the BMPG has been proved in [25]. This problem remains NP-Complete even for simpler graphs such as trees [26] and carterpillars with hair length 3 [27]. In [28] have been shown that it is also NP-Hard to approximate the bandwidth of general graphs with an approximation ratio of 3 2 − ε or less, where ε > 0. Several exhaustive approaches have been developed to solve the BMPG problem. Del Corso and Manzini [29] developed two algorithms based on depth search to solve it, these algorithms are Minimum Bandwidth by Iterative Deepening (MB-ID) and Minimum Bandwidth by Perimeter Search (MB-PS). Caprara and Salazar [20] show two enumeration schemes that solve the BMPG problem, LeftToRight and Both. While Del Corso and Manzini [29] present enumeration schemes based on valid labels on unlabeled nodes according with the labels of labeled neighbor nodes, Caprara and Salazar [20] generalize this definition by choosing valid labels of unlabeled nodes according with the labels of nodes at a distance d [29]. A more recent approach that solves exactly the BMPG problem is presented by Marti, et al., [30]. In this approach the solution φ obtained by the GRASP algorithm proposed in [31] was used as an upper bound in two enumeration schemes, Branch and Bound (BB) and Branch and Bound with Perimeter search (BBP). During the enumeration schemes, when a solution φ is found, then the value is updated by φ = φ − 1 but the search tree is not restarted, as done in the schemes presented in [29] and [20], because if no solution could be found with the actual upper bound, then a solution with a value smaller can not be found either. Besides the exact approaches, several non-exhaustive approaches have been developed to solve it. Among them are Tabu Search [32], GRASP [31] and Simulated Annealing [33]. In the next section we describe the experimental design used to identify the critical behavior on BMPG.
3
Phase Transition in the Bandwidth Minimization Problem for Graphs (BMPG)
The work done on phase transition phenomena shows that hard instances of an optimization problem are found in a region near the critical point of such phenomena [5]. The critical point can be seen in a phase diagram, i.e., a curve that shows the behavior of the order parameter against a control parameter of interest for the optimization problem. In this paper we experimentally studied the critical behavior found when solving BMPG instances using an exact approach. As it was described before, there are six known exact methods that solve the BMPG problem: MB-ID, MBPS, Both, LeftToRight, BB and BBP. For the purpose of our research, we chose the algorithm Both [20] because it is a generalization of the algorithm [20], in average it performs better than the algorithms MB-ID, MB-PS, it also requires less memory than BBP and it has a simpler structure than the algorithms BB and BBP (in the sense that it does not require an heuristic algorithm to build an initial solution).
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Random Connected Graph Generator. The solution of the BMPG problem for unconnected graphs is equivalent to individually solve each of its connected components. Considering this fact, we decided to work only with connected graphs. Therefore, the probabilistic model of graphs known as the Erd¨ os-R´enyi model [34] and denoted by G(n, p), is adapted in this research as the source of random instances for the BMPG problem. The random graph model was implemented using two stages. The first stage generates a random tree using the Pr¨ uffer sequence[35]. The second stage applies the G(n, p) model so that the graph resulting will be always connected. The pseudocode that follows this method to construct random connected graphs is shown in Algorithm 3.1.
Algorithm 3.1. RandomGraphGenerator(n, ρ) ¨fferSequence(n) T ← Pru G←T density ← ρ − (n − 1)/ n2 for i ← ⎧ 1 to n ⎪ ⎪if density ≤0 ⎪ ⎪ return (G) then ⎪ ⎪ ⎪ ⎪ i ⎨for j ← ⎧ + 1 to n do ⎪ ⎪number ← RandomNumber() ⎨ ⎪ ⎪ ⎪ ⎪ do if number ≤ρ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ then Add(G, {i, j}) ⎩ ⎩ density ← density − 1/ n2 Ti ← Ti ∗ α return (G)
Order Parameter. An order parameter is a descriptive measure of an optimization problem instance. Examples of these metrics for graphs G = {V, E} are [15] or the coefficients the number of edges m of the graph [8], the density m (n2 ) m n log n used by [5] to relate the number of edges m with the minimum number of edges required in a graph to be connected n log n, where n is the number of nodes of the graph. Given that the bandwidth of a graph grows with its density, the order parameter selected to control the experiment was the density of the graph G = {V, E}, i.e., the number of edges m = |E| divided by the maximum number of edges that the graph can have n2 , where n = |V |. Complexity Measure. The complexity of solving a BMPG instance G is defined as the number of subproblems spent by the Both Algorithm when solving G. A subproblem is a node that the branch and bound algorithm Both visits in the search tree when solving a BMPG instance. This complexity measure is the most common measure used in the literature [5,36,15,8,14].
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Experimental Evidence. Following the method for constructing random graphs shown in algorithm 3.1, a set of 200000 BMPG instances was generated. The size n of the instances was in the rank from 16 to 25 nodes. The number of edges m for each graph G was set to Δ × n2 , where Δ is the density selected for G. The values of δ can be {0.05, 0.10, 0.15, ..., 1.00}. For each node size n and each density value Δ, a set of 1000 instances of BMPG were generated. The computational cost of solving the set of random instances is shown in Table 1. The first column shows the density value. The rest of the columns represents the different sizes of the instances and contains the average number of subproblems required to solve each instance. Table 1. Complexity results of solving the random set of BMPG instances. The values represent the average number of nodes in the search tree (subproblems) constructed by the algorithm Both before reaching the optimal solution.
Density 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
n 16 17 18 19 20 21 22 23 24 25 20.2 21.6 27.7 27.9 35.6 47.6 53.4 76.8 175.8 336.8 19.7 22.7 26.4 31.2 40.4 42.2 144.5 413.8 465.0 1175.0 27.4 38.9 49.6 91.5 98.1 178.8 218.8 452.6 879.7 1290.3 31.6 45.0 59.6 79.7 123.5 169.3 302.0 308.9 750.7 906.4 38.6 41.5 59.9 63.8 111.3 114.8 167.7 216.8 310.8 460.3 33.1 39.2 49.9 58.7 84.8 107.9 176.8 290.5 390.5 1065.9 30.7 43.2 60.1 77.3 130.1 182.5 423.0 745.5 1071.6 2791.1 41.8 62.4 86.7 175.9 220.9 399.2 724.8 1342.2 2045.6 3355.9 48.6 85.1 137.3 207.9 321.0 473.0 770.7 1231.2 1644.3 2936.1 68.1 100.3 139.8 212.4 318.6 487.4 659.5 953.5 1448.1 1972.4 63.7 92.0 135.9 197.8 279.1 406.1 521.1 703.8 1027.5 1334.0 60.7 84.6 112.6 155.2 211.4 285.2 394.2 506.9 650.4 844.2 51.8 67.6 87.4 120.4 162.6 203.3 253.0 339.5 410.4 541.9 41.8 52.8 69.4 85.9 110.5 138.3 173.5 214.2 279.7 338.7 32.1 40.4 48.4 60.4 75.4 94.1 114.9 140.9 167.1 199.3 24.0 29.0 34.7 41.4 50.1 59.1 68.8 82.1 99.2 118.4 18.6 20.0 22.0 25.2 29.5 34.8 40.5 48.5 57.1 64.2 13.6 14.3 16.5 19.0 20.2 23.0 24.4 26.4 28.4 30.8 17.2 18.1 18.5 18.8 18.6 19.0 18.2 18.0 18.2 18.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
A plot of the data presented in Table 1 (see Figure 1) reveals the existence of a density value where the average number of iterations is maximum. Moreover, the complexity curve is almost bimodal showing an easy-hard-easy-hard-easy pattern but with one of the peak larger than the other. It is precisely in that large peak where a critical behavior in the computational cost of solving the BMPG instances is claimed to exist. In the literature [5,15,8] has been established a relation between a phase diagram describing phase transitions in NP problems and the hardness of the instances in the critical point of that diagram. In our case, the pattern shown in Figure 1 allows us to identify one point where
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the computational cost of the instances grows significantly in comparison with other points, this can be the evidence of the existence of a more general critical behavior on BMPG (the existence of the phase transition phenomena) than can explain such behavior. The complexity results obtained with the set of random instances were contrasted with the set of BMPG instances taken from Harwell-Boeing Sparse Matrix Collection1 (HBSM). In order to understand the complexity nature of such benchmark we take the results given by some state-of-the-art BMPG algorithms. Tables 2 and 3 summarize the results reported by the algorithms Simulated Annealing (SA) [33], GRASP [31] and the Tabu Search [32]. The first three columns show information about the instances. The rest of the columns present the time spent by each algorithm when solving the instances. The solution achieved by each algorithm can be seen in [33]. Figure 2 shows a graph of the complexity results of the BMPG state-of-the-art algorithms solving the HBSM benchmark. This figure presents a behavior in the computational cost (measure as the time that took to each algorithm to reach its best solution) similar to the one presented in the set of random instances, in the sense that when the density increases, the computational cost of solving the instances also increases. The experiments showed that the hardest instances were not found in the set of larger instances but in the set of medium instances of the HBSM benchmark. In addition, the average performance of the algorithms (see last row of Tables 2 and 3) over each different set of instances from the HBSM benchmark indicates that the hardest set was the set of medium size. We can note in Tables 2 and 3 that the density of the hardest instances is close to the value pointed out as the critical point in the experiment with random connected BMPG instances, this suggests that the hardest instances lie at the critical region in a phase transition phenomena for both exact and non-exact algorithms. The analysis of the computational cost presented in this section suggests that there are two types of instances of the BMPG that requires a small computational effort to be solved. The first type corresponds to instances with a low density. The second type is formed by instances with a high density. This behavior suggests the existence of two regions, one where the BMPG instances have a low density and the other where they have a high density. The graphs with low density, as paths and cycles, can be considered under-constrained, a small number of edges are connecting the nodes resulting in a small optimal bandwidth for those graphs. These graphs are easy to solve because almost all the solutions are possible, with the possibility of finding the optimal values in a small set of integers. The graphs with high density, as the complete graph, can be consider over-constrained; the great number of edges in it allows only a few set of possible optimal solutions which also makes them easy to solve. The existence of these separated regions could explain the easy-hard-easy transition in the computational cost of solving BMPG, just as it is presented in the satisfiability problem (SAT)[10]. 1
http://math.nist.gov/MatrixMarket/data/
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Fig. 1. Behavior of the computational cost (measured by the number of solved subproblems) of solving random BMPG instances varying the density of the graphs. A curve for each of the 10 values for the number of nodes is shown. The x axis shows the different values of density analyzed. The y axis shows the average number of subproblems required. The forms of each curve drown follow a non-monotonic increase in the complexity revealing that the BMPG problem presents a bimodal phase transition.
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Fig. 2. Computational time to solve the HBSM benchmark using the algorithms SA, TS and GRASP. This graph relates the density of the instances (shown in x axis) with the time that each algorithm spent to solve them (shown in y axis). In this graph shows two peaks indicating a high computational cost; one of them is located between the densities 0.05 and 0.10, the other one is at the density value of 0.35.
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Table 2. Computational time for solving the small size set of cases of HBSM benchmark. It shows the time (in seconds) spent by TS, GRASP and SA to solve each instance. Also, the number of nodes n, edges m and the density of each instance are presented.
Instance dwt 234 nos1 bcspwr03 west0156 will199 west0167 lns 131 impcol c gre 185 gre 115 fs 183 1 bcsstk22 west0132 can 161 nos4 bcspwr02 can 144 ash85 bcspwr01 will57 arc130 curtis54 bcsstk05 gent113 lund b lund a mcca steam3 bcsstk01 impcol b ibm32 bcsstk04 pores 1 Average
n 117 158 118 156 199 167 123 137 185 115 183 110 132 161 100 49 144 85 39 57 130 54 153 104 147 147 168 80 48 59 32 132 30
TS GRASP m Density Time Time 162 0.02 1.2 1.9 312 0.03 1.3 2.6 179 0.03 1.7 0.9 371 0.03 7.4 8.4 660 0.03 12 26.9 489 0.04 5.8 5.6 275 0.04 3.4 2.6 352 0.04 3.5 4.5 650 0.04 6.2 6.1 267 0.04 2.4 3.2 701 0.04 32.4 11.8 254 0.04 1.1 1.6 404 0.05 3.4 8.5 608 0.05 4 0.7 247 0.05 1.1 1.4 59 0.05 0.2 0.6 576 0.06 1.7 3.1 219 0.06 0.7 0.4 46 0.06 0.1 0.1 127 0.08 0.4 0.4 715 0.09 4.8 1.9 124 0.09 0.7 0.7 1135 0.1 4.7 7.1 549 0.1 6.3 1 1147 0.11 5.3 5.5 1151 0.11 8.8 5.4 1662 0.12 23.9 10.8 424 0.13 0.7 1.O 176 0.16 0.9 1 281 0.16 1.3 1.2 90 0.18 0.2 0.3 1758 0.2 16.7 5.4 103 0.24 0.3 0.3 0.08 4.99 4.12
SA Time 1.1 1.1 1.2 2.8 6.1 4.8 1.8 3.1 6.8 1.6 8.7 5.5 5.4 3.0 0.9 0.2 17.6 1.1 0.4 1.1 23.2 0.5 10.6 3.9 40.8 40.6 81.8 8.9 0.6 1.2 0.3 79.2 3.1 11.18
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Table 3. Computational time for solving the medium and large size cases of HBSM benchmark. It shows the time (in seconds) spent by TS, GRASP and SA to solve each instance. Also, the number of nodes n, edges m and the density of each instance are presented. Medium Instance n 494 bus bcsstk20 impcol d plskz362 west0497 bcspwr05 nnc261 nos5 hor 131 gre 343 saylr1 west0479 bcspwr04 ash292 pores 3 dwt 310 dwt 245 can 445 dwt 419 dwt 361 west0381 dwt 221 can 292 impcol a gre 216a bcsstk06 bcsstm07 str 0 plat362 dwt 209 str 200 impcol e str 600 steam1 mbeause mbeaflw mbeacxc
Average
4
494 467 425 362 497 443 261 468 434 343 238 479 274 292 456 310 245 445 419 361 381 221 292 206 216 420 420 363 362 209 363 225 363 240 492 487 487
m 586 1295 1267 880 1715 590 794 2352 2138 1092 445 1889 669 958 1769 1069 608 1682 1572 1296 2150 704 1124 557 660 3720 3416 2446 2712 767 3049 1187 3244 1761 36209 41686 41686
TS GRASP SA Large Density Time Time Time Instance n 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.3 0.35 0.35
0.05
m
29.2 13.5 24.4 662 bus 662 906 27.8 11 44.6 nos6 675 1290 21.9 30.5 77.5 lns 511 503 1425 27.4 7.1 20.9 jpwh 991 983 2678 78.3 88.7 137.9 nos2 638 1272 24.7 16.3 26.8 nnc666 666 2148 8.6 22.4 8.7 jagmesh1 936 2664 60.8 103.1 121.9 fs 760 1 760 3518 26 27.7 154.1 fs 680 1 680 1464 16.6 21.2 7.4 gre 512 512 1680 4.3 8.5 2.3 gr 30 30 900 3422 81.2 163 40.5 west0989 989 3500 9.6 5.2 28 sh1 200 663 1720 7.9 8.7 34.4 sh1 0 663 1682 16.8 3.2 13 west0655 655 2841 15.2 8.6 13 sh1 400 663 1709 7.5 14.4 9.3 sherman4 546 1341 64.5 47.2 114 orsirr 2 886 2542 23.7 28 59.9 nos7 729 1944 11.8 0.8 8.3 sherman1 681 1373 113 185.4 38.1 bp 200 822 3788 5 7 23.4 bp 1600 822 4809 19 7.9 61.5 bp 0 822 3260 5.5 3.4 5.4 bp 800 822 4518 7.2 9.5 3.5 bp 600 822 4157 47.6 50.6 247.9 bp 400 822 4015 40.3 113.7 215.7 bcsstk19 817 3018 120.8 119.1 39.9 685 bus 685 1282 24.8 3.4 189.4 bp 1400 822 4760 10.8 1.3 29.2 bp 1200 822 4698 90.9 114.1 47.3 bp 1000 822 4635 10.7 4 49.6 dwt 592 592 2256 180.3 92 55.9 dwt 878 878 3285 16.9 12.7 79.1 saylr3 681 1373 2637.6 5494.3 1289.5 dwt 918 918 3233 3409.4 5467.8 1744.9 can 715 715 2975 3409.1 5464.5 1774 can 838 838 4586 fs 541 1 541 2466 nos3 960 7442 dwt 992 992 7876 dwt 503 503 2762 steam2 600 6580 mcfe 731 15086 289.53 480.54 184.9
TS GRASP SA Density Time Time Time 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.06 0.01
113.8 32.9 70.4 70.9 42.8 12.5 74.2 58.3 123.8 312.6 65.1 104.3 43.7 15.7 114.5 138.5 55.8 108.5 150.6 107.8 33.8 101.1 97.9 95.6 43.4 33.6 39.8 77.1 92.7 14.7 282.2 85.4 370.1 372.7 416.9 172.1 161.3 98.4 213.5 153.1 110.2 211.5 150.1 245.8 80.5 188.3 121.2 221.4 33 4.1 11.5 203 42.5 164.5 74.5 89.9 23.9 107.2 78 15.6 315.2 550.3 168.4 546.8 783.7 231.8 386.8 483 140.7 520.9 636.9 219.9 480.9 556.8 193.3 355.7 560.6 180.0 174.3 86.6 222.9 90.4 12.7 62.0 521 741.2 217.4 674.8 897.1 218.2 886.7 886.2 216.2 52.9 106.8 123.7 195 99.6 106.7 107.2 77.8 15.6 290.8 12.6 243.0 183.1 14.2 223.0 158.9 37.4 384.2 54.4 26.5 82.4 143.8 209.1 811.6 272.5 306.4 116.4 99.3 7.9 163.5 242.2 182.5 687.4 800.2 247.2 1868.9 241.98 219.03 211.62
Conclusions and Future Work
In this paper we presented experimental evidence of the existence of a critical behavior in the optimization problem known as Bandwidth Minimization Problem for Graphs (BMPG). We study the computational complexity of the BMPG instances in a set of random connected graphs. The critical behavior was observed in the computational
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cost of solving the BMPG instances taking as the order parameter the density of the instances and as the measure of complexity the number of subproblems required for an state-of-the-art Branch & Bound strategy to find the optimal solution. The complexity results were contrasted with reported results of the well known BMPG benchmark of the Harwell Boeing Sparse Matrix Collection. Keeping the density as the order parameter and the time spent by an algorithm to find its best solution as the complexity measure, we presented graphs showing a similar complexity behavior that the one presented in the set of random connected graphs. Moreover, the results uncover the fact that the hardest instances do not correspond to the ones with the highest number of nodes but the ones with the density close to the point marked as the critical point in the experiment with random connected graph (there are medium sized instances). It seems for the evidence shown that the instances that lie in the critical region are hard even for non-exact algorithms. Finally, given the results presented in this document a guideline to construct harder benchmarks for BMPG must consider the density parameter. Currently we are working on analyzing other order parameters to better understand the hardness of the BMPG instances in the point of maximum difficulty. Also, we are studying new complexity measures for non-exact algorithms. In our future work, we also expect to increase the size of instances. Acknowledgments. This research was partially funded by the following projects: CONACyT 58554-C´alculo de Covering Arrays, 51623-Fondo Mixto CONACyT y Gobierno del Estado de Tamaulipas.
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9. Hartmann, A., Weigt, M.: Statistical mechanics perspective on the phase transition in vertex covering of finite-connectivity random graphs. Theoretical Computer Science 265, 199–225 (2001) 10. Kirkpatrick, S., Selman, B.: Critical behavior in the satisfiability of random Boolean expressions. Artificial Intelligence 81, 273–295 (1994) 11. Smith, B.: Constructing an asymptotic phase transition in random binary constraint satisfaction problems. Theoretical Computer Science 265, 265–283 (2001) 12. Mazure, B., Sais, L., Gr´egoire, E.: Tabu search for sat. In: Proceedings of the Fourteenth Nat´l Conf. on Artificial Intelligence (AAAI 1997), pp. 281–285. AAAI Press, Menlo Park (1997) 13. Monasson, R., Zecchina, R., Kirkpatrick, S., Selman, B., Troyansky, L.: Determining computational complexity from characteristic p ´hase transitions´. Nature 400, 133–137 (1999) 14. Navarro, J.A., Voronkov, A.: Generation of hard non-clausal random satisfiability problems. In: AAAI 2005 (2005) 15. Herroelen, W., De Reyck, B.: Phase transitions in project scheduling. Journal of the Operational Research Society 50, 148–156 (1999) 16. Slaney, J., Walsh, T.: Phase transition behavior: from decision to optimization. In: Proceedings of the 5th International Symposium on the Theory and Applications of Satisfiability Testing, SAT (2002) 17. Achlioptas, D., Naor, Y.P. A.: Rigorous location of phase transitions in hard optimization problems. Nature 435, 759–764 (2005) 18. Gent, I.P., Walsh, T.: Phase transitions and annealed theories: Number partitioning as a case study. In: ECAI, pp. 170–174 (1996) 19. Gent, I., Walsh, T.: Analysis of heuristics for number partitioning. Computational Intelligence 14(3), 430–451 (1998) 20. Caprara, A., Salazar-Gonz´ alez, J.: Laying out sparse graphs with provably minimum bandwidth. Informs Journal on Computing 17(3), 356–373 (2005) 21. Cuthill, E.H., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proc. 24th ACM National Conf. pp. 157–172 (1969) 22. Esposito, A., Catalano, M., Malucelli, F., Tarricone, L.: Sparse matrix bandwidth reduction: Algorithms, applications and real industrial cases in electromagnetics. Advances in the Theory of Computation and Computational Mathematics 2, 27–45 (1998) 23. Bhatt, S., Leighton, F.: A framework for solving VLSI graph layout problems. J. Comput. Sytem Sci. 28, 300–343 (1984) 24. Berry, M., Hendrickson, B., Raghavan, P.: Sparse matrix reordering schemes for browsing hypertext. Lectures in Applied Mathematics 32, 99–123 (1996) 25. Papadimitriou, C.: The np-completeness of the bandwidth minimization problem. Computing 16(3), 263–270 (1976) 26. Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM Journal on Applied Mathematics 34(3), 477–495 (1978) 27. Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 is np-complete. SIAM J. Algebraic Discrete Methods 7(4), 505–512 (1986) 28. Blache, G., Karpinski, M., Wirtgen, J.: On approximation intractability of the bandwidth problem. Electronic Colloquium on Computational Complexity (ECCC) 5(014) (1998) 29. Del Corso, G.M., ManZini, G.: Finding exact solutions to the bandwidth minimization problem. Computing 62, 189–203 (1999)
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30. Marti, R., Campos, V., Pinana, E.: A branch and bound algorithm for the matrix bandwidth minimization. EJORS 186, 513–528 (2008) 31. Pinana, E., Plana, I., Campos, V., Marti, V.: GRASP and path relinking for the matrix bandwdith minimization. European Journal of Operational Research 153, 200–210 (2004) 32. Marti, R., Laguna, M., Glover, F., Campos, V.: Reducing the bandwidth of a sparse matrix with tabu search. European Journal of Operational Research 135(2), 211–220 (2001) 33. Rodriguez-Tello, E., Hao, J., Torres-Jimenez, J.: An improved simulated annealing algorithm for bandwidth minimization. European Journal of Operational Research 185(3), 1319–1335 (2008) 34. Erdos, P., R´enyi, A.: On random graphs. i. Publicationes Mathematicae 6, 290–297 (1959) 35. Pr¨ uffer, P.: Neuer beweis eines satzes u ¨ber permutationen. Arch. Math. Phys. 27, 742–744 (1918) 36. Selman, B., Kirkpatrick, S.: Critical behavior in the computational cost of satisfiability testing. Artificial Intelligence 81, 273–295 (1996)
Abstract. It is known that some NP-Complete problems exhibit sharp phase transitions with respect to some order parameter. Moreover, a correlation between that critical behavior and the hardness of finding a solution exists in some of these problems. This paper shows experimental evidence about the existence of a critical behavior in the computational cost of solving the bandwidth minimization problem for graphs (BMPG). The experimental design involved the density of a graph as order parameter, 200000 random connected graphs of size 16 to 25 nodes, and a branch and bound algorithm taken from the literature. The results reveal a bimodal phase transition in the computational cost of solving the BMPG instances. This behavior was confirmed with the results obtained by metaheuristics that solve a known BMPG benchmark. Keywords: Phase Transition, Bandwidth Minimization Problem, Connected Graphs.
1
Introduction
In general, some computational problems are easier to solve than others. A first approach to understand the hardness of such problems can be obtained using the computational complexity theory [1,2]; a computational problem can be classified as P (can be solved in polynomial time) or N P (there is no known polynomial time algorithm that solves it). Despite the fact that a problem belongs to the class of NP problems, it is possible to find that some particular cases can be easily solved, for example the bandwidth minimization problem for graphs (BMPG) restricted to caterpillars with maximum hair length equal to 2 [3]. In the other side, there are also special cases where the BMPG problem remains NP complete such as in the case of graph caterpillar with maximum hair length equal to 3 [4]. Then, the question: Where the really hard instances are? remains open [5]. The research carried out in [5] presents the transition phenomena as a source of hard typical cases for several computational problems. This work was not pioneering on studying the phase transitions on artificial intelligence (AI). In fact, the application of statistical mechanics to NP-complete optimization problems was presented in [6], and [7] discusses the importance of such phenomena in AI.
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In the last couple of decades there has been an increasing interest in understanding the nature of the hardness of optimization problems. The phase transition phenomena has been studied in some NP-complete problems like Hamiltonian Circuit[5], Traveling Salesman Problem (TSP)[5], Chromatic Number [8], Vertex Covering [9], Independent Set[8], K-Satisfiability (K-SAT) [10] and Constraint Satisfaction Problems [11]. In [12] a work about solving instances that lies in the critical point of the phase transition of K-SAT is presented. The algorithm used was a Tabu Search. In [13] is studied the typical case of complexity of random (2 + p)-SAT formulas, where p varies from 0 to 1. Navarro [14] presents an approach to construct hard instances of the SAT problem; the construction of such instances is based in the location of the phase transition of that hard optimization problem. The location of critical points of phase transitions has been mainly studied for NP-Complete problems. Some works have been performed for NP-Hard problems. Among them are TSP [5], Project Scheduling [15] Chromatic Number and MAX 2-SAT [16,17], and Number Partitioning [18,19]. In this paper we study the existence of a critical behavior in the computational cost of solving instances of BMPG. The evidence was constructed using a set of random connected graphs whose size vary from 16 to 25 nodes. The computational cost of the solution of the instances was determined through the branch and bound algorithm proposed in [20]. The rest of the paper is organized as follows. Section 2 describes the BMPG. Section 3 presents the elements used to identify the phase transition and the experiments performed so that the critical behavior can be identified. Finally, section 4 presents the conclusions of this paper.
2
Bandwidth Minimization Problem
The Bandwidth Minimization Problem for Graphs (BMPG) can be defined as follows. Let G = (V, E) be an undirected graph, where V defines the set of vertices with cardinality n = |V |, and E ⊆ V × V = {{i, j}|i, j ∈ V } is the set of edges with cardinality m = |E|. Given a one-to-one labeling function τ : V → 1, 2, ..., n, the bandwidth β of G for τ is defined according to the Equation (1). βτ = max{|τ (i) − τ (j)| : (i, j) ∈ E}
(1)
∗
Then the BMPG consists in finding a labeling τ for which βτ ∗ (G) is minimum, it is expressed in mathematical terms in the Equation (2). βτ ∗ (G) = min{βτ (G) : τ ∈ T }
(2)
where T is the set of all possible labeling functions. The BMPG has been widely studied since 1960s [21]; and its applications range from sparse systems of equations to applications in electromagnetic industry [22], circuit design [23], hypertext reordering [24] and others.
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The NP-Completeness of the BMPG has been proved in [25]. This problem remains NP-Complete even for simpler graphs such as trees [26] and carterpillars with hair length 3 [27]. In [28] have been shown that it is also NP-Hard to approximate the bandwidth of general graphs with an approximation ratio of 3 2 − ε or less, where ε > 0. Several exhaustive approaches have been developed to solve the BMPG problem. Del Corso and Manzini [29] developed two algorithms based on depth search to solve it, these algorithms are Minimum Bandwidth by Iterative Deepening (MB-ID) and Minimum Bandwidth by Perimeter Search (MB-PS). Caprara and Salazar [20] show two enumeration schemes that solve the BMPG problem, LeftToRight and Both. While Del Corso and Manzini [29] present enumeration schemes based on valid labels on unlabeled nodes according with the labels of labeled neighbor nodes, Caprara and Salazar [20] generalize this definition by choosing valid labels of unlabeled nodes according with the labels of nodes at a distance d [29]. A more recent approach that solves exactly the BMPG problem is presented by Marti, et al., [30]. In this approach the solution φ obtained by the GRASP algorithm proposed in [31] was used as an upper bound in two enumeration schemes, Branch and Bound (BB) and Branch and Bound with Perimeter search (BBP). During the enumeration schemes, when a solution φ is found, then the value is updated by φ = φ − 1 but the search tree is not restarted, as done in the schemes presented in [29] and [20], because if no solution could be found with the actual upper bound, then a solution with a value smaller can not be found either. Besides the exact approaches, several non-exhaustive approaches have been developed to solve it. Among them are Tabu Search [32], GRASP [31] and Simulated Annealing [33]. In the next section we describe the experimental design used to identify the critical behavior on BMPG.
3
Phase Transition in the Bandwidth Minimization Problem for Graphs (BMPG)
The work done on phase transition phenomena shows that hard instances of an optimization problem are found in a region near the critical point of such phenomena [5]. The critical point can be seen in a phase diagram, i.e., a curve that shows the behavior of the order parameter against a control parameter of interest for the optimization problem. In this paper we experimentally studied the critical behavior found when solving BMPG instances using an exact approach. As it was described before, there are six known exact methods that solve the BMPG problem: MB-ID, MBPS, Both, LeftToRight, BB and BBP. For the purpose of our research, we chose the algorithm Both [20] because it is a generalization of the algorithm [20], in average it performs better than the algorithms MB-ID, MB-PS, it also requires less memory than BBP and it has a simpler structure than the algorithms BB and BBP (in the sense that it does not require an heuristic algorithm to build an initial solution).
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Random Connected Graph Generator. The solution of the BMPG problem for unconnected graphs is equivalent to individually solve each of its connected components. Considering this fact, we decided to work only with connected graphs. Therefore, the probabilistic model of graphs known as the Erd¨ os-R´enyi model [34] and denoted by G(n, p), is adapted in this research as the source of random instances for the BMPG problem. The random graph model was implemented using two stages. The first stage generates a random tree using the Pr¨ uffer sequence[35]. The second stage applies the G(n, p) model so that the graph resulting will be always connected. The pseudocode that follows this method to construct random connected graphs is shown in Algorithm 3.1.
Algorithm 3.1. RandomGraphGenerator(n, ρ) ¨fferSequence(n) T ← Pru G←T density ← ρ − (n − 1)/ n2 for i ← ⎧ 1 to n ⎪ ⎪if density ≤0 ⎪ ⎪ return (G) then ⎪ ⎪ ⎪ ⎪ i ⎨for j ← ⎧ + 1 to n do ⎪ ⎪number ← RandomNumber() ⎨ ⎪ ⎪ ⎪ ⎪ do if number ≤ρ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ then Add(G, {i, j}) ⎩ ⎩ density ← density − 1/ n2 Ti ← Ti ∗ α return (G)
Order Parameter. An order parameter is a descriptive measure of an optimization problem instance. Examples of these metrics for graphs G = {V, E} are [15] or the coefficients the number of edges m of the graph [8], the density m (n2 ) m n log n used by [5] to relate the number of edges m with the minimum number of edges required in a graph to be connected n log n, where n is the number of nodes of the graph. Given that the bandwidth of a graph grows with its density, the order parameter selected to control the experiment was the density of the graph G = {V, E}, i.e., the number of edges m = |E| divided by the maximum number of edges that the graph can have n2 , where n = |V |. Complexity Measure. The complexity of solving a BMPG instance G is defined as the number of subproblems spent by the Both Algorithm when solving G. A subproblem is a node that the branch and bound algorithm Both visits in the search tree when solving a BMPG instance. This complexity measure is the most common measure used in the literature [5,36,15,8,14].
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Experimental Evidence. Following the method for constructing random graphs shown in algorithm 3.1, a set of 200000 BMPG instances was generated. The size n of the instances was in the rank from 16 to 25 nodes. The number of edges m for each graph G was set to Δ × n2 , where Δ is the density selected for G. The values of δ can be {0.05, 0.10, 0.15, ..., 1.00}. For each node size n and each density value Δ, a set of 1000 instances of BMPG were generated. The computational cost of solving the set of random instances is shown in Table 1. The first column shows the density value. The rest of the columns represents the different sizes of the instances and contains the average number of subproblems required to solve each instance. Table 1. Complexity results of solving the random set of BMPG instances. The values represent the average number of nodes in the search tree (subproblems) constructed by the algorithm Both before reaching the optimal solution.
Density 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
n 16 17 18 19 20 21 22 23 24 25 20.2 21.6 27.7 27.9 35.6 47.6 53.4 76.8 175.8 336.8 19.7 22.7 26.4 31.2 40.4 42.2 144.5 413.8 465.0 1175.0 27.4 38.9 49.6 91.5 98.1 178.8 218.8 452.6 879.7 1290.3 31.6 45.0 59.6 79.7 123.5 169.3 302.0 308.9 750.7 906.4 38.6 41.5 59.9 63.8 111.3 114.8 167.7 216.8 310.8 460.3 33.1 39.2 49.9 58.7 84.8 107.9 176.8 290.5 390.5 1065.9 30.7 43.2 60.1 77.3 130.1 182.5 423.0 745.5 1071.6 2791.1 41.8 62.4 86.7 175.9 220.9 399.2 724.8 1342.2 2045.6 3355.9 48.6 85.1 137.3 207.9 321.0 473.0 770.7 1231.2 1644.3 2936.1 68.1 100.3 139.8 212.4 318.6 487.4 659.5 953.5 1448.1 1972.4 63.7 92.0 135.9 197.8 279.1 406.1 521.1 703.8 1027.5 1334.0 60.7 84.6 112.6 155.2 211.4 285.2 394.2 506.9 650.4 844.2 51.8 67.6 87.4 120.4 162.6 203.3 253.0 339.5 410.4 541.9 41.8 52.8 69.4 85.9 110.5 138.3 173.5 214.2 279.7 338.7 32.1 40.4 48.4 60.4 75.4 94.1 114.9 140.9 167.1 199.3 24.0 29.0 34.7 41.4 50.1 59.1 68.8 82.1 99.2 118.4 18.6 20.0 22.0 25.2 29.5 34.8 40.5 48.5 57.1 64.2 13.6 14.3 16.5 19.0 20.2 23.0 24.4 26.4 28.4 30.8 17.2 18.1 18.5 18.8 18.6 19.0 18.2 18.0 18.2 18.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
A plot of the data presented in Table 1 (see Figure 1) reveals the existence of a density value where the average number of iterations is maximum. Moreover, the complexity curve is almost bimodal showing an easy-hard-easy-hard-easy pattern but with one of the peak larger than the other. It is precisely in that large peak where a critical behavior in the computational cost of solving the BMPG instances is claimed to exist. In the literature [5,15,8] has been established a relation between a phase diagram describing phase transitions in NP problems and the hardness of the instances in the critical point of that diagram. In our case, the pattern shown in Figure 1 allows us to identify one point where
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the computational cost of the instances grows significantly in comparison with other points, this can be the evidence of the existence of a more general critical behavior on BMPG (the existence of the phase transition phenomena) than can explain such behavior. The complexity results obtained with the set of random instances were contrasted with the set of BMPG instances taken from Harwell-Boeing Sparse Matrix Collection1 (HBSM). In order to understand the complexity nature of such benchmark we take the results given by some state-of-the-art BMPG algorithms. Tables 2 and 3 summarize the results reported by the algorithms Simulated Annealing (SA) [33], GRASP [31] and the Tabu Search [32]. The first three columns show information about the instances. The rest of the columns present the time spent by each algorithm when solving the instances. The solution achieved by each algorithm can be seen in [33]. Figure 2 shows a graph of the complexity results of the BMPG state-of-the-art algorithms solving the HBSM benchmark. This figure presents a behavior in the computational cost (measure as the time that took to each algorithm to reach its best solution) similar to the one presented in the set of random instances, in the sense that when the density increases, the computational cost of solving the instances also increases. The experiments showed that the hardest instances were not found in the set of larger instances but in the set of medium instances of the HBSM benchmark. In addition, the average performance of the algorithms (see last row of Tables 2 and 3) over each different set of instances from the HBSM benchmark indicates that the hardest set was the set of medium size. We can note in Tables 2 and 3 that the density of the hardest instances is close to the value pointed out as the critical point in the experiment with random connected BMPG instances, this suggests that the hardest instances lie at the critical region in a phase transition phenomena for both exact and non-exact algorithms. The analysis of the computational cost presented in this section suggests that there are two types of instances of the BMPG that requires a small computational effort to be solved. The first type corresponds to instances with a low density. The second type is formed by instances with a high density. This behavior suggests the existence of two regions, one where the BMPG instances have a low density and the other where they have a high density. The graphs with low density, as paths and cycles, can be considered under-constrained, a small number of edges are connecting the nodes resulting in a small optimal bandwidth for those graphs. These graphs are easy to solve because almost all the solutions are possible, with the possibility of finding the optimal values in a small set of integers. The graphs with high density, as the complete graph, can be consider over-constrained; the great number of edges in it allows only a few set of possible optimal solutions which also makes them easy to solve. The existence of these separated regions could explain the easy-hard-easy transition in the computational cost of solving BMPG, just as it is presented in the satisfiability problem (SAT)[10]. 1
http://math.nist.gov/MatrixMarket/data/
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Fig. 1. Behavior of the computational cost (measured by the number of solved subproblems) of solving random BMPG instances varying the density of the graphs. A curve for each of the 10 values for the number of nodes is shown. The x axis shows the different values of density analyzed. The y axis shows the average number of subproblems required. The forms of each curve drown follow a non-monotonic increase in the complexity revealing that the BMPG problem presents a bimodal phase transition.
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Fig. 2. Computational time to solve the HBSM benchmark using the algorithms SA, TS and GRASP. This graph relates the density of the instances (shown in x axis) with the time that each algorithm spent to solve them (shown in y axis). In this graph shows two peaks indicating a high computational cost; one of them is located between the densities 0.05 and 0.10, the other one is at the density value of 0.35.
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Table 2. Computational time for solving the small size set of cases of HBSM benchmark. It shows the time (in seconds) spent by TS, GRASP and SA to solve each instance. Also, the number of nodes n, edges m and the density of each instance are presented.
Instance dwt 234 nos1 bcspwr03 west0156 will199 west0167 lns 131 impcol c gre 185 gre 115 fs 183 1 bcsstk22 west0132 can 161 nos4 bcspwr02 can 144 ash85 bcspwr01 will57 arc130 curtis54 bcsstk05 gent113 lund b lund a mcca steam3 bcsstk01 impcol b ibm32 bcsstk04 pores 1 Average
n 117 158 118 156 199 167 123 137 185 115 183 110 132 161 100 49 144 85 39 57 130 54 153 104 147 147 168 80 48 59 32 132 30
TS GRASP m Density Time Time 162 0.02 1.2 1.9 312 0.03 1.3 2.6 179 0.03 1.7 0.9 371 0.03 7.4 8.4 660 0.03 12 26.9 489 0.04 5.8 5.6 275 0.04 3.4 2.6 352 0.04 3.5 4.5 650 0.04 6.2 6.1 267 0.04 2.4 3.2 701 0.04 32.4 11.8 254 0.04 1.1 1.6 404 0.05 3.4 8.5 608 0.05 4 0.7 247 0.05 1.1 1.4 59 0.05 0.2 0.6 576 0.06 1.7 3.1 219 0.06 0.7 0.4 46 0.06 0.1 0.1 127 0.08 0.4 0.4 715 0.09 4.8 1.9 124 0.09 0.7 0.7 1135 0.1 4.7 7.1 549 0.1 6.3 1 1147 0.11 5.3 5.5 1151 0.11 8.8 5.4 1662 0.12 23.9 10.8 424 0.13 0.7 1.O 176 0.16 0.9 1 281 0.16 1.3 1.2 90 0.18 0.2 0.3 1758 0.2 16.7 5.4 103 0.24 0.3 0.3 0.08 4.99 4.12
SA Time 1.1 1.1 1.2 2.8 6.1 4.8 1.8 3.1 6.8 1.6 8.7 5.5 5.4 3.0 0.9 0.2 17.6 1.1 0.4 1.1 23.2 0.5 10.6 3.9 40.8 40.6 81.8 8.9 0.6 1.2 0.3 79.2 3.1 11.18
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Table 3. Computational time for solving the medium and large size cases of HBSM benchmark. It shows the time (in seconds) spent by TS, GRASP and SA to solve each instance. Also, the number of nodes n, edges m and the density of each instance are presented. Medium Instance n 494 bus bcsstk20 impcol d plskz362 west0497 bcspwr05 nnc261 nos5 hor 131 gre 343 saylr1 west0479 bcspwr04 ash292 pores 3 dwt 310 dwt 245 can 445 dwt 419 dwt 361 west0381 dwt 221 can 292 impcol a gre 216a bcsstk06 bcsstm07 str 0 plat362 dwt 209 str 200 impcol e str 600 steam1 mbeause mbeaflw mbeacxc
Average
4
494 467 425 362 497 443 261 468 434 343 238 479 274 292 456 310 245 445 419 361 381 221 292 206 216 420 420 363 362 209 363 225 363 240 492 487 487
m 586 1295 1267 880 1715 590 794 2352 2138 1092 445 1889 669 958 1769 1069 608 1682 1572 1296 2150 704 1124 557 660 3720 3416 2446 2712 767 3049 1187 3244 1761 36209 41686 41686
TS GRASP SA Large Density Time Time Time Instance n 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0.06 0.3 0.35 0.35
0.05
m
29.2 13.5 24.4 662 bus 662 906 27.8 11 44.6 nos6 675 1290 21.9 30.5 77.5 lns 511 503 1425 27.4 7.1 20.9 jpwh 991 983 2678 78.3 88.7 137.9 nos2 638 1272 24.7 16.3 26.8 nnc666 666 2148 8.6 22.4 8.7 jagmesh1 936 2664 60.8 103.1 121.9 fs 760 1 760 3518 26 27.7 154.1 fs 680 1 680 1464 16.6 21.2 7.4 gre 512 512 1680 4.3 8.5 2.3 gr 30 30 900 3422 81.2 163 40.5 west0989 989 3500 9.6 5.2 28 sh1 200 663 1720 7.9 8.7 34.4 sh1 0 663 1682 16.8 3.2 13 west0655 655 2841 15.2 8.6 13 sh1 400 663 1709 7.5 14.4 9.3 sherman4 546 1341 64.5 47.2 114 orsirr 2 886 2542 23.7 28 59.9 nos7 729 1944 11.8 0.8 8.3 sherman1 681 1373 113 185.4 38.1 bp 200 822 3788 5 7 23.4 bp 1600 822 4809 19 7.9 61.5 bp 0 822 3260 5.5 3.4 5.4 bp 800 822 4518 7.2 9.5 3.5 bp 600 822 4157 47.6 50.6 247.9 bp 400 822 4015 40.3 113.7 215.7 bcsstk19 817 3018 120.8 119.1 39.9 685 bus 685 1282 24.8 3.4 189.4 bp 1400 822 4760 10.8 1.3 29.2 bp 1200 822 4698 90.9 114.1 47.3 bp 1000 822 4635 10.7 4 49.6 dwt 592 592 2256 180.3 92 55.9 dwt 878 878 3285 16.9 12.7 79.1 saylr3 681 1373 2637.6 5494.3 1289.5 dwt 918 918 3233 3409.4 5467.8 1744.9 can 715 715 2975 3409.1 5464.5 1774 can 838 838 4586 fs 541 1 541 2466 nos3 960 7442 dwt 992 992 7876 dwt 503 503 2762 steam2 600 6580 mcfe 731 15086 289.53 480.54 184.9
TS GRASP SA Density Time Time Time 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.04 0.06 0.01
113.8 32.9 70.4 70.9 42.8 12.5 74.2 58.3 123.8 312.6 65.1 104.3 43.7 15.7 114.5 138.5 55.8 108.5 150.6 107.8 33.8 101.1 97.9 95.6 43.4 33.6 39.8 77.1 92.7 14.7 282.2 85.4 370.1 372.7 416.9 172.1 161.3 98.4 213.5 153.1 110.2 211.5 150.1 245.8 80.5 188.3 121.2 221.4 33 4.1 11.5 203 42.5 164.5 74.5 89.9 23.9 107.2 78 15.6 315.2 550.3 168.4 546.8 783.7 231.8 386.8 483 140.7 520.9 636.9 219.9 480.9 556.8 193.3 355.7 560.6 180.0 174.3 86.6 222.9 90.4 12.7 62.0 521 741.2 217.4 674.8 897.1 218.2 886.7 886.2 216.2 52.9 106.8 123.7 195 99.6 106.7 107.2 77.8 15.6 290.8 12.6 243.0 183.1 14.2 223.0 158.9 37.4 384.2 54.4 26.5 82.4 143.8 209.1 811.6 272.5 306.4 116.4 99.3 7.9 163.5 242.2 182.5 687.4 800.2 247.2 1868.9 241.98 219.03 211.62
Conclusions and Future Work
In this paper we presented experimental evidence of the existence of a critical behavior in the optimization problem known as Bandwidth Minimization Problem for Graphs (BMPG). We study the computational complexity of the BMPG instances in a set of random connected graphs. The critical behavior was observed in the computational
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cost of solving the BMPG instances taking as the order parameter the density of the instances and as the measure of complexity the number of subproblems required for an state-of-the-art Branch & Bound strategy to find the optimal solution. The complexity results were contrasted with reported results of the well known BMPG benchmark of the Harwell Boeing Sparse Matrix Collection. Keeping the density as the order parameter and the time spent by an algorithm to find its best solution as the complexity measure, we presented graphs showing a similar complexity behavior that the one presented in the set of random connected graphs. Moreover, the results uncover the fact that the hardest instances do not correspond to the ones with the highest number of nodes but the ones with the density close to the point marked as the critical point in the experiment with random connected graph (there are medium sized instances). It seems for the evidence shown that the instances that lie in the critical region are hard even for non-exact algorithms. Finally, given the results presented in this document a guideline to construct harder benchmarks for BMPG must consider the density parameter. Currently we are working on analyzing other order parameters to better understand the hardness of the BMPG instances in the point of maximum difficulty. Also, we are studying new complexity measures for non-exact algorithms. In our future work, we also expect to increase the size of instances. Acknowledgments. This research was partially funded by the following projects: CONACyT 58554-C´alculo de Covering Arrays, 51623-Fondo Mixto CONACyT y Gobierno del Estado de Tamaulipas.
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