Online Supplement to Minimax Models for Diverse Routing

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1 Online Supplement to Minimax Models for Diverse Routing James P. Brumbaugh-Smith Douglas R. Shier Department of Mathematics and Computer Science, Manchester College, North Manchester, IN , USA Department of Mathematical Sciences, Clemson University, Clemson, SC , USA 1 Computational Results: Path-Capture Problem Twenty-four algorithmic variants of the path-generation approach were investigated, based on the formulated LP model (MMP, MMD), the LP solution method (primal, dual), the pricing mechanism (single, multiple, partial), and the selection of dual variables (extreme-point, interior-point). Table I summarizes the codings used to specify these 24 algorithms. In option C, the master problem was solved using CPLEX s CPXoptimize (primal simplex) routine; in option D, the CPXdualopt (dual simplex) routine was used. In addition to using the extreme-point duals (option H) produced by solving the restricted minimax LP, we also considered selecting interior-point duals (option I) to make as many dual variables positive as possible, in effect spreading the penalties over more arcs. To accomplish this we solved the following quadratic program using CPLEX s CPXbaropt routine: min s.t. i π2 i πc R y i π i =1 π 0 Table I. Codes for Algorithmic Variations LP Form LP Solver Pricing Duals A=Primal C=Primal E=Single H=Extreme B=Dual D=Dual F=Multiple I = Interior G=Partial 1.1 Hypercube Networks The path-capture algorithm was investigated on hypercubes H d of dimension d =4, 5,...,10. Experimental data were collected using a Sun Ultra-1 Workstation with 64 MB of main memory and a 143 MHz processor. The total CPU time (in seconds, averaged over the ten replications) was obtained for each algorithmic variant, using the C function clock. 1

2 Figure 1 plots for all 24 algorithms the average (total) CPU time versus the number of nodes n for 16 n The dotted curves correspond to option I, employing interior-point duals; these variants were always less efficient than their counterparts that used extreme-point duals. Such behavior is attributable to the significant overhead of solving the associated quadratic program. Figure 2 shows the average execution times of the five fastest algorithms (ACFH, ADFH, BCFH, BCGH, BDGH). Appropriate matched-pairs tests show a statistically significant difference between BCFH and ACFH (the next most efficient algorithm) for all network sizes. Thus Algorithm BCFH (primal solver applied to the dual LP, using multiple pricing and extreme-point duals) dominates all other variants over the range of hypercube networks considered Total CPU Time (sec.) # Nodes Figure 1. Mean execution time vs. # nodes: all algorithms (hypercube). Several models were considered for expressing the execution time T (n) ofalgorithm BCFH as a function of the network size n. A power model of the form T (n) =β 0 n β1 provided a good fit, resulting in R 2 = The estimated regression parameters are β 0 = and β 1 = A95percent confidence interval for the parameter β 1 is (1.295, 1.391), providing evidence that the time complexity is only mildly superlinear over the range of network sizes studied. Hypercubes on 2048 and 4096 nodes were also constructed to determine whether the apparently modest growth in CPU time by Algorithm BCFH would continue for larger networks. The power model applied to the extended data set still has a large R 2 value but examination of the residuals suggests that an alternative model would be more appropriate, namely a model of the form T (n) = β 0 n β1 e β2n. The fitted model is T (n) = n e n with R 2 = Inclusion of the exponential term e β2n indicates that ultimately the algorithm s execution time will grow rapidly with n; however the small magnitude of β 2 confirms that over the extended range 16 n 4096, 2

3 Algorithm BCFH can nonetheless efficiently solve the path-capture routing problem. 12 ADFH 10 BDGH BCGH 8 ACFH Total CPU Time (sec.) 6 4 BCFH # Nodes Figure 2. Mean execution time vs. # nodes: fastest algorithms (hypercube). 1.2 Shuffle-Exchange and Circulant Networks We also examined the 24 algorithms on shuffle-exchange and circulant networks. The d-dimensional shuffle-exchange network S d, with 2 d nodes and 3 2 d 1 arcs, was studied with d varying from 5 to 12. The circulant network C n (±1, ±k), with n nodes, 2n arcs, and k = n, was studied for n =16, 32, 64,...,1024. In timing data collected for the shuffle-exchange networks, with 32 n 4096, Algorithms ACFH, BCFH, BCGH, and BDFH were consistently the most efficient implementations for the pathcapture problem. Matched-pairs tests could not distinguish these four algorithms from each other. Figure 3 shows the execution times for these algorithms, all of which appear approximately linear in the number of nodes n. In fact a regression performed for Algorithm BCFH indicated the relationship to be slightly sub-linear for network sizes up to The fitted model was T (n) = n with R 2 = Network sizes were then extended to nodes (d = 14) to further examine Algorithm BCFH. As with hypercubes, mildly exponential behavior is now evidenced by the good fit of the model T (n) = n e n with R 2 = A model (with well-behaved residuals) for the number of paths generated versus the number of nodes was the very low-order power model P (n) =5.220n

4 2.5 2 Total CPU Time (sec.) # Nodes Figure 3. Mean execution time vs. # nodes: fastest algorithms (shuffle exchange). Initial tests involving circulant networks, with 16 n 1024, showed that Algorithms ACFH and BCFH were the most efficient for solving the path-capture problem. Matched-pairs tests could not distinguish between these two algorithms; however they were distinguishable from the next fastest algorithm, BCGH. For Algorithm BCFH a near-linear power model fit the data well for networks up to 1024 nodes. However when networks were extended to 4096 nodes a better model was T (n) = n e n with R 2 = The best model for P (n) was higher order than logarithmic but still sub-linear: P (n) =0.8804n Table II summarizes the estimated model parameters for the growth in CPU time for Algorithm BCFH on the three structured topologies. Table II. Estimated Parameters for T (n) =β 0 n β1 e β2n Path Capture, Algorithm BCFH Topology β0 β1 β2 Hypercube Shuffle Exchange Circulant

5 1.3 Random Networks The various path-capture algorithms were also studied on random undirected networks R(n, m) with n nodes and m arcs, n 1 m ( n 2). Random networks were initially generated with 128, 256, 512 nodes and average degrees of 5, 10, 15, 20, 25, 30. Note that the arc density q = m n is one-half the average degree. For each of the eighteen possible combinations of size and density, ten replications were generated where each replication has both a unique set of arcs and arc lengths. Empirical results showed that Algorithm BCFH dominated the other path-capture algorithms for the range of densities investigated. For Algorithm BCFH, various models were examined relating execution time T to arc density q and the number of nodes n. While the empirical model T (n, q) = n q has a high R 2 value (0.9397), the model T (n, q) = n q e nq, with R 2 =0.9482, has better behaved residuals. Models were also fit to express the total number of paths generated as a function of both n and q. Inall cases the parameter associated with n was not significant at the.05 level, giving rise to the empirical model P (q) =1.585q with R 2 = Heuristics for the Path-Capture Problem We now examine three heuristic methods for obtaining approximate solutions to the path-capture problem. Specifically, these methods produce disjoint solutions consisting of relatively short paths. The first heuristic is the arc-elimination shortest path (AESP) approach. Namely, we begin by determining a shortest (minimum length) st-path. All arcs in this path are then eliminated from the network and a new shortest st-path is found. This process is repeated until no st-path exists in the reduced network. While the arc-elimination method does not necessarily produce a maximum set of arc-disjoint st-paths, such a set can be obtained by solving a maximum st-flow problem with unit capacities on all arcs. Notice that since the maximum flow problem does not explicitly consider the arc lengths, there is no guarantee of obtaining short paths. However, an augmenting-path maximum flow algorithm can be used to build up a set of paths which are locally short either in cardinality (number of arcs) or in cost (length). Specifically, at each step an augmenting path of least cardinality or minimum cost can be selected in the current residual network. The maximum flow heuristics based on these two implementations are denoted MFLC and MFMC, respectively. 2.1 Computational Results The heuristics AESP, MFLC, and MFMC were studied using the same experimental design described in Section 1, except that the focus is primarily on larger networks. Specifically, we examined hypercubes having from 16 to 4096 nodes, circulants with 256 to 4096 nodes, shuffle-exchange networks with 1024 to nodes, and random networks with 1024 nodes and an average node degree ranging from 5 to 30. Thus a total of 25 network topology/size combinations were considered, each with ten replications, yielding a total of 250 test networks. Our empirical study of the three disjoint-path heuristics gathered information on two important quantities: speed and solution quality. The former is measured by the percent of CPU time required 5

6 relative to the exact algorithm BCFH; the latter is measured by the optimality gap, the percent above the optimal objective value achieved by the heuristic. While MFLC is very fast (consistent with its low-order theoretical complexity), the solution quality is quite poor, with objective values often 20 50% above optimal. Consequently, we have omitted this heuristic from any further mention. Table III compares the optimality gaps of AESP and MFMC for the largest random networks (n = 1024). Bolded figures indicate superiority when comparing AESP and MFMC. The sample standard deviation ±s for each set of ten replications is also given. Heuristic AESP is seen to be superior to MFMC for networks having an average node degree of 15 and greater. For lower densities, the solution quality is identical. Table IV shows that for all densities AESP is the faster heuristic. Table III. Percent Optimality Gap for Random Networks Heuristic Average Node Degree AESP ±0.40 ±1.22 ±0.95 ±1.05 ±0.90 ±0.92 MFMC ±0.40 ±1.22 ±2.36 ±2.43 ±1.35 ±2.77 Table IV. Percent CPU Time Required for Random Networks Heuristic Average Node Degree AESP ±2.11 ±2.12 ±1.94 ±0.51 ±0.60 ±0.58 MFMC ±2.29 ±2.25 ±1.99 ±0.62 ±0.54 ±0.67 Table V shows the clear dominance (in solution quality) of Heuristic MFMC over AESP for circulants of all sizes. Furthermore, the execution times turned out to be quite comparable for these two heuristics. The dominance of Heuristic MFMC is also observed for shuffle-exchange networks in Table VI; in addition MFMC was significantly faster than AESP in execution time. Finally, hypercube networks were examined over the range d = 4, 5,...,12. Similar tables indicate that for node degrees (or dimension d) exceeding 5, Heuristic AESP produces more accurate solutions than MFMC, while for smaller densities (d = 4, 5) Heuristic MFMC yields better quality solutions. Solution times were comparable for both heuristics. In summary, AESP is generally preferred for networks having average node degree greater than five, while MFMC is preferred for lower-density networks. Using this adaptive rule for heuristic selection produces a solution within three percent of optimality for 85% of our test cases and within five percent of optimality in 97% of the cases. 6

7 Table V. Percent Optimality Gap for Circulant Networks Heuristic Number of Nodes AESP ±5.53 ±3.38 ±2.23 ±1.29 ±2.65 MFMC ±3.70 ±2.06 ±1.11 ±0.83 ±0.93 Table VI. Percent Optimality Gap for Shuffle-Exchange Networks Heuristic Number of Nodes AESP ±0.32 ±2.81 ±2.19 ±2.46 ±5.18 MFMC ±0.26 ±2.46 ±1.13 ±2.06 ± Improvement Phase for Disjoint Heuristics An improvement phase can be added to enhance the quality of the solutions given by Heuristic AESP or MFMC. The improvement phase consists of using a previously obtained set of disjoint paths as the starting point for the path-generation algorithm. In order to retain the polynomial-time complexity of the underlying construction heuristic (AESP or MFMC), we iterate the improvement phase until an amount of time has elapsed that is proportional to the heuristic construction time. In particular, we examined the efficacy of using an expansion factor (EFactor) ofeither 10 or 15. As we are primarily concerned with the efficacy of heuristics for larger networks, we consider only those instances requiring at least one second of CPU time to compute the exact solution: namely, random (n = 1024; m = 10240, 12800, 15360), hypercube (n = 512, 1024, 2048, 4096), circulant (n = 1024, 2048, 4096), and shuffle-exchange (n = 4096, 8192, 16384) networks. Table VII summarizes the solution qualities by network type using EFactor = 10. The values in parentheses are the comparable values for the adaptively chosen heuristic (AESP or MFMC) used in the construction phase. To further test the effectiveness of using adaptive heuristic selection with an improvement phase, five very large test networks were generated, each containing over 30,000 arcs. Results are summarized in Table VIII. Times are given in seconds, whereas values in parentheses indicate the percent CPU time, compared to Algorithm BCFH, and the percent optimality gap. For the four random networks, use of the construction phase alone resulted in optimality gaps ranging from 0.17% to 1.35% and required approximately 2.5% of the CPU time needed to obtain the exact solution. Use of the improvement phase with EFactor = 10 involved an increased CPU time of approximately 28% while achieving reduced optimality gaps of 0.08% to 0.90%. Although these results represent 7

8 Table VII. Solution Quality After Improvement Phase (EFactor = 10) Percent Percent Within Given Average Topology Optimal Percentage of Optimal Optimality Gap (%) Random (0) (8) (30) (88) (100) (1.69) Hypercube (5) (18) (28) (80) (98) (2.08) Circulant (0) (20) (37) (97) (100) (1.32) Shuffle (63) (73) (73) (83) (90) (1.35) only a single replication for each network size, they suggest that the adaptive construction approach, possibly augmented with an improvement phase, can obtain high quality heuristic solutions to large problems in a reasonable amount of time. 3 Computational Results: Backward-Capture Problem This section investigates the empirical performance of several variants of the column-generation approach for the backward-capture problem. It is expected that optimal solutions will exhibit significant disjointness only near the destination node t, inview of the much larger costs associated with arc failures occurring close to t. The computational data reported here are collected using the same computing environment and the same test networks described for the path-capture algorithms, except that some of the smaller networks have been omitted and several larger networks have been added. In view of the expense of obtaining interior-point duals and their limited effectiveness in speeding up the solution process, we consider only the 12 algorithmic variations that use extremepoint dual values: Algorithms ACEH BDGH. 3.1 Exact Algorithms While the empirical results cited here primarily relate to hypercube networks, many of the observations and conclusions apply to the other three network topologies. A first observation is that variants using the dual form of the LP (BCEH BDGH) have faster execution times than the corresponding variants using the primal LP (ACEH ADGH). Moreover, for Algorithms BCEH BDGH the multiple-pricing schemes (BCFH and BDFH) turned out to be preferable to single and partialpricing for all network sizes examined. In addition, the dual solver (BDFH) is generally preferred to the primal solver (BCFH), particularly for larger network sizes. Figure 4 plots the mean execution time versus network size n for hypercube networks; Algorithms BCFH and BDFH have been applied 8

9 Table VIII. Exact and Heuristic Path-Capture Solutions for Five Very Large Networks Exact Construction Improvement Phase Test Network Solution Phase (EFactor = 10) Time Obj. Time Obj. Time Obj. Random n = 3000, m = (2.67%) (0.17%) (28.0%) (0.08%) n = 3000, m = (2.41%) (1.35%) (28.3%) (0.90%) n = 4000, m = (2.44%) (0.71%) (28.2%) (0.41%) n = 5000, m = (2.61%) (0.84%) (26.7%) (0.84%) Shuffle Exchange n = 32768, m = (4.27%) (4.18%) (59.3%) (0.00%) to both the path-capture and backward-capture problems. Inspection of this figure produces the following conclusions, which also apply to the other three topologies: Algorithm BCFH is more efficient for solving the path-capture problem; Algorithm BDFH is more efficient for the backward-capture problem; and exact solutions can be obtained significantly faster for the backward-capture problem. Figure 5 compares the number of paths generated and the number utilized during solution of the path-capture and backward-capture problems using Algorithm BCFH. For the path-capture problem, on average ten times as many paths are generated than are actually utilized in the final routing solution. However when solving the backward-capture problem, only about 20% more paths are generated than are utilized. This fairly constant factor is illustrated in Figure 5 by the nearly uniform gap (on a logarithmic scale) between corrresponding curves. For backward capture. a goodfitting model representing the number of paths generated as a function of the number of nodes n is P (n) =2.017 ln n with R 2 = This demonstrates the significant reduction in the number of paths generated compared to path capture, where a similar model has a coefficient of associated 9

10 with the logarithmic term; again an order of magnitude gain in efficiency is noted PATH CAPTURE BACK CAPTURE BDFH 50 Total CPU Time (sec.) BCFH BCFH # Nodes BDFH Figure 4. Mean execution time vs. # nodes: Algorithms BCFH and BDFH (hypercube). 2.2 Generated Common Log of Average # of Paths Generated Utilized Utilized 0.8 PATH CAPTURE BACK CAPTURE # Nodes Figure 5. Log of # paths vs. # nodes: Algorithm BCFH (hypercube). 10

11 Table IX. Estimated Parameters for T (n) =β 0 n β1 e β2n Backward Capture, Algorithm BDFH Topology β0 β1 β2 Hypercube Shuffle Exchange Circulant For the backward-capture problem, we model the execution time T for Algorithm BDFH as a function of network size n using T (n) =β 0 n β1 e β2n, just as in analyzing the empirical complexity of path-capture algorithms. Table IX summarizes the estimated model parameters for hypercube, shuffle-exchange, and circulant networks. Notice that the estimates for β 1 are significantly and consistently lower than those seen in the path-capture case (Table II). This reflects the significantly lower execution times, since in this model the factor n β1 dominates for the network sizes under consideration. For random networks, a model for T as a function of n and the density q was fitusing T (n, q) = n q Again both estimated exponents are significantly lower than those for the analogous path-capture model in Section 1.3. The modest execution times required for this problem (particularly compared to solution times for the path-capture problem) are attributable to the relatively small number of paths that must be generated to achieve and verify optimality. 3.2 Backward-Capture Heuristics First we examine how well the backward-capture problem can be approximated using disjoint paths. The primary reasons for exploring disjoint-path solutions are the ease with which such paths can be obtained (Algorithm AESP) and the simplicity in obtaining the optimal routing and objective value algebraically. While the disjoint-path heuristic AESP may offer speed, it is not expected to necessarily provide high-quality solutions since it enforces disjointness at both terminals (s and t). Thus we also investigate an arc-elimination heuristic TDIS tailored to the backward-capture problem; it greedily finds deg(t) paths such that each arc incident to t is used exactly once. The effectiveness of these two heuristics was studied empirically using the four network topologies. In particular, percent CPU times were computed relative to the best exact algorithm (BDFH). Using totally disjoint paths (AESP) produces optimality gaps of 35 55% for random networks, 20 25% for the larger hypercubes, and 40 50% for shuffle-exchange networks. The optimality gaps for circulants were generally lower, averaging about 10 20%. Although this heuristic had low execution times compared to BDFH, as expected it did not yield high-quality approximations. Solution qualities for TDIS were reasonably good (less than 10 15% above optimality) for low-density random and hypercube networks but the quality degraded as the density increased. Optimality gaps for shuffleexchange networks were rather inconsistent (ranging from 10 40%) while the solution quality for circulants was not as good as that achieved by AESP. The times required to generate the TDIS solution were comparable to those for AESP. However, because evaluation of the objective function requires setting up and solving an LP, the total amount of time taken by the TDIS heuristic was 11

12 generally much higher than the AESP computation time. In view of the large overhead (initial LP solution) involved in TDIS, we investigated applying an improvement phase to the Heuristic TDIS. Using EFactor = 10 yielded quite small optimality gaps: essentially 0% for random, hypercube, and shuffle-exchange networks, whereas about 10 20% for circulants. Typical CPU times ranged from 65% to 95% (of that required by BDFH) for random, circulant, shuffle-exchange, and high-density hypercube networks. In fact, using EFactor = 5 still provided reasonably good solution qualities (for random, hypercube, and shuffle-exchange networks) but with only a modest decrease in CPU time. We also investigated using the TDIS solution as a starting point for the exact solution method BDFH; this approach was beneficial for the denser random and hypercube networks, reducing the average CPU time by about 15 35% compared to BDFH. However, it was not particularly effective for circulant and shuffle-exchange networks. Finally the TDIS heuristic was tested on the same very large networks defined at the end of Section 2.2, with results summarized in Table X. Note that values in parentheses represent the percent CPU time (compared to BDFH) and the percent optimality gap. Times indicated for the improvement phase also include time spent in the construction phase (Heuristic TDIS). For three of the four random networks examined, TDIS obtains a good quality initial solution in about half the exact solution time. Furthermore, solution quality for the four random networks decreases with increasing average degree (density); however the denser networks required less CPU time (percentage wise). The use of an improvement phase with EFactor = 10 also shows promise for large random networks as we generally achieve optimality in approximately 50% of the exact CPU time. Application of the improvement phase to the very large shuffle-exchange network achieved optimality using 87% of the exact CPU time. It is worth pointing out that even in these very large networks, the exact solution times are not particularly large, so in absolute terms the use of TDIS together with an improvement phase is fairly successful in making such large problems practically solvable. 12

13 Table X. Exact and Heuristic Backward-Capture Solutions for Five Very Large Networks Exact Construction Improvement Phase Test Network Solution Phase (TDIS) (EFactor = 10) Time Obj. Time Obj. Time Obj. Random n = 3000, m = (45.6%) (4.18%) (54.0%) (0.00%) n = 3000, m = (32.9%) (64.7%) (116%) (0.00%) n = 4000, m = (41.3%) (0.28%) (48.7%) (0.00%) n = 5000, m = (52.8%) (0.00%) (55.3%) (0.00%) Shuffle Exchange n = 32768, m = (50.7%) (30.2%) (86.6%) (0.00%) 13