Graph Bisection Modeled as Binary Quadratic Task Allocation and Solved via Tabu Search

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1 Graph Bisection Modeled as Binary Quadratic Task Allocation and Solved via Tabu Search Mark Lewis a* and Gary Kochenberger b a Steven Craig School of Business, Missouri Western State University, Saint Joseph, MO b School of Business, University of Colorado, Denver, CO Abstract Balanced graph bisection is an NP-complete problem which partitions a set of nodes in the graph G = (N,E) into two sets with equal cardinality such that a minimal sum of edge weights exists between the nodes in the two separate sets. In this paper we transform graph bisection to capacitated task allocation using variable substitutions to remove most capacity and assignment constraints. The resulting problem is in the form of a generic xqx unconstrained quadratic binary problem, except for a single cardinality constraint. Problems are solved using tabu search employing strategic oscillation around critical events created when the single constraint is satisfied. On a set of benchmark graphs, we improve the best known solution for several problems. Comparison results with the fast, freely available multilevel balanced graph partitioning program METIS are presented on a set of random graphs. Our approach works well when balanced graph bisection is augmented to the 2-processor task allocation problem which adds node preferences for a set, as well as edge weights, to the objective function. For these problems, our approach compares favorably to Cplex and Gurobi, providing better solutions in a much shorter time. Keywords: Combinational analysis; Graph bisection, Heuristics; task allocation * Corresponding author: addresses: mlewis14@missouriwestern.edu, Gary.kochenberger@ucdenver.edu 1 of 17

2 1 Introduction Partitioning a graph G = (N, E) of nodes and edges into k segments is useful in many areas, such as parallel computing implementations, [1], numerical solution of differential equations, [2], designing circuit boards and VLSI design, [3] and data mining, [4]. Balanced graph bisection attempts to partition a graph into two sets each with equal node cardinality, while minimizing total edge costs between the nodes of the two sets. Balanced partitioning of a graph is an NP-complete problem with even small instances difficult to solve exactly. State of the art methods, such as described in [5] are able to exactly solve balanced partitioning problems with up to 90 nodes, while [6] have used semidefinite programming relaxations to find nearly optimal bisections of graphs with 250 nodes. To find good solutions to larger problems, heuristic techniques are employed, for which there are excellent techniques, see for example [7]. Polynomial and logarithmic time bisection algorithms having good approximation ratios are discussed in [8]. Many tabu search and evolutionary algorithms are based on the iterative approach for graph bisection presented in the seminal work of [9]. In practice, the objective of the balanced bisection problem is to create two sets of nodes with a minimum edge cut between them and approximately the same number of nodes in each set. Thus, some programs trade-off having exactly equal node cardinality for achieving a smaller total cut cost, particularly when the number of nodes is very high. 1.1 Multiple Partitioning Multiple partitions of a graph can be created by applying bisection repeatedly to the resulting sets, see [10], [11], [12]. When working with very large graph bisections, multilevel algorithms are employed. Multilevel schemes reduce the original graph to an easily solvable size by combining nodes and edges followed by node/set swap refinements (see [13], [14]). Multilevel algorithms for graph partitioning consist of three phases: coarsening the graph until it is an appropriate size, performing a partition on 2 of 17

3 the coarsened graph, then making adjustments as the coarsened graph is projected backwards to create the desired number of partitions. In a graph that has been coarsened to an appropriate size, a good graph bisection algorithm provides a quick, high quality starting solution for the recombining and refinement portion. Multilevel implementations, such as METIS, see [15], are of near linear complexity. Thus, they are fast and scale well to very large graphs and also have been shown to provide good objective function values. 1.2 Two Processor Task Allocation Hendrickson and Kolda [16] provide a survey of partitioning models showing that the typical objective function for graph partitioning is lacking in expression for the underlying problem and suggest several alternative partitioning models. Alternative models allowing an edge to connect more than two nodes, multi-objective partitioning and skewed partitioning can more accurately model a situation. For example, the mapping of activities in a business process into organizational functional areas having preferences for those activities is very close to a partitioning problem, but more accurately modeled as task allocation. Partitioning social networks where the people (nodes) have preferences for the partition into which they might be placed, or the partitioning of air traffic control sectors into airspace blocks that have preferences for which controllers they utilize are other examples where preferences affect partitioning. Skewed partitioning [16-18] includes in the objective function preference values of nodes for sets, essentially creating a two-processor task allocation problem. Skewed partitioning also has application in modeling VLSI design, optimizing the placement of circuit components and in multi-processor task allocation [18]. For example, in multiprocessor task allocation, a task which is represented by a node, may be math intensive and would best be assigned to a math-friendly processor, but in order to complete the task calculations, it must communicate with a large number of other tasks that are not math intensive, e.g. database or communication intensive tasks. The trade-off between assigning this task to be with the other tasks it communicates with versus assigning it to a 3 of 17

4 compatible processor is more accurately modeled as a task allocation problem rather than a balanced graph bisection [19-21]. The task allocation problem (TAP) minimizes the inter-task communication plus the assignment costs of tasks to processors. TAP is an NP-Hard, binary integer quadratic program [22] appearing in uncapacitated (UTAP) and processor capacitated (CTAP) forms. Balanced graph bisection can be modeled as a CTAP problem involving assigning nodes (tasks) to sets (processors) where the two sets (two processors) have capacity constraints of an equal number of nodes (# tasks divided by two). Common heuristics for TAP include multi-start genetic algorithms [6, 23] and multistart tabu search strategies for binary quadratic optimization [24]. Lewis and Kochenberger [25] use a continuous approximation to a quadratic binary problem to solve CTAP instances up to 100 tasks and 10 processors and they show that as the number of processors is decreased, problem difficulty increases. To our knowledge, we are the first to pursue graph bisection using techniques typically applied to task allocation. As a side note, METIS can solve graph bisections that include node weights. It creates partitions such that the node weights (rather than the count of nodes) are evenly balanced between the two sets. For example, [26] used node weighting to quantify the ability of a cell phone to infect other phones via connections, proposing a balanced partition of a social interaction graph to help contain and treat the infection. However, a single weight on a node does not accurately quantify a preference of a node for inclusion in various sets. In this paper we propose an approach that is applicable to both balanced graph bisection as well as two-processor task allocations (2-TAP). We employ variable substitution to take advantage of the binary nature of these problems, recast them within a generic xqx unconstrained binary quadratic optimization framework, then solve them using a strategic oscillation tabu search algorithm. Using problem instances with up to 3000 nodes as well as a benchmark set of graphs from [27], we compare our approach to METIS [15] and the starting heuristics employed in the commercial solvers Cplex and Gurobi. 4 of 17

5 The remaining sections of this paper are organized as follows. Section 2: presentation of mathematical models. Section 3: discussion of tabu search metaheuristic. Section 4: presentation and discussion of computational experience. Section 5: summary and conclusions. 2 Mathematical Model We first formulate the k-tap model, then 2-TAP, which is an augmentation of balanced graph bisection. Let P = {P 1, P 2,, P m } be the set of m distributed processors (sets in graph partitioning) and T = {T 1, T 2,, T n } be the set of n tasks (nodes) to be run on the processors. Let U = {(i,j) : T i and T j interact at some cost} and let c ij be this edge cost between nodes T i and T j. Let q tp be the assignment cost of task (node) t to processor (set) p. This is the node preference cost that differentiates 2-TAP from graph bisection. Therefore, if all q tp = 0, then the problem is a balanced graph bisection. Let x tp be the binary decision variable that is equal to 1 if node t is assigned to set p, and is otherwise 0. In the initial model, 1 - x tp is denoted by x which indicates that tp task t is not assigned to set p. Each task t is required assignment to one of the processors p. The standard model is: n m Min t 1 p 1 q tp x tp m + c x x ij ip jp i, ju, p 1 i j s.t. m p 1 x tp = 1 for t = 1,, n (1) n t 1 x tp = n / m for p = 1,, m x tp {0, 1} for t = 1,, n and p = 1,, m The first term of the minimization objective function is the preference of a task for a processor and is equal to zero for balanced graph bisection. The second term sums 5 of 17

6 the edge costs between tasks that are not allocated to the same processor, that is, they represent nodes in separate sets with edge costs across the partition. The first constraint requires that each task be assigned to only one processor and the second constraint requires that each processor should have an equal number of tasks for a balanced partition into m processors. Note that partitions may not be perfectly balanced, e.g. if there are an odd number of nodes, then a bisection algorithm adds a dummy node to the graph. By making the variable substitution (1 - x tp ) = x we can sum up the benefits tp associated with assigning two nodes to the same set and subtract that constant from the costs associated with assigning nodes to the same set. In other words, let l tp = q tp + t, t ' U, t t ' c t t be the cost associated with assigning a node t to a set p plus all edge costs between the two sets. Thus the second term in the new objective quantifies the benefits associated with having nodes i and j in the same set p. When the number of partitions is two, i.e., m = 2, we have the 2-TAP problem and the following model is used. This model does not change the number of variables needed but the unconstrained part of the model is a maximum flow problem (see [22]), which should be a benefit for commercial optimization packages with excellent presolvers, such as Cplex. The standard 2-TAP model with m = 2 is: n Min t 1 2 p 1 l tp x tp 2 - c x x ij ip jp i, ju, i j p 1 (2) s.t. x x = 1 for t = 1,, n (3) t1 t 2 n t 1 x tp = n / 2 for p = 1, 2 (4) x tp {0, 1} for t = 1,, n and p = 1, 2 6 of 17

7 Lastly, since a graph bisection considers exactly two sets, the variable substitution x i2 = (1 - x i1 ) dramatically reduces the number of variables needed and eliminates the n assignment constraints (3) and the p capacity constraints (4). A single constraint is now needed to specify that half the binary variables be assigned to the first processor. Except for this single cardinality constraint, the model is now in the form of an xqx unconstrained minimization problem. This reduced model, which we employed in our testing, uses variables with a single subscript to indicate whether a task is assigned to set 1. n Min t 1 l t x t - c x x ij i j i, ju, i j s.t. n t 1 x t = n / 2 x t {0, 1} for t = 1,, n A small example using this reduced model is provided below. Minimize 16 x x x x 4-8 x 1 x 2-4 x 1 x 3-20 x 1 x 4-14 x 2 x 3-28 x 2 x 4-20 x 3 x 4 st x 1 + x 2 + x 3 + x 4 = 2 x t is binary The corresponding Q matrix for this problem is Q = The optimal solution is to partition the graph such that tasks 1 and 4 are in set 1, therefore tasks 2 and 3 are in set 2. The resulting objective value is of 17

8 3 Tabu Search Strategic Oscillation Strategic oscillation constitutes one of the primary strategies of tabu search. The variant of strategic oscillation we employ consists of cycles of varying amplitude around critical events. For the class of problems considered here, a critical event is said to occur when the number of variables equal to one satisfies the cardinality constraint and there is an improvement to the objective function value. The method alternates between constructive phases that progressively set variables to 1 (whose steps we call add moves ) and destructive phases that progressively set variables to 0 (whose steps we call drops moves ). To control the underlying search process, memory structures that are updated at critical events are used. Solutions corresponding to critical events are called critical solutions. A parameter span is used to indicate the amplitude of oscillation about a critical event, where span amplitude is a parameter measured by the number of variables added or dropped in a cycle. We begin with span equal to 1 and gradually increase it to a limiting value. For each value of span, a series of alternating constructive and destructive phases is executed until the span limit is reached. In other words, during a constructive phase all the variables currently equal to 0 are evaluated, one at a time, for their effect on being set to 1 and similarly for the destructive phase. At the upper limit of the current span, variables equal to 1 are selected to be set to 0 during the destructive phase, continuing until the lower span limit is reached. The basic oscillation process is shown in Figure 1. In this set of tests, the number of span cycles around a critical event was set to three and the upper span limit was set to seven. Information stored at critical events is used to influence the tabu search process by generating tabu evaluation criteria based on short term and long term frequency information. For example, a short term "recency" vector is used to record the frequency at which variables have been assigned the value 1 in the most recent set of critical events. Long term memory stores all critical event information and is used to introduce a subtle long term bias into the search process that is useful for breaking ties between candidate variables with equal selection evaluation criteria. A detailed description of the method with pseudocode is given in [28]. In the computational section that follows, we refer to this tabu search heuristic by KL. 8 of 17

9 Figure 1. Oscillation around a critical event # of variables equal to 1 in current solution Start of a destructive phase span cycle n / 2 Critical Event Span limit iterations 4 Computational Results We compare our graph bisection approach with the results of other approaches using a benchmark set of graphs and to METIS, Cplex and Gurobi using a set of randomly generated graphs. Cplex is a widely available, well known and high performance optimization benchmark, which uses branch-and-cut to find exact solutions to mixed integer and quadratic programs. Cplex incorporates pre-solvers to discover useful cliques, networks and variable substitutions prior to starting its search for an exact solution using branch-and-cut. Cplex also employs sophisticated heuristics to quickly construct feasible solutions at node zero of the branch-and-cut tree. Gurboi is a branchand-cut solver that provides free licenses for academic use. Its performance compares favorably with Cplex on MIQP benchmark tests. Gurobi takes advantage of multiple core processors and uses heuristics in its optimization engine. Proving optimality of a graph bisection solution is very difficult, with Cplex and Gurobi able to prove optimality on only the smallest problems, and more often than not, running out of memory before finding solutions of the same quality as our approach. For these problems, Cplex often quickly finds a good solution, then spends a long time in branch-and-cut trying to prove the optimality of that solution, only slightly improving the 9 of 17

10 solution along the way. For example, on a dense thirty node balanced bisection, Cplex heuristics were within 1% of the optimal solution within 0.1 seconds, found the optimal in 2900 seconds and proved optimality in 4500 seconds. For a more reasonable comparison with the other heuristics in this paper, we report the objective values and times of the best heuristic solution found by Cplex and Gurobi. A standard Dell desktop processor having a single core 3.2 GHz processor and 2 GB of RAM using Windows XP was used for all comparisons. Our tabu search parameters used default settings derived from other quadratic binary problems previously solved using our unified xqx model. These problems include set partitioning [29], task allocation [20] and linear ordering [30]. Thus, no problem specific parameter tailoring was used for the results reported in Tables 1, 2 and 4. The primary parameters we employed were span = 7 and cycles = 3, although Table 3 reports solution improvements found when letting the heuristic run for a greater number of cycles and spans. These two parameters also define the stopping criteria for the heuristic. The results reported in the tables from our tabu search method are denoted by KL. 4.1 Bisection Comparisons using Randomly Generated Graphs The problems reported in Table 1 were randomly generated graphs with varying edge densities and no node preference weighting. The objective function is to minimize the total edge weights between two sets having an equal number of nodes. In general, all three solution techniques produced objective function values within a few percent of each other. METIS is a near linear time complexity algorithm and the results verify that it is only minimally impacted by increases in the number of nodes or edge density and is very fast. Table 1 shows that our KL algorithm is faster than Cplex and generally finds a better heuristic solution. KL does not scale to problems with a higher number of nodes as well as METIS. For a given number of nodes, increasing edge density seems to reduce problem difficulty. Table 1a provides a comparison to Gurobi on a representative sample of the problems from Table 1. For these problems, Cplex found a better starting solution faster than Gurobi with KL faster than both to consistently better solutions. The stopping 10 of 17

11 criteria for both Cplex and Gurobi was the incumbent feasible solution at the end of their heuristic search at node zero, in other words, the first node of the start of branch-and-cut. For some of the problems, branch-and-cut was left to run until a normal termination condition was encountered, but it was unable to prove optimality for any of these problems. Table 1 Results from randomly generated graph bisections (no node preferences) # nodes % edge density time (sec) Cplex KL METIS KL time factor improvement over over Cplex METIS KL % solution improvement over over Cplex METIS 50 10% % % % % % % % % % % Table 1a Comparison between KL and Gurobi using selected Table 1 instances. # nodes time (sec) % edge density Gurobi Cplex KL KL time factor improvement over Gurobi over Cplex KL % solution improvement over Gurobi over Cplex % % 0.74% % % 0.34% % % 0.12% % % 1.05% % % 3.0% % % 5.6% 11 of 17

12 4.2 Bisection Results Using Benchmark Graphs In Table 2 we present the results of applying our approach to the benchmark set of graph bisections found in [27]. As mentioned previously, no tailoring of tabu search parameters were applied to obtain these results. The solutions marked with an asterisk are known to be optimal and our approach quickly found these solutions. It should be noted that the best known solutions were often generated using days of computing time, while our times are measured in seconds. Our approach is strongest on those graphs with higher edge densities, weaker on very sparse graphs. For example, we quickly find the best known solution, or a new best solution, on all the densest graphs, but have some trouble with the 500 node and 1000 node graphs with low average node degree. As a side note, Cplex was also used on these problems; however the Cplex results are not reported because the times were not competitive, for example, Cplex ran for 2 hours on the 250 node 8% average node degree problem before running out of memory with an objective function value of 840 (versus 828 in under 4 seconds by KL). Encouraged by our good results on the benchmarks, we did further tests on these graphs to see if we could improve the best known solutions. To this end, we let problems run for extended numbers of cycles, and for larger problems increased the span size. By doing so, we were able to match the best known solution to the 500 node 0.01 degree graph and improve the solutions on the 1000 node graphs with 0.05 and 0.1 average degree (see Table 3). Thus, our approach improved the best known solution for the three largest, densest graphs. 12 of 17

13 Table 2 Results on Benchmark Johnson Graphs # nodes Average node degree Best known solution KL time (sec) Delta from best * * * * * *indicates proven optimal Table 3 Johnson Graph Results with Extended Run Times Average node Previous best KL time Solution # nodes degree known solution solution (min) improvement Comparison Results based on 2-TAP Bisection Model Table 4 compares our results to Cplex on a set of randomly generated 2-TAP graphs of varying edge density, including node preferences for set inclusion. Note that METIS does not solve this category of problem. The table illustrates that our approach generated solutions that improved the Cplex heuristic solution and did so in a shorter period of time. 13 of 17

14 The results also show that our approach scales better as node count and edge density increases. For example we are about 10 times faster for the smaller problems and 100 times faster to a better solution for the largest problems. Table 4a illustrates that for the problems with 2500 nodes and more, the Gurobi optimizer spends less time generating an initial heuristic starting solution than Cplex does, but their starting solution is not as good. KL is about 3x faster than Gurobi on the larger 2-TAP problems and found better solutions to all problems. Based on the typical MIP gap of 100% between incumbent solution and current branch and cut relaxation, for both Gurobi and Cplex, 2-TAP is more difficult than balanced graph bisection. Table 4 Results for the 2-TAP graphs (having edge weights & node preferences) Time (sec) % % solution Cplex Cplex (sec) / KL KL # nodes density improvement* (heuristic) (sec) % 5.0% % 2.8% % 3.0% % 2.1% % 3.0% % 1.6% % 3.8% % 3.2% % 3.2% % 3.0% % 3.5% % 2.6% % 3.1% % 3.0% % 5.6% % 2.6% * percent reduction in objective function of KL over Cplex, computed as 1 KL / Cplex 14 of 17

15 Table 4a Comparison between KL, Cplex and Gurobi using selected Table 4 instances. # nodes time (sec) % edge density Gurobi Cplex KL KL time improvement over Gurobi Over Cplex KL solution improvement Over Gurobi Over Cplex % % 5.0% % % 3.0% % % 1.6% % % 3.1% % % 3.0% % % 5.6% 5 Summary & Conclusions In this paper we augmented the balanced graph bisection model by incorporating preferences between nodes and sets, thus creating capacitated task allocation problems. These 2-TAP problems were modeled as binary quadratic programs. High quality graph bisections were generated using tabu search with strategic oscillation around critical events. Results on a benchmark set of graph bisections were very competitive in solution quality and time to solution and we improved the best known solution for three of the fifteen problems. On a set of randomly generated graph bisection problems, our approach provided better solutions than either METIS, Cplex or Gurobi, although METIS was faster. Our approach to solving 2-TAP was shown to be computationally effective when compared to the heuristically generated solutions of Cplex and Gurobi and our approach scaled better than either as graph size increased. Future research involves integrating our approach as the partitioning phase of a multi-level scheme for k-partitioning much larger graphs. 15 of 17

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