Evolutionary Algorithms and Combinatorial Optimization

Evolutionary algorithms have often been shown to be successful for difficult combinatorial optimization problems appearing in various industrial, economical, and scientific domains.

Our goal is to understand evolutionary algorithms in the context of combinatorial optimiza-tion problems in a rigorous way. To achieve this goal, we analyze evoluoptimiza-tionary algorithms for problems from combinatorial optimization with respect to their runtime until they have found an optimal solution or a good approximation. These new insights should help practitioners to develop better algorithms for important difficult combinatorial optimization problems.

Shortest Path Problems

Investigators: Benjamin Doerr, Tobias Friedrich, Edda Happ, Christian Klein, and Frank Neumann in cooperation with Surender Baswana (IIT Kanpur), Somenath Biswas (IIT Kanpur), and Piyush P. Kurur (IIT Kanpur)

The first combinatorial optimization problem where rigorous runtime results have been achieved is the well-known single source shortest path (SSSP) problem [11]. Computing shortest paths in a given graph is one of the fundamental problems in computer science and still an important field of research [10, 1]. In the area of bio-inspired computation re-lated problems such as vehicle routing [8] and routing problems in networks [7, 9] have been tackled. Therefore, it seems to be important to understand the basic SSSP problem from a theoretical point of view to gain new insights that will help practitioners solving related problems arising in applications.

In [11], the authors examined a simple EA together with a multi-objective fitness function which makes the EA mimic Dijkstra’s algorithm for the SSSP problem [3]. The upper bound on the number of fitness evaluations given in that paper is O(n²` log n), where n is the number of vertices of the input graph and ` is the maximum over all vertices of the number of edges of a shortest path having a minimum number of edges. In [4], we improved this bound to O(n²max{log n, `}) and showed that this bound is tight, that is we give for each value of ` an example graph for which the expected optimization time is Ω(n²max{log n, `}).

Our bounds not only hold in expectation, but also with high probability, which is with

probability 1−O(n^−c) for an arbitrary constant c. For the analysis, we used that the expected time needed to find a shortest path having `<sup>0</sup> edges is O(n<sup>2</sup>`⁰). The actual time needed is that sharply concentrated on this mean, that using a Chernoff bound and a union bound argument we get the bound of O(n²`). On the other hand, we get a O(n²log n) bound by using arguments similar to the ones used in the Coupon Collector’s theorem. The lower bound can be obtained by similar arguments.

Additionally, a single-objective approach which is supposed to be efficient has been pre-sented in [11]. However, the authors state that they were not able analyze their approach with respect to the runtime behavior. Answering the question whether the SSSP problem can be solved be a single-objective approach further insights into the optimization process of bio-inspired computation methods for this problem are needed. In [2], we showed that the single-objective approach solves the problem after O n³· (log n + log w_max)
fitness evalua-tions with high probability, where w_max is the largest weight of the given input graph. Our proof examines the different possibilities of local changes that can decrease the distance of the current solution to an optimal one. The distance is measured as the sum of the differ-ent path lengths of the currdiffer-ent solution minus the sum of the differdiffer-ent paths lengths of an optimal one. We have shown that there exists at each time step a set of operations which shortens the distance by a factor of (1 − 1/n). Using these insights the stated upper bounded is proven. Our analyses are complemented with a lower bound for a certain class of graphs which shows that our results are almost tight.

A generalization of the SSSP problem is the All-Pairs Shortest Path (APSP) problem.

For this problem, we examined in [5] an evolutionary algorithm that uses as representation of an individual a sequence of edges. In the population we have for each pair of vertices at most one edge sequence starting in the first and ending in the second vertex. A mutation chooses one individual and adds or deletes a Poisson distributed number of times an edge at either end of the individual. We proposed three different crossover operators, which all choose two individuals at random and combine (parts of) them. If an individual is a walk, its fitness value is the length of this walk, otherwise it is infinity. A new individual is accepted, if no other individual in the population has the same start and end vertex or if the fitness of the new individual is not worse than the fitness of the other one connecting the same vertices. By arguments similar to the ones used for the SSSP problem, we showed that if only mutation is used, the optimization time of the algorithm is Θ(n⁴). A rigorous analysis shows that the upper bound drops to O(n^3.5+ε) for an arbitrary ε > 0 if we use with any constant probability any of the crossover operators instead of the mutation operator. Later on, we improved the bound on the optimization time for the crossover-based approach to O(n^3.25log^1/4n) and showed that this bound is asymptotically tight [6]. These analyses are an important step towards understanding how crossover works and how it can be analyzed with rigorous methods. This is the first time that the usefulness of a crossover operator could be shown for a natural combinatorial problem.

References

• [1] H. Bast, S. Funke, P. Sanders, and D. Schultes. Fast routing in road networks using transit nodes. Science, 316(5824):566, 2007.

• [2] S. Baswana, S. Biswas, B. Doerr, T. Friedrich, P. P. Kurur, and F. Neumann. Computing

single source shortest paths using single-objective fitness functions. In T. Jansen, I. Garibay, R. Wiegand, and A. S. Wu, eds., Proceedings of the 10th International Workshop on Foundations of Genetic Algorithms (FOGA 2009), Orlando, USA, 2009. ACM. To appear.

[3] E. W. Dijkstra. A note on two problems in connexion with graphs. In Numerische Mathematik, vol. 1, pp. 269–271. Mathematisch Centrum, Amsterdam, The Netherlands, 1959.

• [4] B. Doerr, E. Happ, and C. Klein. A tight bound for the (1+1)-ea on the single source shortest path problem. In IEEE Congress on Evolutionary Computation 2007, Singapore, 2007, pp.

1890–1895. IEEE.

• [5] B. Doerr, E. Happ, and C. Klein. Crossover can provably be useful in evolutionary computation.

In C. Ryan and M. Keijzer, eds., Genetic and Evolutionary Computation Conference 2008, Atlanta, USA, 2008, Proceedings of the 10th annual conference on Genetic and evolutionary computation, pp. 539–546. ACM. Best paper award.

• [6] B. Doerr and M. Theile. Improved analysis methods for crossover-based algorithms. In G. Raidl and F. Rothlauf, eds., Genetic and Evolutionary Computation Conference 2009, Montreal, Canada, 2009. ACM. To appear.

[7] M. Dorigo and T. St¨utzle. Ant Colony Optimization. MIT Press, 2004.

[8] A. El-Fallahi, C. Prins, and R. W. Calvo. A memetic algorithm and a tabu search for the multi-compartment vehicle routing problem. Computers & OR, 35(5):1725–1741, 2008.

[9] S. J. Kim and M. K. Choi. Evolutionary algorithms for route selection and rate allocation in multirate multicast networks. Appl. Intell., 26(3):197–215, 2007.

[10] P. Sanders and D. Schultes. Engineering highway hierarchies. In Proc. of the 14th Annual European Symposium on Algorithms (ESA ’06), 2006, pp. 804–816.

[11] J. Scharnow, K. Tinnefeld, and I. Wegener. The analysis of evolutionary algorithms on sorting and shortest paths problems. Journal of Mathematical Modelling and Algorithms, 3(4):349–366, 2004.

Sorting and Ordering

Investigators: Benjamin Doerr and Edda Happ

We introduced a new representation for sorting and ordering problems in [1]. In contrast to a previous evolutionary algorithm which uses permutations as representation [2], our repre-sentation is based on directed trees given by an array of predecessors. Given n comparable elements, we add an artificial element a0 which is smaller than all other elements as the root of the tree. An element is supposed to be a descendant of another, if it is smaller. Thus a good initial solution is to set the predecessors of all elements to a0. A natural mutation operator is to choose two elements having the same predecessor and to make one the new predecessor of the other. One possible fitness function is to count the number of vertex pairs for which the smaller one is a predecessor of the bigger one. If one additionally punishes vertex pairs where the opposite is the case, we assure that no such pairs are ever part of an individual. Thus, even before the algorithm has finished, a partial correct solution can be obtained.

The analysis of the (1 + 1)-evolutionary algorithm that arises from these parts shows that the expected optimization time is bounded by O(n²) (whereas an earlier algorithm

for the sorting problem [2] has an upper bound of O(n²log n)). Our algorithm can be effi-ciently implemented so that the optimization time is up to a constant factor equal to the runtime. Experiments imply that the true expected optimization time is even lower (be-tween O(n log n) and O(n log²n)) whereas the algorithm in [2] seems to have an expected optimization time matching the upper bound of O(n²log n).

References

• [1] B. Doerr and E. Happ. Directed trees: A powerful representation for sorting and ordering prob-lems. In Proceedings of CEC 2008, Hong Kong, 2008, pp. 3606–3613. IEEE.

[2] J. Scharnow, K. Tinnefeld, and I. Wegener. The analysis of evolutionary algorithms on sorting and shortest paths problems. Journal of Mathematical Modelling and Algorithms, 3(4):349–366, 2004.

Minimum Cuts

Investigators: Frank Neumann in cooperation with Joachim Reichel (TU Berlin) and Martin Skutella (TU Berlin)

Minimum cut problems belong to the class of basic network optimization problems that occur as crucial subproblems in many real-world optimization problems and have a variety of applications in several different areas. In [3], we studied the minimum s-t-cut problem in graphs with costs on the edges in the context of evolutionary algorithms. We proved that there exist instances of the minimum s-t-cut problem that cannot be solved by standard single-objective evolutionary algorithms in reasonable time. On the other hand, we devel-oped a bicriteria approach based on the famous MaxFlow-MinCut Theorem that enables evolutionary algorithms to find an optimum solution in expected polynomial time. The bi-criteria approach takes the cost of a subset of edges as well as the remaining s-t-flow value into account that can be sent after removing the chosen edges. This trick helps to somehow enlarge the actual search space by enhancing infeasible edge sets (whose removal does not disconnect t from s). The enlarged search space no longer allows for the undesired situation in the single-objective approach discussed above.

While an explicitly given maximum s-t-flow (specified by the flow value on every edge of the graph) directly exposes a minimum s-t-cut, the maximum flow value alone does not contains any structural information about a minimum cut besides the minimum cut capacity.

In particular, having access to such an oracle does not render the minimum cut problem entirely trivial. From a more practical point of view, having access to such a maximum flow oracle seems reasonable in certain situations. Consider, for example, a network of water or oil pipelines. When a leak occurs at some point t of the network, enough pipeline connections have to be cut off by using stop-cocks such that no more liquid leaks from the system. On the other hand, it is desirable to keep the number of inactivated pipeline connections at a minimum in order to keep the negative impact small. In the described scenario, after cutting off some edges, the remaining flow out of the leak can be easily observed and is actually the crucial basis for further decision-making.

Finally, in contrast to the basic minimum s-t-cut problem considered here, in more com-plex settings the comcom-plexity of a minimum cut computation and the related maximum flow

computation can be considerably different. Consider for a example a multi-commodity flow setting with k source-sink pairs (si, ti), i = 1, . . . , k. Here, a maximum multi-commodity flow can be computed in polynomial time while the minimum multicut problem where it is the task to find a set of edges of minimum cost that disconnects every sink t_i from its associ-ated source si, i = 1, . . . , k, is NP-hard [1]. In [2], we generalized our ideas to the NP-hard minimum multicut problem. Given a set of k terminal pairs, we proved that evolutionary algorithms in combination with a natural multi-objective model of the problem are able to obtain a k-approximation for this problem in expected polynomial time.

References

[1] E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis. The complexity of multiterminal cuts. SIAM J. on Comp., 23:864–894, 1994.

• [2] F. Neumann and J. Reichel. Approximating minimum multicuts by evolutionary multi-objective algorithms. In G. Rudolph, T. Jansen, S. M. Lucas, C. Poloni, and N. Beume, eds., Parallel Problem Solving from Nature (PPSN X), Dortmund, Germany, 2008, LNCS 5199, pp. 72–81.

Springer. Best Paper Award.

• [3] F. Neumann, J. Reichel, and M. Skutella. Computing minimum cuts by randomized search heuristics. In C. Ryan and M. Keijzer, eds., Genetic and Evolutionary Computation Conference 2008, Atlanta, USA, 2008, pp. 779–786. ACM Press.

Covering Problems

Investigators: Tobias Friedrich, Nils Hebbinghaus, Stefan Kratsch, and Frank Neumann in cooperation with Jun He (University of Wales) and Carsten Witt (DTU Copenhagen) Covering problems occur frequently in combinatorial optimization. We mainly investigated the Vertex Cover problem. The input is given by a undirected graph G = (V, E) and the task is to compute a set of vertices V⁰ ⊆ V of minimal size such that for each edge e, e ∩ V⁰ 6= ∅ holds, i.e. each edge has at least one vertex in V⁰. First, some simple evolutionary algorithms for single-objective optimization have been investigated. It is shown in [1] that a natural single-objective approach which minimizes the number of vertices and penalizes the number of uncovered edges has an exponential optimization even on simple bipartite graphs.

One property of these bipartite graphs is that they consists of a local optimum with a large inferior neighborhood which makes it hard to obtain the global optimal solution. The other property is that these two optima differ significantly with respect to the value of their solu-tions. Based on these ideas it is shown that simple evolutionary algorithms can only obtain an approximation on this class of instances which is almost trivial. Based on these negative results the combination of evolutionary algorithms which classical approximation algorithms has been studied in [2]. The idea is to start with a solution which is produced by an approxi-mation algorithm for the vertex cover problem and to improve it over time by the stochastic search process of the evolutionary algorithm. The combination of evolutionary algorithms with different approximation algorithms is investigated and the benefits and limitations of this approach are pointed out. On the other hand, we investigated in [1] how multi-objective models can enhance the optimization process for covering problems. Considering the much broader class of set covering problems, we have shown that simple evolutionary algorithms

working with multi-objective models achieve a factor log n-approximation in expected poly-nomial time.

In [3], we investigated the multi-objective model for the vertex cover problem in greater detail and related the runtime of our algorithms to the input size and the cost of an optimal solution. For the first time, it has been pointed out that the search process of evolutionary algorithms using multi-objective models creates partial solutions that are similar to the effect of a kernelization (i. e. a special type of preprocessing from parameterized complexity). Based on this, we showed that evolutionary algorithms solve the vertex cover problem efficiently if the size of a minimum vertex cover is not too large, i.e. the expected runtime is bounded by O(f (OP T ) · n^c), where c is a constant and f a function that only depends on OPT. This shows that evolutionary algorithms are randomized fixed-parameter tractable algorithms for the vertex cover problem.

References

• [1] T. Friedrich, J. He, N. Hebbinghaus, F. Neumann, and C. Witt. Approximating covering problems by randomized search heuristics using multi-objective models. In D. Thierens, ed., Genetic and Evolutionary Computation Conference 2007, London, UK, 2007, pp. 797–804. ACM.

• [2] T. Friedrich, J. He, N. Hebbinghaus, F. Neumann, and C. Witt. Analyses of simple hybrid algorithms for the vertex cover problem. Evolutionary Computation,, 17(1), 2009.

• [3] S. Kratsch and F. Neumann. Fixed-parameter evolutionary algorithms and the vertex cover problem. In G. Raidl and F. Rothlauf, eds., Genetic and Evolutionary Computation Conference 2009, Montreal, Canada, 2009. ACM.