We introduce the CliqueTreeMk algorithm to construct tree decomposition (TD) Mk Landscapes and to compute their global optimum efficiently. TD Mk Landscapes are well suited to serve as benchmark functions for blackbox genetic algorithms that are not given a priori the structural problem information as specified by the tree structure and their associated codomain fitness values. Specifically, for certain types of codomains the use of linkage learning might prove to be necessary in order to be able to solve these type of fitness functions

Chicano, Francisco

Driessel, van, Tobias

Krawiec, Krzysztof

Thierens, Dirk

Utrecht University Repository

A Benchmark Generator of Tree Decomposition Mk LandscapesDirk ThierensUtrecht UniversityUtrecht, Netherlandsd.thierens@uu.nlTobias van DriesselUtrecht UniversityUtrecht, Netherlandstobiasvandriessel@startmail.comABSTRACTWe introduce the CliqueTreeMk algorithm to construct tree decom-position (TD) Mk Landscapes and to compute their global optimumefficiently. TD Mk Landscapes are well suited to serve as bench-mark functions for blackbox genetic algorithms that are not givena priori the structural problem information as specified by the treestructure and their associated codomain fitness values. Specifically,for certain types of codomains the use of linkage learning mightprove to be necessary in order to be able to solve these type offitness functions.CCS CONCEPTS• Computing methodologies→Heuristic function construc-tion;KEYWORDSBenchmarking, Decomposable Landscapes, Dynamic ProgrammingACM Reference Format:Dirk Thierens and Tobias van Driessel. 2021. A Benchmark Generator ofTree Decomposition Mk Landscapes. In Proceedings of the Genetic and Evo-lutionary Computation Conference 2021 (GECCO ’21). ACM, New York, NY,USA, 2 pages. https://doi.org/10.1145/3449726.34594271 INTRODUCTIONSuitable benchmark functions are vital to test the effectiveness andperformance of evolutionary algorithms. Ideally, these benchmarkfunctions should be completely understood in the sense that weknow their structure and, importantly, their global optimum (or op-tima) so that we can check if a given EA has actually found the bestpossible solution. A problem with designing benchmark functionsis that for many interesting problem classes it is not possible to com-pute the global optimum efficiently. Not knowing whether an EAhas found the best solution limits the practical use of the benchmarkand only allows relative comparisons between different algorithms- or different parameter settings of a given algorithm - but it doesnot allow to evaluate the overall performance and effectiveness. Forexample, [1] propose an interesting class of benchmark functions,but unfortunately there is no way to efficiently compute the globaloptimum. Similarly, the well known NK Landscapes does not allowto compute the global optimum. For this reason, EA researchers of-ten use the Adjacent NK Landscapes where the interaction betweenPermission to make digital or hard copies of part or all of this work for personal orclassroom use is granted without fee provided that copies are not made or distributedfor profit or commercial advantage and that copies bear this notice and the full citationon the first page. Copyrights for third-party components of this work must be honored.For all other uses, contact the owner/author(s).GECCO ’21, July 10–14, 2021, Lille, France© 2021 Copyright held by the owner/author(s).ACM ISBN 978-1-4503-8351-6/21/07. . . $15.00https://doi.org/10.1145/3449726.3459427the variables is limited to adjacent problem variables, allowing theuse of dynamic programming to compute the global optimum.2 TREE DECOMPOSITION MK LANDSCAPESAlthough (Adjacent) NK Landscapes are popular as a benchmarkfor optimization algorithms, its constraints (M = N , k = K + 1,and variable xi must appear in subfunction fi ) are unnecessary formost benchmark purposes, as they turn out not to be importantfor most fundamental theoretical properties of NK Landscapes [2].Whitley et al.[3] therefore introduced the term Mk Landscapes torefer to any k-bounded pseudo-Boolean optimization problem, thusa generalization of NK Landscapes without these constraints. M isthe number of subfunctions and k is a constant that provides anupper bound on the interaction order size of the subfunctions, withM being polynomial in n.Whitley introduced an important subclass of Mk Landscapes:the Tree Decomposition (TD) Mk Landscapes. The global optimumof Adjacent NK Landscapes can be computed efficiently becausethey control the tree-width of the variable interaction graph byconsidering only adjacent variables for the subfunctions. Similarly,Whitley et al. created Tree Decomposition Mk Landscapes thatcontrol the tree-width of the interaction graph in a more generalway, thereby allowing the use of dynamic programming to do opti-mization in polynomial time. A Tree Decomposition Mk Landscape(TD Mk Landscape) is any Mk Landscape which has a known andbounded tree-width of k [3]. Tree Decomposition Mk Landscapescan be optimized in polynomial time if k ∈ O(logn) when oneknows the tree decomposition and the associated fitness values ofthe subfunctions of the clique variables.2.1 CliqueTreeMkWhitley et al.[3] introduced a construction algorithm for TD MkLandscapes, however, it limits the construction to TD Mk Land-scapes which have a chain as tree decomposition, just like AdjacentNK Landscapes. Obviously, chains are restricted trees, they cannotrepresent the added complexity that arises from TD Mk landscapecomposed of a branching tree. Here, we introduce an algorithm toconstruct branching - or tree-based - TD Mk Landscapes, and applydynamic programming to efficiently compute the global optimum(or multiple global optima).The CliqueTreeMK algorithm constructs a TD Mk Landscape byprogressively selecting subsets of the problem variables that areput in a random order, and each subset becomes a clique in theclique tree. The first k variables from the shuffled variable list areassigned to become the root of the clique tree C0. Next, for everyclique Ci , b children cliques Cj ∈childreni are selected until we haveconstructedM cliques. Each childCj overlaps with its parentCi foro variables, described by the separator Sj between Ci and Cj , and229GECCO ’21, July 10–14, 2021, Lille, France Dirk Thierens and Tobias van Driesselthe remaining k − o variables are taken from the shuffled variablelist to complete Cj .A dynamic programming algorithm then uses this clique treestructure with its cliques and separators to calculate the globaloptimum. Starting at the leaves of the tree, for each separator Sjwe store for each of the instances of the separator variables themaximizing variable values for its child clique Cj and the resultingscore. Then, we can iterate in the reverse direction and assign valuesto the clique variables in Cj based on the maximizing values for itsvariables stored in its parent separator Cj .We illustrate the CliqueTreeMK algorithm during these phasesusing an example instance with number of subfunctions M = 7,subfunction size k = 3, and overlap o = 2. Together, these definelength N = 9. Furthermore, we choose a branching factor b = 2.The construction algorithm uses fixed values for k , o, and b, butthe algorithm can be extended to allow for non-fixed values duringconstruction. In the example the variables are randomly ordered:(x4,x2,x7,x5,x1,x9,x3,x8,x6).2.1.1 CliqueTreeMK Algorithm.(1) At the start of the algorithm, take next k variables as cliqueC0. Otherwise take next already constructed clique Ci .(2) Choose random o variables from parent clique Ci , assign toseparator Sj(3) Take next (k − o) unchosen variables and add the variablesfrom Sj to construct child clique Cj(4) Go to step 2 until b branches have been built(5) Go to 1 to build the whole treeFollowing the algorithm with the given example instance couldresult in the following list of cliques: (x4,x2,x7), (x4,x7,x5), (x4,x2,x1),(x7,x5,x9), (x4,x7,x3), (x4,x1,x8), (x2,x1,x6)Figure 1 shows the constructed clique tree with its separators.x4   x2  x7x4   x7  x5 x4   x2  x1x7   x5  x9 x4   x7  x3 x4   x1  x8 x2   x1  x6x4  x7 x4  x2x7  x5 x4  x7 x4  x1 x2  x1S1 S2S3S4 S5 S6C0C1 C2C3 C4 C5 C6Figure 1: Clique tree with Cliques Ci and Separators Si .3 EXPERIMENTAL RESULTSWe have implemented the CliqueTreeMk algorithm and a dynamicprogram to generate and compute the global optimum (or globaloptimal) of TDMk landscapes with a given number of subfunctions,length of the subfunctions, overlap size, and branching factor. Thesefunctions are very well suited to act as a benchmark1 for blackboxgenetic algorithms that are not given the structural information ofthe optimization problem - this is, the position of the subfunctions.1Available at https://github.com/tobiasvandriessel/problem-generatorAfter specifying the tree structure of the TD Mk landscape acodomain for the subfunctions needs to be chosen. Possible valuescould be random real numbers between 0 and 1 as in the original NKlandscape, quantized values that enforce the fitness contributionvalues to take one of Q possible discrete values as in the NK-Qlandscape, deceptive trap functions, etc . . . .As an example we generated TD Mk landscapes with randomreal values and with quantized values (Q = 2) for a tree withoverlap o = 1, branching factor b = 2, clique sizes k = 3, 4, 5, andnumber of cliques M = 4, 5, 6, 7, 8, 9, 10. We applied a multi-start,single bit-flipping, local search algorithm (MLS) to explore theselandscapes. The results shown are median values averaged over25 runs of 100,000 fitness function evaluations with MLS. Clearly,the landscapes generated are very different for the two types ofcodomain, especially the actual number of global optima which canall be computed efficiently by the CliqueTreeMK algorithm.k M4 53 60/60-28(28) 179/179-64(64)4 773/756-587(587) 2548/1670-1638(1910)5 5237/2007-4547(8254) 5685/267-4876(61960)Table 1: NK-Q (o = 1, b = 2). #unique local optima found /#unique local optima foundmore than once - #unique globaloptima found (exact number of unique global optima).k M6 7 8 9 103 16/16-1 22/22-1 28/28-1 46/46-1 64/64-14 128/122-1 271/217-1 423/260-1 678/238-1 689/173-15 853/284-1 930/138-0 858/59-0 790/17-0 730/4-0Table 2: Random-NK (o = 1, b = 2). #unique local optimafound / #unique local optima found more than once -#unique global optima found (there is 1 global optimum).4 CONCLUSIONSWe have outlined the CliqueTreeMk algorithm to generate tree de-composable Mk landscapes, and to efficiently compute their globaloptimum (or optima). By varying the codomain of the landscape,multiple types of problems can be created. TD Mk Landscapes arewell suited to serve as benchmark functions for blackbox GeneticAlgorithms that are not given the structural problem informationas specified by the clique tree. Specifically, for particular codomains- for instance (partially) deceptive functions - linkage learning tech-niques will be necessary to be able to find the global optimum (oroptima) reliably and efficiently.REFERENCES[1] Kei Ohnishi, Shota Ikeda, and Tian-Li Yu. 2020. A Test Problem with Difficultyin Decomposing into Sub-Problems for Model-Based Genetic Algorithms. In Pro-ceedings of the 2020 Genetic and Evolutionary Computation Conference Companion(GECCO ’20). Association for Computing Machinery, New York, NY, USA, 221?222.https://doi.org/10.1145/3377929.3389993[2] Darrell Whitley. 2015. Mk landscapes, NK landscapes, MAX-kSAT: A proof thatthe only challenging problems are deceptive. In Proceedings of the 2015 AnnualConference on Genetic and Evolutionary Computation. 927–934.[3] L Darrell Whitley, Francisco Chicano, and Brian W Goldman. 2016. Gray boxoptimization for Mk landscapes (NK landscapes and MAX-kSAT). Evolutionarycomputation 24, 3 (2016), 491–519.230

A benchmark generator of tree decomposition Mk landscapes

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