Analysis of Evolutionary Diversity Optimisation for Permutation Problems
AAnalysis of Evolutionary Diversity Optimisation for PermutationProblems
Anh Viet Do
Optimisation and LogisticsThe University of Adelaide, Adelaide, Australia
Mingyu Guo
Optimisation and LogisticsThe University of Adelaide, Adelaide, Australia
Aneta Neumann
Optimisation and LogisticsThe University of Adelaide, Adelaide, Australia
Frank Neumann
Optimisation and LogisticsThe University of Adelaide, Adelaide, Australia
ABSTRACT
Generating diverse populations of high quality solutions has gainedinterest as a promising extension to the traditional optimizationtasks. We contribute to this line of research by studying evolutionarydiversity optimization for two of the most prominent permutationproblems, namely the Traveling Salesperson Problem (TSP) andQuadratic Assignment Problem (QAP). We explore the worst-caseperformance of a simple mutation-only evolutionary algorithm withdifferent mutation operators, using an established diversity measure.Theoretical results show most mutation operators for both problemsensure production of maximally diverse populations of sufficientlysmall size within cubic expected run-time. We perform experimentson QAPLIB instances in unconstrained and constrained settings, andreveal much more optimistic practical performances. Our resultsshould serve as a baseline for future studies.
KEYWORDS
Evolutionary algorithms, diversity maximization, traveling salesper-son problem, quadratic assignment problem, run-time analysis
Evolutionary diversity optimization (EDO) aims to compute a setof diverse solutions that all have high quality while maximally dif-fering from each other. This area of research started by Ulrich andThiele [20, 21] has recently gained significant attention within theevolutionary computation community, as evolution itself is increas-ingly regarded as a diversification device rather than a pure objectiveoptimizer [17]. After all, in nature, deviating from the predecessorsleads to finding new niches, which reduces competitive pressureand increases evolvability [11]. This perspective challenges the no-tion that evolutionary processes are mainly adaptive with respectto some quality metrics, and that population diversity is only inservice of adapting its individuals and is without intrinsic worth. Inoptimization, diversity optimization is a useful extension to the tra-ditional optimization tasks, as a set of multiple interesting solutionshas more practical value than a single very good solution.Along this line of research, there have been studies that exploredifferent relationships between quality and diversity. A trend emerg-ing from the evolutionary robotics is Quality Diversity, which fo-cuses on exploring diverse niches in the feature space and maximiz-ing quality within each niche [3, 5, 9, 17]. This approach maximizesdiversity via niches discovery, meaning the what constitutes a nichein the solution space needs to be well-defined beforehand. Otherstudies place more importance on diversity measured directly fromsolutions, applying evolutionary techniques to generate images withvarying features [2], or to compute diverse Traveling SalespersonProblem (TSP) instances [4, 8] useful for automated algorithm se-lection and configuration [10]. Different indicators for measuringthe diversity of sets of solutions in EDO algorithms such as the stardiscrepancy [14] or popular indicators from the area of evolutionarymulti-objective optimization [15] have been investigated to createhigh quality sets of solutions. The study [6] explores EDO for the TSP, the first study on solution diversification for a combinatorialoptimization problem.In this study, we contribute to the understanding of evolution-ary diversity optimization on combinatorial problems. Specifically,we focus on TSP and Quadratic Assignment Problem (QAP), twofundamental NP-hard problems where solutions are representedas permutations, and the latter of which has also been attemptedwith genetic algorithms [1, 13, 18, 19]. The structures of the solu-tion spaces associated with these problems are similar, yet differentenough to merit distinct diversity measures. We use two approachesto measuring diversity: one based on the representation frequenciesof βobjectsβ (edges or assignments) in the population, and one basedon the minimum distance between each solution and the rest. Weconsider the simple evolutionary algorithm that only uses mutation,and examine its worst-case performances in diversity maximizationwhen various mutation operators are used. Our results reveal howproperties of a population influence the effectiveness of mutationsin equalizing objectsβ representation frequencies. Additionally, wecarried out experimental benchmark on various QAPLIB instancesin unconstrained (no quality threshold) and constrained settings,using a simple mutation-only algorithm with 2-opt mutation. Theresults indicate optimistic run-time to maximize diversity on QAPsolutions, and show maximization behaviors when using differentdiversity measures in the algorithm. With this, we extend the in-vestigation in [6] theoretically, and experimentally with regard toQAP.The paper is structured as follows. In Section 2, we introduce theTSP and QAP in the context of evolutionary diversity optimizationand describe the algorithm that is the subject of our analysis. InSection 3, we introduce the diversity measures for both problems.Section 4 consists of the run-time analysis of the introduced algo-rithm. We report on our experimental investigations in Section 5and finish with some conclusions.
Throughout the paper, we use the shorthand [ π ] = { , . . . , π } . Thesymmetric TSP is formulated as follow. Given a complete undirectedgraph πΊ = ( π , πΈ ) with π = | π | nodes, π = π ( π β )/ = | πΈ | edgesand the distance function π : π Γ π β R β₯ , the goal is to computea tour of minimal cost that visits each node exactly once and finallyreturns to the original node. Let π = [ π ] , the goal is to find a tourrepresented by the permutation π : π β π that minimizes the tourcost π ( π ) = π ( π ( π ) , π ( )) + π β βοΈ π = π ( π ( π ) , π ( π + )) . The QAP is formulated as follow. Given facilities πΉ = { π , . . . , π π } ,locations πΏ = { π , . . . , π π } , weights π€ : πΉ Γ πΉ β R β₯ , flows π : πΏ Γ πΏ β R β₯ , find a 1-1 mapping π : πΉ β πΏ that minimizes the costfunction π ( π ) = βοΈ π,π β πΉ π€ ( π, π ) π ( π ( π ) , π ( π )) . a r X i v : . [ c s . N E ] F e b lgorithm 1 ( π + ) -EA for diversity optimization π β initial population while stopping criteria not met do πΌ β ππππππππππππ‘ ( π ) πΌ β² β ππ’π‘ππ‘π ( πΌ ) if π ( πΌ β² ) β€ ( + πΌ ) πππ then π β π βͺ { πΌ β² } πΌ β²β² β argmin π½ β π { πππ£πππ ππ‘π¦ ( π \ { π½ })} π β π \ { πΌ β²β² } end ifend whilereturn π A problem instance is encoded with two π Γ π matrices: one for π€ andone for π . Similar to TSP, we can abstract πΉ and πΏ like we do π : πΉ = [ π ] and πΏ = [ π ] . Therefore, each mapping is uniquely defined by a [ π ] β [ π ] permutation. Given that there is a 1-to-1 correspondencebetween all permutations and all mappings, the solution space is thepermutation space. This is an important distinction between TSPand QAP from which low-level differences between the diversitymeasures in each case emerge. On the other hand, the high levelstructure of a tour is identical to that of a mapping, so the notionslike distance or diversity are the same for both above a certain layerof abstraction.In this paper, we consider diversity optimization for the TSP andthe QAP. For each problem instance, we are to find a set π of π = | π | solution that is diverse with respect to some diversity measure, whileeach solution meets a given quality threshold. Let πππ is the valueof an optimal solution, a solution πΌ satisfies the quality threshold iff π ( πΌ ) β€ ( + πΌ ) πππ , where πΌ > ( + πΌ ) approximationsfor a problem instance. We assume that the optimal tour is knownfor a given TSP or QAP instance.We consider ( π + ) -EA algorithm which was used to diversifyTSP tours [6]. The algorithm is described in Algorithm 1. It uses onlymutation to introduce new genes, and tries to minimize duplicationin the gene pool with elitist survival selection. The algorithm slightlymodifies the population in each step by mutating a random solution,essentially performing random local search in the population space.As with many evolutionary algorithms, it can be customized fordifferent problems, in this case by modifying the mutation operatorand the diversity measure. In this work, we are interested in worst-case performances of the algorithm under the assumption that anyoffspring is acceptable. The structure of a TSP tour is similar to that of a QAP mapping inthe sense that they are both each defined by a set of objects: edges intours and assignments in mappings. In fact, the size of such a set isalways equal to the instance size. For this reason, diversity measuresfor populations of tours, and those for populations of mappingsshare many commonalities. In particular, we describe two measuresintroduced in [6], customized for TSP and QAP. For consistency,we use the same notations for the same concepts between the twoproblems unless told otherwise. We also refer to [6] for more in-depth discussion on the measures, and fast implementations of thesurvival selection for Algorithm 1 based on these measures, whichcan be customized for QAP solutions.
In this approach, we consider diversity in terms of equal representa-tions of edges/assignments in the population. It takes into account, for each object, the number of solutions containing it, among the π solutions in the population.For TSP, given a population of tours π and an edge π β πΈ , wedenote by π ( π, π ) its edge count, which is defined, π ( π, π ) = |{ π β π | π β πΈ ( π )}| β { , . . . , π } where πΈ ( π ) β πΈ is the set of edges used by tour π . Then in order tomaximize the edge diversity we aim to minimize the vector N ( π ) = sort ( π ( π , π ) , π ( π , π ) , . . . , π ( π π , π )) , in the lexicographic order where sorting is performed in descendingorder. As shown in [6], this maximizes the pairwise distances sum π· ( π ) = βοΈ π β π βοΈ π β π | πΈ ( π ) \ πΈ ( π )| . Similarly for QAP, given a population of mappings π , we denoteby π ( π, π, π ) its assignment count as follow, π ( π, π, π ) = |{ π β π |( π, π ) β π΄ ( π )}| β { , . . . , π } where π΄ ( π ) β [ π ] Γ [ π ] is the set of assignments used by solution π . The corresponding vector to be minimized in order to maximizeassignment diversity is then N ( π ) = sort ( π ( π, π, π )) π,π β[ π ] , in the lexicographic order where sorting is performed in descendingorder. Similar, this maximizes the following quantity π· ( π ) = βοΈ π β π βοΈ π β π | π΄ ( π ) \ π΄ ( π )| . While this diversity measure is directly related to the notionof diversity, using it to optimize populations has its drawbacks.As mentioned in [6], populations containing clustering subsets ofsolutions can have high π· score, which is undesirable. For thisreason, we also consider another measure that circumvents thisissue. Instead of maximizing all pairwise distances at once, this approachfocuses on maximizing smallest distances, potentially reducing largerdistances as a result. Optimizing for this measure reduces clusteringphenomena, as well as tends to increase the distance sum. In thisapproach, we minimize the following vector lexicographically D( π ) = sort (cid:16)(cid:0) π π,π (cid:1)
π,π β π (cid:17) , where sorting is performed in descending order, and π ππ = | πΈ ( π ) β© πΈ ( π )| if π and π are TSP tours, and π ππ = | π΄ ( π ) β© π΄ ( π )| if theyare QAP mappings. Doing this would also maximize the followingquantity π· ( π ) = βοΈ π β π min π β π \{ π } {| πΈ ( π ) \ πΈ ( π )|} , or π· ( π ) = βοΈ π β π min π β π \{ π } {| π΄ ( π ) \ π΄ ( π )|} . We can see that for any TSP tour population π of size at most (cid:4) π β (cid:5) ,we have argmin π {N ( π )} = argmin π {D( π )} = argmin π { π· ( π )} = argmin π { π· ( π )} . One of the results in this study implies that the same is true forany QAP mapping population of size at most π . On the other hand,when π > π , π β β argmax π { π· ( π )} doesnβt necessarily imply π β β argmin π {D( π )} , as shown by the following example. xample 1. For a QAP instance where π = and π = , let π = ( , , , ) , π = ( , , , ) , π = ( , , , ) , π = ( , , , ) , π = ( , , , ) , π = ( , , , ) , π = ( , , , ) , π = { π , π , π , π , π } , π β² = { π , π , π , π , π } , we have π· ( π ) = π· ( π β² ) = which is themaximum. However, D( π ) = ( , , , , , , , , , ) > D( π β² ) = ( , , , , , , , , , ) . Because of this, it is tricky to determine the maximum achievablediversity D in such cases. For now, we rely on the upper bound ππ of π· , which is relevant to our experimentation in Section 5. We investigate the theoretical performance of Algorithm 1 in op-timizing for N without the quality criterion. For TSP, we considerthree mutation operators: 2-opt, 3-opt (insertion) and 4-opt (ex-change). For QAP, we consider the 2-opt mutation where two assign-ments are swapped. In particular, we are interested in the number ofiterations until a population with optimal diversity is reached. Ourderivation of results is predicated on the lack of local optima: it isalways possible to strictly improves diversity in a single step of thealgorithm. Let π π = max π β πΈ { π ( π, π )} and π π = | π β πΈ | π ( π, π ) = π π | . For eachnode π , let ππ ( π ) be the set of edges incident to π , and π ( π, π ) = (cid:205) π β ππ ( π ) π ( π, π ) . For each tour πΌ , let 2 πππ‘ ( πΌ, π, π ) be the tour resultedfrom applying 2-opt to πΌ at positions π and π in the permutation, and4 πππ‘ ( πΌ, π, π ) be the tour from exchanging π -th and π -th elements in πΌ .We assume π β₯ π β₯ π π or π π , asaligned with the algorithmβs convergence path.Lemma 1. Given a population of tours π such that β€ π β€ (cid:4) π + (cid:5) and π π β₯ , there exist a tour πΌ β π and a pair ( π, π ) , such that π β² = ( π \ { πΌ }) βͺ { πππ‘ ( πΌ, π, π )} satisfies, ( π π > π π β² β§ π π = π π β² ) β¨ π π > π π β² . (1) Moreover, in each iteration, the Algorithm 1 with 2-opt mutationand πΌ = β makes such an improvement with probability at least [( π β ) ( π π β )+ ] ππ ( π β ) . Proof. There must be π π tours πΌ in π such that β π β πΈ ( πΌ ) , π ( π, π ) = π π , let πΌ be one such tour. W.l.o.g, let πΌ be represented by a permuta-tion of nodes ( π , π , . . . , π π ) where π (( π , π ) , π ) = π π . The operation2 πππ‘ ( πΌ, , π ) trades edges ( π , π ) and ( π π , π π + ) in πΌ for ( π , π π ) and ( π , π π + ) . If π (( π , π π ) , π ) < π π β π (( π , π π + ) , π ) < π π β
1, then π β² = ( π \ { πΌ }) βͺ { πππ‘ ( πΌ, , π )} satisfies (1) since π (( π , π π ) , π β² ) and π (( π , π π + ) , π β² ) cannot reach π π . We show that there is always such aposition π . Since π can only go from 3 to π β
1, there are π β π . Itβs the case that π ( π, π ) = π for any π since each tour con-tributes 2 to π ( π, π ) , and that π (( π π , π ) , π ) β₯ π (( π , π ) , π ) β₯ πΌ uses them, thus π β βοΈ π = π (( π , π π ) , π ) β€ π β π π β β€ (cid:106) π (cid:107) β π π , and (2) π βοΈ π = π (( π , π π ) , π ) β€ (cid:106) π (cid:107) β π π . According to the pigeonhole principle, (2) implies there are at least πΏ elements π from 3 to π β π (( π , π π ) , π ) < π π β
1, where πΏ = π β β (cid:22) β π / β β π π π π β (cid:23) . Likewise, there are at least πΏ elements π from 4 to π such that π (( π , π π ) , π ) < π π β
1. This implies that there are at least 2 πΏ β π + π from 3 to π β π (( π , π π ) , π ) < π π β π (( π , π π + ) , π ) < π π β
1. We have2 πΏ β π + = π β β (cid:22) β π / β β π π π π β (cid:23) β₯ ( π β )( π π β ) + π π β β₯ , proving the first part of the lemma. In each iteration, the Algorithm1 selects a tour like πΌ with probability at least π π / π . There are at least [( π β )( π π β ) + ]/( π π β ) different 2-opt operations on such atour to produce π β² . Since there are π ( π β )/ π β² from π is atleast ( π β )( π π β ) + π π β π π ππ ( π β ) β₯ [( π β )( π π β ) + ] ππ ( π β ) . β‘ In Lemma 1, only one favorable scenario is accounted for whereboth edges to be traded in have counts less than π π β
1. However,there are other situations where strict improvements would be madeas well, such as when both swapped-out edges have count π π . Fur-thermore, a tour to be mutated might contain more than 2 edgeswith such count, increasing the number of beneficial choices dramat-ically. Consequently, the derived probability bound is pessimistic,and the average success rate might be much higher. It also meansthat the bound of the range of π is pessimistic and the lack of lo-cal optima is very probable at larger population sizes, albeit withreduced diversity improvement probability.Intuitively, larger population sizes present more complex searchspaces where local search approaches are more prone to reachingsub-optimal results. It is reasonable to infer that small populationsizes make diversity maximization easier for Algorithm 1. However,for 3-opt mutation, local optima can still exist even with popula-tion size being as small as 3. Next, we show a simple constructionof supposedly easy cases where 3-opt fails to produce any strictimprovement.Example 2. For any TSP instance of size π β₯ where π is a multipleof 4, we can always construct a population of 3 tours having sub-optimal diversity, such that no single 3-opt operation on any tour canimprove diversity. Let the first tour be πΌ = ( π , π , . . . , π π ) , we derivethe second tour πΌ sharing only 2 edges with πΌ and containing edgesthat form a βcrisscrossingβ pattern on πΌ , πΌ = ( π , π π β , . . . , π π + , π π β π β , . . . , π π / β , π π / + ,π π / , π π / + , . . . , π π / β π , π π / + π , . . . , π , π π ) . The third tour πΌ shares no edge with πΌ or πΌ and contains many edgesthat βskip one nodeβ on πΌ . πΌ = ( π , . . . , π π + , . . . , π π / β , π π / + , . . . , π π / + π , . . . , π π π π / , . . . , π π / β π , . . . , π , π π β , . . . , π π β π β , π π / + ) . In order to improve diversity, the operation must exchange, on eithertour, at least one edge with count 2. However, any 3-opt operation withsuch restriction ends up trading in at least another edge used by theother tours, nullifying any improvement it makes. This populationpresents a local optimum for algorithms that uses 3-opt as the onlysolution generating mechanism. Figure 2 illustrates two examples ofthe construction with π = and π = . We speculate that in many cases, the insertion 3-opt suffers fromits asymmetrical nature. Both 2-opt and 3-opt operations are eachdefined by two decisions. For 2-opt, the two decisions are which igure 1: Examples of constructed tours with π = and π = where no single 3-opt operation on any tour improves diver-sity. two edges to be exchanged, and only after both are made will thetwo new edges be fixed. For 3-opt, one decision determines whichset of two adjacent edges to exchanged, and the other defines thethird edge. Unlike 2-opt, after only one decision, one out of thethree new edges is already fixed. Such limited flexibility makesit difficult to guarantee diversity improvements via 3-opt withoutadditional assumptions about the population. In contrast, 4-opt isnot subjected to this drawback, as the two decisions associated withit are symmetric. For this reason, we can derive another result for4-opt similar to Lemma 1.Lemma 2. Given a population of tours π such that β€ π β€ (cid:4) π + (cid:5) and π π β₯ , there exist a tour πΌ β π and a pair ( π, π ) , such that π β² = ( π \ { πΌ }) βͺ { πππ‘ ( πΌ, π, π )} satisfies (1) . Moreover, in each iteration,the Algorithm 1 with 4-opt mutation and πΌ = β makes such animprovement with probability at least [( π β ) ( π π β )+ ] ππ ( π β ) . Proof. There must be π π tours πΌ in π such that β π β πΈ ( πΌ ) , π ( π, π ) = π π , let πΌ be one such tour. W.l.o.g, let πΌ be represented by a permu-tation of nodes ( π , π , . . . , π π ) where π (( π , π ) , π ) = π π . The opera-tion 4 πππ‘ ( πΌ, , π ) trades edges ( π , π ) , ( π , π ) , ( π π β , π π ) , ( π π , π π + ) in πΌ for ( π , π π ) , ( π , π π ) , ( π , π π β ) , ( π , π π + ) . If π (( π , π π ) , π ) < π π β π (( π , π π ) , π ) < π π β π (( π , π π β ) , π ) < π π β π (( π , π π + ) , π ) < π π β
1, then π β² = ( π \ { πΌ }) βͺ { πππ‘ ( πΌ, , π )} satisfies (1) followingsimilar reasoning in the proof of Lemma 1. We show that there isalways such a position π . Since π can only go from 5 to π β
1, thereare π β π . We use the fact that π ( π, π ) = π for any π , andthat π (( π , π ) , π ) β₯ πΌ uses them, thus π β βοΈ π = π (( π , π π β ) , π ) β€ π β π π β β€ (cid:106) π (cid:107) β π π , and (3) π β βοΈ π = π (( π , π π + ) , π ) β€ (cid:106) π (cid:107) β π π . According to the pigeonhole principle, (3) implies there are at least πΏ elements π from 5 to π β π (( π , π π β ) , π ) < π π β
1, where πΏ = π β β (cid:22) β π / β β π π π π β (cid:23) . Likewise, there are at least πΏ elements π from 5 to π β π (( π , π π + ) , π ) < π π β
1. This implies that there are at least 2 πΏ β π + π from 5 to π β π (( π , π π β ) , π ) < π π β π (( π , π π + ) , π ) < π π β
1, which we will call condition 1. We denotethe number by ΞΞ = πΏ β π + = π β β (cid:22) β π / β β π π π π β (cid:23) , Using π (( π , π π ) , π ) β₯
1, we similarly derive that there are at least πΏ element π from 5 to π β π (( π , π π ) , π ) < π π β
1. However, we only have π (( π , π ) , π ) β₯
1, meaning there are at least πΏ β² element π from 5 to π β π (( π , π π ) , π ) < π π β πΏ β² = π β β (cid:22) β π / β π π β (cid:23) . From this, we have that there are at least πΏ + πΏ β² β π + π from5 to π β π (( π , π π ) , π ) < π π β π (( π , π π ) , π ) < π π β Ξ β² Ξ β² = πΏ + πΏ β² β π + = π β β (cid:22) β π / β β π π π π β (cid:23) β (cid:22) β π / β π π β (cid:23) , Finally, we can infer that there are at least Ξ + Ξ β² β π + π such that both condition 1 and condition 2 are met. We have Ξ + Ξ β² β π + β₯ π + β β π / β β π π π π β β₯ ( π β )( π π β ) + π π β β₯ , proving the first part of the lemma. In each iteration, the Algorithm1 selects a tour like πΌ with probability at least π π / π . There are at least [( π β )( π π β ) + ]/( π π β ) different 4-opt operations on such atour to produce π β² . Since there are π ( π β )/ π β² from π is atleast ( π β )( π π β ) + π π β π π ππ ( π β ) β₯ [( π β )( π π β ) + ] ππ ( π β ) . β‘ Like in Lemma 1, only one out of many favorable scenarios isconsidered in Lemma 2, so the lower bound is strict. The range of thepopulation size is smaller to account for the fact that the conditionfor such a scenario is stronger than the one in Lemma 1. With theseresults, we derive run-time results for 2-opt and 4-opt, relying onthe longest possible path from zero diversity to the optimum.Theorem 1.
Given any TSP instance with π β₯ nodes and π β₯ ,the Algorithm 1 with πΌ = β obtains a π -population with maximumdiversity within expected time O( π π ) if β’ it uses 2-opt mutation and π β€ (cid:4) π + (cid:5) , β’ it uses 4-opt mutation and π β€ (cid:4) π + (cid:5) . Proof. In the worst case, the algorithm begins with π π = π and π π = π . At any time, we have π π β€ ππ / π π . Moreover, in theworst case, each improvement either reduces π π by 1, or reduces π π by 1 and sets π π to its maximum value. With 2 β€ π β€ (cid:4) π + (cid:5) , themaximum diversity is achieved iff π π = π βοΈ π = πππ ππ ( π β ) [( π β )( π β ) + ] = O( π π ) . On the other hand, Lemma 2 implies that when 2 β€ π β€ (cid:4) π + (cid:5) , Al-gorithm 1 with 4-opt mutation needs at most the following expectedrun time π βοΈ π = πππ ππ ( π β ) [( π β )( π β ) + ] = O( π π ) . β‘ As expected, the simple algorithm requires only quadratic ex-pected run-time to achieve optimal diversity from any starting pop-ulation of sufficiently small size. The quadratic scaling with π comesfrom two factors. One is the fact that Algorithm 1 needs to selectthe βcorrectβ tour to mutate out of π tours. The other is the fact thatup to π β π comesfrom the quadratic number of possible mutation operations, and thenumber of edges to modify in each tour. Additionally, most of the un-time is spent on the βlast stretchβ when reducing π π from 2 to1, as the rest only takes up O( π π ) expected number of steps. Let π π = max π,π β[ π ] { π ( π, π, π )} and π π = | π, π β [ π ]| π ( π, π, π ) = π π | .We denote the 2-opt operation by π π,π (Β·) where π and π are twopositions in the permutation to be exchanged. For each π β [ π ] , let π ( π, π ) = (cid:205) π β[ π ] π ( π, π, π ) and π§ ( π, π ) = (cid:205) π β[ π ] π ( π, π, π ) . Let π be ashift operation such that for all permutation π : [ π ] β [ π ] , π = π ( π ) = β β π β [ π β ] , π ( π ) = π ( π + ) β§ π ( π ) = π ( ) . Also, for convenience, we use the notation π΄ ( π ) = {( π, π )|β π β π, π ( π ) = π } . We first show the achievable maximum diversity for anypositive π , which will be the foundation for our run-time analysis.Theorem 2. Given π, π β₯ , there exists a π -size population π ofpermutations of [ π ] such that max π,π β[ π ] π ( π, π, π ) β min π,π β[ π ] π ( π, π, π ) β€ . (4)Proof. We prove by constructing such a π . Let π : [ π ] β [ π ] be some arbitrary permutation and π = { π π ( π )| π β [ π ]} where π π is π applied π times. Note that π π ( π ) = π . It is the case that notwo solutions in π share assignments, so for all π, π β [ π ] , we have π ( π, π, π ) =
1, and π΄ ( π ) = [ π ] Γ [ π ] . Let π = ππ + π where π, π β N and π < π , and π΅ β π where | π΅ | = π , we include in π π + π΅ and π copies of each solution in π \ π΅ . Then π satisfies (4) since β( π, π ) β π΄ ( π΅ ) , π ( π, π, π ) = π + , and β( π, π ) β π΄ ( π \ π΅ ) , π ( π, π, π ) = π. β‘ With maximum diversity well-defined, we can determine whetherit is reached with population π using only information from N ( π ) .Therefore, we can show the guarantee of strict diversity improve-ment with a single 2-opt on some sub-optimal population, and theprobability that Algorithm 1 makes such an improvement, similarto Lemma 1. For brevityβs sake, we reuse the expression (1) withnotations defined in the QAP context.Lemma 3. Given a population of permutations π such that β€ π β€ (cid:4) π + (cid:5) and π π β₯ , there exist a permutation π β π and pair ( π, π ) where β€ π < π β€ π , such that π β² = ( π \ { π }) βͺ { π π,π ( π )} satisfies (1) . Moreover, in each iteration, the Algorithm 1 with 2-opt mutationand πΌ = β makes such an improvement with probability at least [( π + ) ( π π β )+ ] ππ ( π β ) . Proof. There must be π π permutations π in π such that β π β[ π ] , π ( π, π ( π ) , π ) = π π , let π be one such permutation, and π β [ π ] such that π ( π, π ( π ) , π ) = π π . The operation π π,π ( π ) trades assignments π β π ( π ) and π β π ( π ) in π for π β π ( π ) and π β π ( π ) . Regardlessof π ( π, π ( π ) , π ) , if π ( π, π ( π ) , π ) < π π β π ( π, π ( π ) , π ) < π π β π β² = ( π \ { π }) βͺ { π π,π ( π )} satisfies (1) since π ( π, π ( π ) , π β² ) and π ( π, π ( π ) , π β² ) cannot reach π π . We show that there is always such aposition π . There are π β π since π β π . Itβs the case that π ( π, π ) = π§ ( π, π ) = π , thus βοΈ π β π ( π ) π ( π, π, π ) β€ π β π π β€ (cid:22) π + (cid:23) β π π , and (5) βοΈ π β π π ( π, π ( π ) , π ) β€ (cid:22) π + (cid:23) β π π . According to the pigeonhole principle, (5) implies there are at least πΏ elements π β π such that π ( π, π ( π ) , π ) < π π β
1, where πΏ = π β β (cid:22) β( π + )/ β β π π π π β (cid:23) . Likewise, there are at least πΏ elements π β π such that π ( π, π ( π ) , π ) < π π β
1. This implies that there are at least 2 πΏ β π + π β π where π ( π, π ( π ) , π ) < π π β π ( π, π ( π ) , π ) < π π β
1. We have2 πΏ β π + = π β β (cid:22) β( π + )/ β β π π π π β (cid:23) β₯ ( π + )( π π β ) + π π β β₯ , proving the first part of the lemma. In each iteration, the Algorithm1 selects a tour like πΌ with probability at least π π / π . There are atleast [( π + )( π π β ) + ]/( π π β ) different 2-opt operations on sucha tour to produce π β² . Since there are π ( π β )/ π β² from π is atleast ( π + )( π π β ) + π π β π π ππ ( π β ) β₯ [( π + )( π π β ) + ] ππ ( π β ) . β‘ Compared to Lemma 1, the range of π in Lemma 3 is about twiceas large, which coincides with the fact that the maximum numberof disjoint solutions (sharing no assignment/edge) for any giveninstance size is also twice as large in QAP than it is in TSP. Theresult lends itself to the following run-time bound for Algorithm 1,similar to Theorem 1.Theorem 3. Given any QAP instance with π β₯ and β€ π β€ (cid:4) π + (cid:5) , the Algorithm 1 with 2-opt mutation and πΌ = β obtains a π -population with maximum diversity within expected time O( π π ) . Proof. In the worst case, the algorithm begins with π π = π and π π = π . At any time, we have π π β€ ππ / π π . Moreover, in the worstcase, each improvement either reduces π π by 1, or reduces π π by1 and sets π π to its maximum value. With 2 β€ π β€ (cid:4) π + (cid:5) , themaximum diversity is achieved iff π π = π βοΈ π = πππ ππ ( π β ) [( π + )( π β ) + ] = O( π π ) . β‘ The results in Theorem 1 and 3 are identical due to similaritiesbetween structures of TSP tours and QAP mappings, and the sameintuition applies. Of note is that according to the proofs, the proba-bility of making improvements drops as the population is closer tomaximum diversity. This is a common phenomenon for randomizedheuristics in general, which we expect to see replicated in experi-mentation.
We perform two sets of experiments to establish baseline results forevolving diverse QAP mappings. These involve running Algorithm1 separately using two described measures: N and D . We denotethese two variants by π· and π· . The mutation operator used is2-opt. Firstly, we consider the unconstrained case where no qualityconstraint is applied. Then, we impose constraints with varyingquality thresholds πΌ on the solutions.For our experiments, we use three QAPLIB instances: Nug30 [16],Lipa90b [12], Esc128 [7]. The optimal solutions for these instancesare known. We vary the population size among 3, 10, 20, 50. Werun each variant of the algorithm 30 times on each instance, andeach run is allotted ππ maximum iterations. It is important tonote that any reported diversity score is normalized with the upperbound appropriate to the instance. For π· , the bound is derivedfrom Theorem 2, while it is ππ for π· as mentioned. We specify thedifferences in settings between unconstrained case and constrainedcase in the following sections. .1 Unconstrained diversity optimization In the unconstrained case, we are interested in how optimizing forone measure affect the other, and how many iterations are neededto reach maximum diversity from zero diversity. To this end, we setthe initial population to contain only duplicates of some randomtour. Furthermore, we apply a stopping criterion that holds whenthe measure being optimized for reaches its upper bound. However,for π > π , the bound is unreachable, so we expect that the algorithmdoes not terminate prematurely while minimizing D .Figure 2 shows the mean diversity scores and their standard devi-ations throughout the runs, and the average numbers of iterationstill termination. Overall, when π β€ π , Algorithm 1 maximizes both π· and π· well within the run time limit. The ratios between neededrun-times and corresponding total run-times seem to correlate withthe ratio π / π . Additionally, the algorithm seems to require similarrun-time to optimize for both measures, as no consistent differencesare visible.The figure also shows a notable difference in the evolutionary tra-jectories resulted from using N and D for survival selection. When D is used, Algorithm 1 improves π· about as efficiently as when N is used. On the other hand, when N is used, it increases π· poorlyduring the early stages, and even noticeably decreases it in shortperiods. In fact, in many cases, π· only starts to increase quicklywhen π· reaches a certain threshold. That said, this particular dif-ference is not observable for π =
3. Nevertheless, it indicates thateven in easy cases ( π β€ π ), highly even distributions of assignmentsin the population are unlikely to preclude clustering. In contrast,separating each solution from the rest of the population tends toimprove overall diversity effectively. In the constrained case, we look for the final diversity scores acrossvarying πΌ and the extent to which optimizing for π· mitigate clus-tering, especially at small πΌ . Therefore, we consider πΌ values 0.05,0.2, 0.5, 1, and run the algorithm for ππ steps with no additionalstopping criterion. Furthermore, we initiate the population withduplicates of the optimal solution to allow flexibility for meaningfulbehaviors. Aside from diversity scores, we also record the percent-age of assignments belonging to exactly one solution (unique) outof ππ assignments in each final population.Table 1 shows a comparison in terms of π· and π· scores aswell as unique assignment percentages. Overall, maximum diver-sity is achieved reliably in most cases when πΌ = . ,
1. For Lipa90b,there are tremendous gaps in final diversity scores when πΌ changesfrom 0 .
05 to 0 .
2. The differences are much smaller in other QAPLIBinstances. Also, at πΌ = .
5, maximum diversity is not reached asfrequently for Esc128 as for other instances. These suggest signifi-cantly different cost distributions in the solution spaces associatedwith these QAPLIB instances.Comparing the diversity scores from the two approaches, we cansee trends consistent with those in the unconstrained case. Eachapproach predictably excels at maximizing the its own measure overthe other. That said, the π· approach does not fall far behind in π· scores, even in cases where statistical significance is observed (atmost 7% difference). Meanwhile, the π· approachβs π· scores aremuch lower than those of the other, especially in hard cases (small πΌ and large π ). The same differences can be seen in the percentages ofunique assignments, which seem to strongly correlate with π· . Thisindicates that using the measure D , Algorithm 1 significantly re-duces clustering, and equalizes assignmentsβ representations almostas effectively as when using the measure N . We studied evolutionary diversity optimization in the TravelingSalesperson Problem and Quadratic Assignment Problem. In thistype of optimization problem, the goal is to maximize diversity asquantified by some metric, and the constraint involves the solutionsβqualities. We described the similarity and difference between thestructure of a TSP tour and that of a QAP mapping, and customizedtwo diversity measures to each problem. We considered a baseline ( π + ) evolutionary algorithm that incrementally modifies the pop-ulation using traditional mutation operators on one solution at atime. We showed that for any sufficiently small π , the algorithmguarantees maximum diversity in TSP within using 2-opt and 4-optwithin O( π π ) expected iterations, while 3-opt suffers from localoptima even with very small π . We derived the same result in QAPwith 2-opt, where the upper bound of π is more generous. Additionalexperiments on QAPLIB instances shed light on differences on evolu-tionary trajectories when optimizing for the two diversity measures.Our results show heterogeneity in the correlation between the qual-ity constraint threshold and the achieved diversity across differentinstances, and that the average practical performance is much moreoptimistic than the worst-case suggests. ACKNOWLEDGMENTS
This work was supported by the Phoenix HPC service at the Univer-sity of Adelaide, and by the Australian Research Council throughgrant DP190103894.
REFERENCES [1] Ravindra K. Ahuja, James B. Orlin, and Ashish Tiwari. 2000. A greedy geneticalgorithm for the quadratic assignment problem.
Computers & Operations Research
27, 10 (Sept. 2000), 917β934. https://doi.org/10.1016/s0305-0548(99)00067-2[2] Brad Alexander, James Kortman, and Aneta Neumann. 2017. Evolution of artisticimage variants through feature based diversity optimisation. In
Proceedings ofthe Genetic and Evolutionary Computation Conference . ACM, 171β178. https://doi.org/10.1145/3071178.3071342[3] Alberto Alvarez, Steve Dahlskog, Jose Font, and Julian Togelius. 2019. EmpoweringQuality Diversity in Dungeon Design with Interactive Constrained MAP-Elites.In . IEEE, 1β8. https://doi.org/10.1109/cig.2019.8848022[4] Jakob Bossek, Pascal Kerschke, Aneta Neumann, Markus Wagner, Frank Neumann,and Heike Trautmann. 2019. Evolving diverse TSP instances by means of novel andcreative mutation operators. In
Proceedings of the 15th ACM/SIGEVO Conferenceon Foundations of Genetic Algorithms - FOGA '19 . ACM Press, 58β71. https://doi.org/10.1145/3299904.3340307[5] Antoine Cully and Yiannis Demiris. 2018. Quality and Diversity Optimization: AUnifying Modular Framework.
IEEE Transactions on Evolutionary Computation
Parallel Problem Solving from Nature β PPSN XVI . Springer Interna-tional Publishing, 588β603. https://doi.org/10.1007/978-3-030-58115-2_41[7] B. Eschermann and H.-J. Wunderlich. 1990. Optimized synthesis of self-testablefinite state machines. In [1990] Digest of Papers. Fault-Tolerant Computing: 20thInternational Symposium . IEEE Comput. Soc. Press, 390β397. https://doi.org/10.1109/ftcs.1990.89393[8] Wanru Gao, Samadhi Nallaperuma, and Frank Neumann. 2020. Feature-BasedDiversity Optimization for Problem Instance Classification.
Evolutionary Compu-tation (June 2020), 1β22. https://doi.org/10.1162/evco_a_00274[9] Daniele Gravina, Ahmed Khalifa, Antonios Liapis, Julian Togelius, and Georgios N.Yannakakis. 2019. Procedural Content Generation through Quality Diversity. In . IEEE, 1β8. https://doi.org/10.1109/cig.2019.8848053[10] Pascal Kerschke, Holger H. Hoos, Frank Neumann, and Heike Trautmann. 2019.Automated Algorithm Selection: Survey and Perspectives.
Evolutionary Computa-tion
27, 1 (March 2019), 3β45. https://doi.org/10.1162/evco_a_00242[11] Joel Lehman and Kenneth O. Stanley. 2013. Evolvability Is Inevitable: IncreasingEvolvability without the Pressure to Adapt.
PLoS ONE
8, 4 (April 2013), 1β9.https://doi.org/10.1371/journal.pone.0062186[12] Yong Li and Panos M. Pardalos. 1992. Generating quadratic assignment testproblems with known optimal permutations.
Computational Optimization andApplications
1, 2 (Nov. 1992), 163β184. https://doi.org/10.1007/bf00253805[13] Alfonsas Misevicius. 2004. An improved hybrid genetic algorithm: new results forthe quadratic assignment problem.
Knowledge-Based Systems
17, 2-4 (May 2004),65β73. https://doi.org/10.1016/j.knosys.2004.03.001[14] Aneta Neumann, Wanru Gao, Carola Doerr, Frank Neumann, and Markus Wagner.2018. Discrepancy-based evolutionary diversity optimization. In
Proceedingsof the Genetic and Evolutionary Computation Conference . ACM, 991β998. https://doi.org/10.1145/3205455.3205532
00 1000 1500 2000 250000.51
200 400 60000.51
200 400 600 800 100000.51
20 40 60 8000.51
100 200 300 40000.51 N o m a li z e d d i ve r s i t y sc o r e D : D scoreD : D scoreD : D scoreD : D scoreD stop timeD stop time Nug30Lipa90bEsc128 = 10 = 3 Steps = 50 = 20
Figure 2: Means and standard deviations of normalized π· and π· scores from both approaches over time. For visibility, theX-axis range is scaled to the maximum number of steps till termination from all runs, missing data points are extrapolatedfrom the final scores. The total run-time is ππ . The dashed lines denote the average numbers of steps till termination. [15] Aneta Neumann, Wanru Gao, Markus Wagner, and Frank Neumann. 2019. Evo-lutionary diversity optimization using multi-objective indicators. In Proceed-ings of the Genetic and Evolutionary Computation Conference . ACM, 837β845.https://doi.org/10.1145/3321707.3321796[16] Christopher E. Nugent, Thomas E. Vollmann, and John Ruml. 1968. An Experi-mental Comparison of Techniques for the Assignment of Facilities to Locations.
Operations Research
16, 1 (Feb. 1968), 150β173. https://doi.org/10.1287/opre.16.1.150[17] Justin K. Pugh, Lisa B. Soros, and Kenneth O. Stanley. 2016. Quality Diversity: ANew Frontier for Evolutionary Computation.
Frontiers in Robotics and AI
Computers & Operations Research
22, 1 (Jan. 1995), 73β83.https://doi.org/10.1016/0305-0548(93)e0020-t [19] Umut Tosun. 2014. A New Recombination Operator for the Genetic AlgorithmSolution of the Quadratic Assignment Problem.
Procedia Computer Science
Proceedings ofthe 12th annual conference on Genetic and evolutionary computation - GECCO '10 .ACM Press, 455β462. https://doi.org/10.1145/1830483.1830569[21] Tamara Ulrich and Lothar Thiele. 2011. Maximizing population diversity insingle-objective optimization. In
Proceedings of the 13th annual conference onGenetic and evolutionary computation - GECCO '11 . ACM Press, 641β648. https://doi.org/10.1145/2001576.2001665 able 1: Diversity scores and the ratios of unique assignments in the output populations. The highlights denote greater valuesbetween the two approaches with statistical significance, based on Wilcoxon rank sum tests with 95% confidence level. π πΌ Optimizing π· Optimizing π· π· π· unique percentage π· π· unique percentage mean std mean std mean std mean std mean std mean std N u g L i p a b E s c0.37%1 100.00% 0.00% 100.00% 0.01% 100.00% 0.01% 100.00% 0.00% 99.99% 0.02% 99.99% 0.02%