A Coevolutionary Variable Neighborhood Search Algorithm for Discrete Multitasking (CoVNS): Application to Community Detection over Graphs
aa r X i v : . [ c s . N E ] S e p A Coevolutionary Variable Neighborhood SearchAlgorithm for Discrete Multitasking (CoVNS):Application to Community Detection over Graphs
Eneko Osaba †∗ , Esther Villar-Rodriguez †∗ and Javier Del Ser †‡† TECNALIA, Basque Research and Technology Alliance (BRTA), 48160 Derio, Bizkaia, SpainEmail: [eneko.osaba, esther.villar, javier.delser]@tecnalia.com ‡ University of the Basque Country (UPV/EHU), 48013 Bilbao, Bizkaia, Spain ∗ Corresponding authors. These authors contributed equally to this work.
Abstract —The main goal of the multitasking optimizationparadigm is to solve multiple and concurrent optimization tasksin a simultaneous way through a single search process. Forattaining promising results, potential complementarities andsynergies between tasks are properly exploited, helping eachother by virtue of the exchange of genetic material. This paper isfocused on Evolutionary Multitasking, which is a perspective fordealing with multitasking optimization scenarios by embracingconcepts from Evolutionary Computation. This work contributesto this field by presenting a new multitasking approach namedas Coevolutionary Variable Neighborhood Search Algorithm,which finds its inspiration on both the Variable NeighborhoodSearch metaheuristic and coevolutionary strategies. The secondcontribution of this paper is the application field, which isthe optimal partitioning of graph instances whose connectionsamong nodes are directed and weighted. This paper pioneerson the simultaneous solving of this kind of tasks. Two differentmultitasking scenarios are considered, each comprising 11 graphinstances. Results obtained by our method are compared tothose issued by a parallel Variable Neighborhood Search andindependent executions of the basic Variable NeighborhoodSearch. The discussion on such results support our hypothesisthat the proposed method is a promising scheme for simultaneoussolving community detection problems over graphs.
Index Terms —Transfer Optimization, Evolutionary Multitask-ing, Variable Neighborhood Search, Community Detection.
I. I
NTRODUCTION
Transfer Optimization is an incipient research stream withinthe general field of optimization. Currently, this area is gath-ering a significant momentum from the related community,leading to an intense scientific production during the lastyears [1]. The main inspiration behind this paradigm is toexploit what has been learned through the optimization of oneproblem or tasks for the solving of another related or unrelatedtask. Due to its relatively youth, efforts dedicated to thetransferability of knowledge among optimization problems hasnot been remarkable until recent years, when this concept hasbecome a priority for a wider research community. Arguably,the ever-growing complexity and dimensionality of optimiza-tion scenarios has made researchers to turn their attention onmethods that allow efficiently harnessing knowledge acquiredbeforehand. In this regard, three different categories can be distinguishedin Transfer Optimization [2]: sequential transfer [3], multi-tasking [4] and multiform optimization. In this paper, weput our attention on the second of these categories. In anutshell, multitasking is devoted to the simultaneous tacklingof different tasks of equal priority by dynamically exploitingexisting complementarities and synergies among them.More concretely, the present paper is focused on Evolu-tionary Multitasking (EM, [5]), which deals with multitaskingoptimization scenarios by embracing concepts, operators andsearch strategies from the area of Evolutionary Computation[6], [7]. Related to this specific branch, a particular flavorof EM has shown a remarkable performance when dealingwith multitasking environments: Multifactorial Optimizationstrategy (MFO, [8]). Until now, MFO has been successfullyadopted for solving different continuous, discrete, multi- andsingle-objective optimization tasks [9]–[12]. Furthermore, aspecific method has garnered most of the literature around thisconcept: the Multifactorial Evolutionary Algorithm (MFEA,[8]). Unfortunately, alternative methods that populate the EMcommunity are still scarce.This lack of competitive EM methods is one of the mainmotivations for the development of this research work. Specif-ically, this paper proposes a novel EM metaheuristic algo-rithm based on the well-known Variable Neighborhood Search(VNS, [13]) for solving discrete multitasking environments.The Coevolutionary Variable Neighborhood Search Algorithm(CoVNS) herein presented takes a step further beyond the stateof the art in two different directions. Firstly, we contributeto the EM field by proposing a new competitive algorithmwhich, unlike most works published so far in this specifictopic, does not hinge on the MFO paradigm. Secondly, CoVNSis a pioneering attempt at exploring the applicability of VNSto the Transfer Optimization paradigm.Besides the novelty of the method itself, a second contribu-tion of this work relates to the application scenario to whichit is applied. It is relevant to first underscore that we focuson discrete optimization In particular, the problem tackledin this work is the detection of communities in weighteddirected graphs [14], namely, the optimal partitioning ofraph instances whose connections among nodes are directedand weighted. This scenario has been less addressed in theliterature than other networks of simpler nature [15], [16].This being said, to the best of our knowledge this studyis the first of its kind dealing with multitasking for solvingseveral community detection problems at the same time. Tothis end, the discovery of optimal partitions is formulated asan optimization problem, which is driven by a measure ofmodularity adapted to the directional and weighted nature ofthe edges of the network [17], [18]. Results from an extensiveexperimental setup are presented and discussed to show thatthe proposed CoVNS excels at solving such multitasking sce-narios, outperforming non-multitasking variants of the samealgorithm and, hence, providing informed evidence of thebenefits of knowledge exchange among tasks.The remainder of the article is organized as follows. SectionII provides background and related work. Section III posesthe mathematical formulation of the community detectionproblems in weighted directed networks. Next, Section IVexposes in detail the main features of the proposed CoVNS.The experimentation setup and discussion of the results aregiven in Section V. Finally, Section VI concludes the paperwith an outlook towards further research.II. B
ACKGROUND
In order to contextualize this work and properly assess itsscientific contribution, this section provides a short overviewof the EM research area. In recent years, this scientificbranch has emerged as a competitive paradigm for tacklingsimultaneous optimization tasks. The adoption of evolutionarycomputation concepts to multitasking (giving rise to EM) hasbecome the de facto search strategy: by designing a unifiedsearch space, these population-based algorithms allow for aninherent parallel evolution of the whole set of tasks, and forthe transfer of genetic material among individuals to exploitinter-task synergies [1], [8].There is a solid consensus that EM was only materializedthrough the perspective of MFO until late 2017 [19]. Sincethen, this incipient research field is gathering a notable corpusof literature focused on new algorithmic schemes, such asthe multitasking multi-swarm optimization introduced in [20],the coevolutionary multitasking scheme proposed in [21] orthe coevolutionary bat algorithm detailed in [22]. Furtheralternatives to MFEA have also emerged, partly inspired bythe concepts of this influential method. Some examples arethe multifactorial differential evolution proposed in [23], themultifactorial cellular genetic algorithm in [24], the particleswarm optimization-firefly hybridization introduced in [25], orthe multifactorial brain storm optimization algorithm presentedin [26]. Although in this work the EM environment underconsideration is not addressed by using the MFO strategy, werefer interested readers to [27]–[29] for a recent overview onthese methods.We can mathematically formulate an EM scenario as anenvironment comprised by K concurrent problems or tasks T k ,which must be simultaneously optimized. Thus, the scenario could be characterized by the existence of as many searchspaces as tasks. Furthermore, each of the K problems to besolved has a fitness function (objective) f k : Ω k → R , where Ω k denotes the search space of task T k . We define the mainobjective of EM as the discovery of a group of solutions { x , ∗ , . . . , x K, ∗ } such that x k, ∗ = arg max x ∈ Ω k f k ( x ) .An aspect of paramount importance for adequately under-standing the above formulation and the EM paradigm itself isthat each solution x p in the population P = x pPp =1 is evolvedover an unified search space Ω U , which relates Ω to Ω K via an encoding/decoding function ξ k : Ω k Ω U . For thisreason, each individual x p ∈ Ω U in P should be decoded toyield a task-specific solution x kp for each of the K tasks.III. P ROBLEM S TATEMENT
We now proceed by defining the community detectionproblem over weighted graphs. First, we model the networkas a graph G . = {V , E , f W } , where V represents the group of |V| = V nodes or vertices of the network, E stands for the setof edges connecting every pair of vertices, and f W : V × V 7→ R + is a function assigning a non-negative weight to eachedge. Furthermore, we consider that f W ( v, v ) = 0 (i.e. no selfloops), and that f W ( v, v ′ ) = 0 if nodes v and v ′ are not linked.For notation purposes we define f W ( v, v ′ ) . = w v,v ′ , yielding a V × V adjacency matrix W given by W . = { w v,v ′ : v, v ′ ∈ V} and fulfilling Tr ( W ) = 0 , with Tr ( · ) denoting trace ofa matrix. Lastly, the directed characteristic of the graph isguaranteed by not imposing any requirement on the symmetryof the adjacency matrix, that is, w v,v ′ is not necessarily equalto w v ′ ,v for any v = v ′ .Using this notation, the task of detecting communities ina network G can be defined as the partition of the vertexset V into a number of disjoint, arbitrarily-sized, non-emptygroups. Let us denote M as the amount of partitions e V . = {V , . . . , V M } , such that ∪ Mm =1 V m = V and V m ∩ V m ′ = ∅∀ m ′ = m (i.e., no overlapping communities). Under thisformulation, the community to which node v ∈ V belongscan be represented as V v ∈ e V .With all this, we should bear in mind that the weighteddirected feature of the graphs used in this paper enforcesthe reformulation of the in-degree and out-degree values thatparticipate in conventional modularity formulations. A way toredefine such measures is to formulate the so-called input andoutput strengths of node v , which are given by: s inv = X v ′ ∈V w v ′ ,v , s outv = X v ′ ∈V w v,v ′ , (1)that is, as the sum of the weight of the incident (outgoing)edges to (from) node v . It is worth noting here that thesevalues represent both the directivity and the weighted natureof adjacency matrix W . Therefore, these two quantities are ofparamount importance for properly redefining the concept ofcommunities, in an analogous way to the role played by in-and out-degree values when clustering undirected, unweightednetworks.earing all the above formulation in mind, a quality measurefor a given partition e V can be furnished from the maindefinition of the classical modularity for undirected graphsintroduced in [15], [18]. By defining a binary function δ : V × V 7→ { , } , so that δ ( v, v ′ ) = 1 if V v = V v ′ as perthe partition set by e V (and otherwise), the modularity inweighted directed networks can be calculated as: Q ( e V ) . = 1 | P W | X v ∈V X v ′ ∈V (cid:20) w v,v ′ s inv s outv ′ | P W | (cid:21) δ ( v, v ′ ) , (2)where | P W | represents the sum of the weights of every edgeof the graph [30]. Thus, detecting a high-quality partition e V ∗ of a weighted directed network G can be defined as: e V ∗ = arg max e V∈B V Q ( e V ) , (3)where B V stands for the whole set of possible partitionsof V elements into nonempty subsets. It is interesting topoint out that the cardinality of this set is given by the V -th Bell number [31]). As a brief example, a small graphcomposed by V = 20 nodes amounts up to . · possible partitions. Assuming now that the computation ofthe modularity in (2) takes just microsecond, a practitionerwould need more than six months to exhaustively evaluateall the possible partitions. This example is illustrative ofthe convenience of using heuristics and meta-heuristics forefficiently solving this complex combinatorial problem, andthe adoption of multi-tasking approaches when solving severalinstances of the problem at the same time.IV. P ROPOSED V ARIABLE N EIGHBORHOOD S EARCH FOR D ISCRETE M ULTITASKING
Inspired by concepts from previous solvers [21], [22], oneof the remarkable features of the proposed CoVNS is its multi-population nature. Thus, CoVNS comprises a fixed number ofsubpopulations or demes [32], composed by the same amountof candidates. The number of subpopulations is equal to thenumber of tasks K to be solved. Furthermore, each of the K demes { P k } Kk =1 is devoted to the optimization of a specifictask T k , meaning that individuals belonging to subpopulation P k are only evaluated on task T k as per its objective f k ( x ) .The coevolutionary strategy of CoVNS implies the mi-gration of individuals across subpopulations. Therefore, theconsideration of an unified representation Ω U becomes neces-sary. To realize this, the same philosophy of MFEA has beenadopted. Nonetheless, one of the main innovative feature ofCoVNS is that each deme has its partial view (often restrictedby the problem size) of the common search space, potentiallyrequiring a size adjustment when different subpopulationsshare their individuals.Let us focus on the community finding problem for ex-emplifying this noted size adjustment. First, we encode eachindividual x ki using a label-based representation [33]. In thisway, each solution x kp belonging to a subpopulation k isdenoted as a combination of V integers from the range [1 , . . . , V ] , where V represents the number of edges in the graph. The value of the v -th component of x kp represents thecluster label to which node v belongs. For instance, if weassume a network composed by V = 10 nodes, a possibleindividual for task k could be x kp = [1 , , , , , , , , , .The communities represented by this individual would be e V = {V , V , V } , where V = { , , } , V = { , , } and V = { , , , } . Furthermore, the use of this encodingstrategy requires a repairing procedure to avoidance of ambi-guities in the representation. To this end, we design a similarprocedure to the repairing function proposed in [34]: ambigu-ities such as those present in x ki = [2 , , , , , , , , , and x ki = [3 , , , , , , , , , (representing both thesame partition) are solved by standardizing the solution to x ki = [1 , , , , , , , , , .Turning our attention again to the unified representation Ω U used in CoVNS, we denote the dimension of each task T k (i.e. the number of nodes ) as D k . Thus, once an individual x kp ∈ Ω k is about to be migrated to a deme in which thedimension of the tasks T k ′ to be optimized is D k ′ < D k , onlythe first D k elements are considered, reducing in this fashionthe phenotype of the solution. In the opposite case, i.e. if D k ′ > D k , the reverse procedure is carried out. In such a case,and taking into account that when a solution x kp is transferredto another subpopulation it replaces another individual x k ′ p ′ , allelements from D k to D k ′ are introduced in x kp respecting theorder as in x k ′ p ′ . Algorithm 1:
Proposed CoVNS multitasking solver Randomly generate P individuals (initial population) Evaluate each individual for all the K tasks Arrange K subpopulations ( demes ) Set it = 0 while termination criterion not met do Update iteration counter: it = it + 1 for each deme k do for each individual x kp in the subpopulation do Generate new solution succFun = rand( CE , CE , CC , CC ) x new,kp ← succFun ( x kp ) if f k ( x new,kp ) > f k ( x kp ) then Accept the new solution f k ( x new,kp if it mod migr = 0 then for each deme k do for j = 1 , . . . , migr prop do k ′ = rand (1 , . . . , k − , k + 1 , . . . , K ) Replace the worst solution in deme k bythe best solution in deme k ′ Return the best individual in P for each task T k With all this, Algorithm 1 shows the pseudo-code of the pro-posed CoVNS. As can be seen in this high-level description, inthe initialization phase P individuals are randomly generated.Then, each solution is assessed over all the considered K tasks.fter this evaluation phase, each subpopulation is generatedby choosing the best P/K individuals for the task at hand.This means that the same solution can be chosen for beingpart of different demes. Once all subpopulations are built,each evolves independently by following the main conceptsof a basic discrete VNS. More concisely, each individual,at each iteration, undergoes a successor generation procedureby applying a movement operator on a random basis ( CE , CE , CC or CC ). These operators have been introducedin previous studies [16]. For each of these functions, thesubscript indicates the amount of randomly chosen nodes,which are extracted from its assigned community. In CE ∗ , thechosen elements are re-inserted in already existing communi-ties, whereas in CC ∗ they can be also introduced in newlygenerated partitions.Furthermore, every migr iterations, each deme transfers migr prop number of individuals to a randomly chosensubpopulation. It should be pointed here that migr = E × f req migr , where E represents the number of function eval-uations per execution. Furthermore, we set migr prop propor-tional to the population size as P × prop . In our study, and asa result of a thorough empirical process, f req migr = 0 . and prop = 0 . . Moreover, individuals chosen to be migratedare the migr prop best ones, replacing the migr prop worstof the destination subpopulation. Lastly, CoVNS completesits search process after E objective function evaluations, afterwhich the best individual of each deme is returned.V. E XPERIMENTAL S ETUP AND R ESULTS
For properly gauging the performance of the proposedCoVNS, an extensive set of experiments has been conducted,which is detailed in this section. First, in Section V-A weelaborate on the benchmark problems used for the proposedalgorithm, along with the rest of details of the experimentationsetup. Next, in Section V-B we examine and discuss on theresults from such experiments.
A. Benchmark Problems and Experimentation Setup
As has been mentioned in preceding sections, the benefitsof the proposed method will be showcased by considering,as tasks, the optimal partitioning of weighted and directedgraphs. Accordingly, the performance of CoVNS has beentested over two multitasking scenarios, each composed by different graph instances. In order to assess the advantage ofexchanging genetic material between demes, the performanceof our method has been compared to that yielded by twoapproaches: a separated VNS (sVNS) and a parallel VNS(pVNS). The first approach solves each problem separately byusing a single VNS search. For these executions, a fair con-figuration has been applied for the operators and parameters.The second of the approaches is a parallel implementationof VNS, with no coevolution strategy (each subpopulationevolves independently). Even though no relevant algorithmicdifferences exist between sVNS and pVNS, the considerationof the parallel approach permits to quantify the contribution of the exchange of knowledge among demes to the convergenceof the overall solver.Having said that, each multitasking scenario is composed by11 synthetically generated network instances, which shouldbe optimized in a simultaneous fashion by the three afore-mentioned methods. Specifically, both benchmarks consist ofnetworks of sizes from 50 to 100 nodes. Each graph hasa number of ground truth communities, which are modeledby first creating a partition of the network (with randomsizes for its constituent communities {V m } Mm =1 ), and thenby connecting nodes within every community with probability p in and nodes of different communities with probability p out .Weights w v,v ′ for every link ( v, v ′ ) are modeled as uniformlydistributed random variables with support R [10 . , . (intra-community edges) and R [0 . , . (inter-community edges).The first environment is called ordered incremental ( OI ),and all the tasks included in this scenario has been namedas OI_V_M , where V is the number of nodes populating thegraph and M the amount of underlying partitions as per theground truth partition of the network at hand. Regarding p in and p out , all datasets have an assigned value of . and . , respectively. The main characteristic of this OI sce-nario is that instances have been generated in an incrementaland ordered way. In other words, new instances have beenbuilt by extending the precedent smaller instance respectingthe predecessor’s graph structure and node identifiers. Forinstance, all the nodes belonging to the instance OI_60_8 arealso present in the subsequent
OI_65_8 instance, in identicalorder. Furthermore, the new 5 nodes are added in the 61 st to65 th positions of the adjacency matrix. By imposing theseconditions our intention is to maintain the order of nodesin the matrix adjacency, guaranteeing that the best solution(partitions) of each instances will share most of their structure.The second scenario has been coined as unordered incre-mental ( UI ), naming all the cases as UI_V_M , following thesame criterion as with the previous OI environment. In theseinstances, we keep p in = 0 . and p out = 0 . . Therefore,the main difference between the two environments is that thenew incremental nodes in UI are inserted in the first positions,i.e. the new 5 nodes introduced in UI_65_8 in comparison to
UI_60_8 are added in 1 st to 5 th positions. This apparentlyslight modification alters significantly the adjacency matrixand thereby the structure of the best solution corresponding toeach incrementally generated graph instance.The rationale behind this experimental setup follows frominfluential works [35], [36], which emphasize that one of themost critical aspects when dealing with EM environments isthe analysis of the mutual information among the optimizedtasks. In fact, it is widely acknowledged that this synergybetween tasks is of crucial importance for reaching profitablegenetic material exchanges. For this reason, the explorationof what features and characteristics should share differenttasks for being synergistic is also valuable in this researchcontext. Therefore, these experiments will help gain a deeperunderstanding about the conditions that should be met and theperformance boundaries when opting for Transfer Optimiza- ABLE IP
ARAMETER VALUES SET FOR C O VNS, P VNS
AND S
VNS.
CoVNS pVNS sVNSParameter Value Parameter Value Parameter ValuePopulation size P ×
10 Population size P ×
10 Population size P CE , CE CC , CC Successor functions CE , CE CC , CC Successor functions CE , CE CC , CC Function evaluations 10 × × × × × freq migr prop tion in the context of community detection over graphs.Finally, independent executions have been carried outfor each test case, aiming at shedding light on the statisticalsignificance of eventually discovered performance gapss. Re-garding the ending criterion of each method, every run endsafter E = K × N × objective function evaluations, where N represents the number of individuals per subpopulation.Using this formula, we ensure fairness in comparisons betweenCoVNS, pVNS and sVNS, dedicating to each approach thesame amount of computational resources [37]. To supportthe replicability of this work, parameters employed for theimplemented techniques are shown in Table I. B. Results and Discussion
Table II depicts the results obtained by CoVNS, pVNSand sVNS. Outcomes obtained for each dataset and test case( OI and UI ) are given in terms of fitness average, bestsolution found and standard deviation. It should be mentionedhere that the measure used for comparison is the modularityvalue attained by the solvers (as described in Section III).In addition, we ease the visualization of the outcomes byhighlighting the best average results in bold. Furthermore,in order to ascertain the statistical relevance of differencesamong algorithms, two different hypothesis tests have beencarried out for both OI and UI environments [38]. Results ofthese tests can be analyzed in Table III. First, the Friedman’snon-parametric test for multiple comparison permits provingif differences in performances among the techniques can becataloged as statistically significant. Thus, first column ofTable III depicts the mean ranking returned by this test foreach of the compared methods in both test cases (the lowerthe rank, the better the performance). Furthermore, to assessthe statistical significance of the better performance method(CoVNS in both test cases), a Holm’s post-hoc test has beenperformed using our proposal as control solver. This way, theresulting unadjusted and adjusted p -values have been includedin the second and third columns of Table III.Several interesting conclusions can be drawn from TableII. To begin with, CoVNS dominates as the best performingmethod in all the instances that compose the OI multitaskingenvironment. Furthermore, Table III supports the significanceof these results at a 99% confidence level, taking into accountthat all the p -values of the Holm’s post-hoc test are lowerthan . . These findings statistically conclude that solving OI instances in a simultaneous way and sharing knowledgeamong different subpopulations contributes to reaching better results. More specifically, since CoVNS has demonstratedto be statistically superior than pVNS, we can confirm thatjust the simultaneous solving of the tasks is not enoughfor attaining higher performances. The competitive advantagearises from the efficient sharing of genetic material throughindividuals belonging to synergistic tasks. As expected, pVNSand sVNS perform similarly, as the only difference betweenthem is the parallelization of the search process (at the levelof deme and entire search process, respectively).The second important fact is that the structure of thenetworks is of paramount importance for leveraging genetictransfer. This conclusion becomes evident in the results at-tained for the DI multitasking environment. In this test case,CoVNS performs best in 6 out of 11 instances, althoughthe overall performance gap is not statistically significant asobserved in Table III. These outcomes clearly brings us tothe conclusion that the genetic material sharing among non-complementary instances does not provide any competitiveadvantage for the search process. We recall at this pointthat, as opposed to OI instances, in UI tasks the structureof incrementally generated graphs changes considerably asmore nodes are added to the graphs. Therefore, we concludethat although CoVNS seemingly outperforms both pVNS andsVNS, there is no statistical evidence that the sharing ofknowledge leads to significant better outcomes.In fact, this analysis leads to the two main conclusions ofthis paper. This first one regards the composition of com-plementary graphs. As observed in this experimentation, formaterializing positive genetic transfer among tasks, networkinstances should share their structure in an incremental wayas explained in the case of OI so as to enforce a degree ofoverlap between their optimal partitions. Secondly, CoVNShas demonstrated to be a promising method for simultaneoussolving community detection problems over graphs, obtainingsignificant competitive advantages whenever the networks areinterrelated.VI. C ONCLUSIONS AND F UTURE W ORK
This paper has elaborated on the design, implementation andvalidation of a novel Coevolutionary Variable NeighborhoodSearch algorithm for dealing with evolutionary multitaskingscenarios. The proposed method relies on a discrete adaptationof the VNS heuristic, incorporating further elements from co-evolutionary multitasking algorithms [21], [22]. In addition tothe method itself, an equally important contribution of thiswork is the first attempt at applying Transfer Optimization to
ABLE IIR
ESULTS OBTAINED BY C O VNS, P VNS
AND S
VNS
FOR ALL BOTH TEST ENVIRONMENTS . B
EST AVERAGE RESULTS HAVE BEEN HIGHLIGHTED IN BOLD .E ACH ( ALGORITHM , INSTANCE ) CELL INDICATES AVERAGE ( TOP ), BEST ( MIDDLE ) AND STANDARD DEVIATION ( BOTTOM ) OF THE MODULARITY FITNESSCOMPUTED OVER INDEPENDENT RUNS . OI_50_8 OI_55_8 OI_60_8 OI_65_8 OI_70_8 OI_75_8 OI_80_8 OI_85_8 OI_90_8 OI_95_8 OI_100_8 O r d e r e d I n c r e m e n t a l CoVNS
UI_50_8 UI_55_8 UI_60_8 UI_65_8 UI_70_8 UI_75_8 UI_80_8 UI_85_8 UI_90_8 UI_95_8 UI_100_8 U no r d e r e d I n c r e m e n t a l CoVNS 0.299 0.279
TABLE IIIR
ESULTS OF THE F RIEDMAN ’ S NON - PARAMETRIC TESTS , ANDUNADJUSTED AND ADJUSTED p - VALUES OBTAINED THROUGH THEAPPLICATION OF H OLM ’ S POST - HOC PROCEDURE USING C O VNS
ASCONTROL ALGORITHM .Friedman’s Test Holm’s Post HocRank Unadjusted p Adjusted p O I CoVNS 1 – –pVNS 2.7273 0.000051 0.000102sVNS 2.2727 0.002838 0.002838 D I CoVNS 1.7273 – –pVNS 2.0455 0.240955 0.481909sVNS 2.2273 0.455545 0.481909 community detection over weighted and directed graphs. Inthis way, we have compared the results attained by CoVNSover two test cases composed of 11 datasets with the onesfurnished by a parallel (not coevolutionary) VNS and byindependent executions of the VNS. The obtained resultsvalidate our hypothesis: the knowledge sharing that lies at theheart of CoVNS is crucial for reaching better results whensimultaneously solving complementary tasks.Several research lines have been arranged as future work.In the short term, we plan to evaluate the scalability of theproposed method by analyzing its computational efficiencywhen simultaneously dealing with a high number of cases. Wewill also explore the adaptation of the method to other combi-natorial optimization problems stemming from other researchfields [39]. In a longer term, we plan to endow this methodwith enhanced adaptive mechanisms so as to automaticallydefine the optimal strategy for sharing knowledge accordingto the detected level of relationship amongst tasks. To this end,we plan to design schemes for automatically detecting the level synergy of the optimizing graphs during the search process,in order to autonomously boost the transfer of knowledge. Weexpect that these methods, currently under active investigationin other related works [40], will help the solver adaptivelyharness positive knowledge transfers, and stay resilient againstnegative (hence, counterproductive) genetic shares.A
CKNOWLEDGMENTS
The authors would like to thank the Spanish Centro parael Desarrollo Tecnologico Industrial (CDTI, Ministry of Sci-ence and Innovation) through the “Red Cervera” Programme(AI4ES project), as well as by the Basque Government throughEMAITEK and ELKARTEK (ref. 3KIA) funding grants. J.Del Ser also acknowledges funding support from the Depart-ment of Education of the Basque Government (ConsolidatedResearch Group MATHMODE, IT1294-19).R
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