Ant Colony Optimization Applied to Minimum Weight Dominating Set Problem

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1 Ant Colony Optmzaton Appled to Mnmum Weght Domnatng Set Problem Raa JOVANOVIC Mlan TUBA Dana SIMIAN Texas AM Unversty Faculty of Computer Scence Department of Computer Scence at Qatar Megatrend Unversty Belgrade Lucan Blaga Unversty of Sbu PO Box 23874, Doha Bulevar umetnost dr. I. Ratu str. QATAR SERBIA ROMANIA Abstract: - In ths paper we present an applcaton of ant colony optmzaton (ACO) to the Mnmum Weghted Domnatng Set Problem. We ntroduce a heurstc for ths problem that taes nto account the weghts of vertexes beng covered and show that t s more effcent than the greedy algorthm usng the standard heurstc. Further we gve mplementaton detals of ACO appled to ths problem. We tested our algorthm on graphs wth dfferent szes, edge denstes, and weght dstrbuton functons and shown that t gves greatly mproved results over these acqured by the greedy algorthms. Key-Words: - Ant Colony Optmzaton, Domnatng Set Problem, Optmzaton Problems, Populaton Based Algorthms Introducton A domnatng set for graph G= (V, E) s a subset of vertces W V, such that every vertex n V \ W s adjacent to some vertex n W. We call vertexes u, v adjacent f there exsts an edge (u, v) E. The mnmum domnatng set problem (MDSP) s to fnd a smallest possble domnatng set n a graph. The mnmum weght domnatng set problem (MWDSP) s defned n the case that weghts c are added to vertexes v and nstead of fndng W wth the mnmum number of elements we search for W that has the mnmum sum of weghts. The exstence of a domnatng set of elements s one of the classcal NP-complete decson problems []. MDS for general graphs s equvalent to the coverng set problem. A wde range of practcal problems can be transformed to ths form, from postonng retal stores to Sensor Networs and Moble ad hoc Networs (MANETs). Dfferent extensons of the the MDS le connected or weghted, are used to more precsely defne the propertes of the problem beng solved [2]. Due to the fact that fndng the optmal soluton for MDS problem and ts varatons cannot be done n polynomal tme a wde range of methods have been used to acqure near optmal solutons le greedy algorthm [3], Constant-factor approxmaton [4], energy functon algorthm [5], collaboratve cover heurstc [6] and Polynomal Kernels [7]. Ant colony optmzaton (ACO) s another metaheurstc for solvng combnatoral problems, that was frst used on the Travelng Salesman Problem by M. Dorgo. The MDSP has also been solved usng populaton based algorthms le ant colony optmzaton (ACO) and genetc algorthms wth ACO gvng better results [8]. In ths paper we extend the applcaton of ACO to the MWDSP. A heurstc functon that taes nto account the weghts of vertexes beng covered s also ntroduced. We show advantages of usng ACO nstead of the greedy algorthms for MWDSP. Ths paper s organzed as follows. In the second secton we explan heurstc functons that are used for ths problem. In the next secton we gve detals on applyng ACO on MWDSP. In the fourth secton we show our test and results. 2 Greedy Algorthms for MWDSP The dea of the basc greedy algorthm as presented n [3] for the MDS s to add a new node to W n each teraton, untl W forms a domnatng set. We use the term that a node s covered f j W or j s adjacent to some node W, and uncovered n the opposte case. In teraton n, we put a new node V n nto W that covers the maxmum number of uncovered vertexes. V n s the set on vertexes that have not been selected n the frst n teratons. The algorthm s fnshed when all the nodes have been covered. Ths heurstc needs to be extended by tang nto account the weghts of vertexes. To mplement ths algorthm the frst step s to represent the problem n a way that dynamcally calculates the heurstc functon, and maes tracng ISSN: ISBN:

2 the covered vertexes smple. We use a smlar approach as presented by Shyu for the Mnmum Weghted Vertex Cover Problem [9] whch s a weghted verson of the heurstc used n [3]. In t a fully connected graph G c (V,E c ) s derved from G. In the artcle [9] they propose addng weghts of f the edge exsts n G or f t does not exst n the orgnal graph. We have adopted ths approach whch s llustrated by Fg.. Fgure : Transton from the orgnal graph to a fully connected one. Blac edges represent edges wth a value one and red ones have a value zero. As we mentoned before we also have to update ths graph as we add new vertexes to our result set. Ths s done n the followng way when we add vertex a all edges n G c that are connected to a and to ts neghbors, are set to. Ths s llustrated by Fg. 2. Cov j (, j) Ec Cov (, j) (2) The heurstc n Equaton 2 states that vertexes wth low weght and have a large number of unused edges are hghly desrable. The addton of to the sum s used to mae a dfference between vertexes that have no connectons but are not yet covered. The heurstc γ taes nto account only the weght of the vertex that s currently beng selected but not the weghts of the vertexes that are beng covered. We propose a new heurstc (Greedy2) that taes ths nto account Cov( w( ) (, )) (, j) E j c j (3) η j states that we prefer vertexes that have low weght, a large number of connectons and cover nodes wth a hgh sum of weghts. For the two versons of the greedy algorthm to wor, we have to be able to recognze f all the vertexes have been covered. Ths s done by addng set Y = V at the start of the algorthm and at each step, when vertex a s selected, we have Y { v ( v, a) E ( a, v) E}/ Y (4) The algorthm s fnshed when Y =. Fgure 2. Correcton of a graph when vertex a s added to the soluton set. Blac edges represent edges wth a value one and red ones have a value zero. Now we can defne G (V,E c, ) as the state of the graph after vertexes have been added to the soluton set, and a correspondng functons n Equaton. (, j) Value( E (, j)) () c Ths update rule maes t possble to dynamcally evaluate the preference of vertexes wth functon ψ κ. Now we can defne a dynamc heurstc (Greedy) 3 ACO for MWDSP The use of ACO has proven to be effectve on varous types of problems from Economc Load Dspatch [], Schedulng problems [], even mage processng [2]. The applcaton of ACO on MWDSP dffers n two man aspects to the orgnal applcaton on the Travelng Salesman Problem (TSP). In TSP our soluton s an array of all the ctes appearng n the problem or n other words the soluton s the permutaton of the set of vertexes. In the case of MWDSP the soluton s a subset of the graph vertexes set, n whch the order s unmportant. In the case of TSP the heurstc functon beng used s statc n the sense that t represents the dstance between vertexes and does not change durng the generaton of the soluton. Contrary, for MWDSP, the heurstc functon s the rato between the weght and the sum of weghts of neghborng vertexes, whch s dynamc. Ths sum ISSN: ISBN:

3 changes as we add new vertexes to the soluton set because more vertexes become covered. These two dfferences affect the basc algorthm n two drectons. Frst ants leave the pheromone on vertexes nstead of edges. Second we dynamcally update the graph, and wth t, the heurstc functon as shown n the prevous secton. Usng the heurstc defned wth η j n Equaton 3. we can setup the state transton rule for ants., q q & j arg max A p, q q & j arg max j j, q q j j A Y (5) In Equaton 5 q s the standard parameter that appears n ACO that specfes the explotaton / exploraton rate of ndvdual ant searches. q s a random varable that decdes the type of selecton on each step. Y s a lst of avalable vertexes. We pont out that opposte to the TSP, transton rule does not depend on the last selected vertex and that s why we have τ nstead of τ j. The next step s to defne the global (when an ant fnshes ts path) and a local (when an ant chooses a new vertex) update rules. The role of the global update rule s to mae paths that create better solutons to become more desrable, or n other words, t ntensfes explotaton. jw ( p), W (6) (7) Equatons 6, 7 defne the global update rule. In t Δτ ι s a qualty measure of soluton subset W that contans vertex, and wth t we defne an global update rule n Equaton 7. Ths measure s nverse proportonal to the weght of a soluton. Parameter p s used to set the nfluence of newly found soluton on the pheromone tral. The local update rule purpose s to shuffle solutons and to prevent all ants from usng very strong vertexes. The dea s to mae vertexes less desrable as more ants vst them. In ths way, exploraton s supported. The formula for the local update rule has the standard form ( ) (8) For the value of t we tae the qualty measure of the soluton acqured wth the greedy algorthm (Greedy2). Parameter φ s used to specfy the strength of the local update rule. 4 Tests and Results In ths secton we compare the results of applyng the standard greedy algorthm for MWDSP (Greedy), our mproved verson of ths algorthm (Greedy2) and ACO optmzaton usng the same heurstc as Greedy2. The program for our experments was wrtten n C#, usng the framewor from artcle [3]. We have created a plug-n for ths system that mplements ACO for MWDSP. We used the followng parameters for the ACO algorthm. Colones conssted of ants. The exploraton rate was q =., evaporaton rates were φ=. and p=.. For the nfluence factor of the heurstc we used α=4. We mplemented ACO n the MMAS varaton. The ntal best soluton V was gven by Greedy2 and the ntal value of the pheromone tral s gven by Equaton 9. (9) n jv' In Equaton 9, n s the number of nodes n the V. Each colony had teratons. We generated two types of random problem nstances for our tests. In the frst one weghts for nodes where randomly selected for vertexes from the nterval [2, 7]. In the second group the weghts where dependent of the number of connectons vertex v had and t would be randomly selected from the [, e(v) 2 ]. e gves the number of connectons for v. We used graphs from 5 to nodes wth dfferent number of randomly created edges but always mang the graph connected. For all the tested vertex-node pars, we created dfferent problem nstances and we observed the average soluton values for each of the three methods. We show our results n Tables, 2, 3, 4. When comparng the two greedy algorthms, we frst notce that the mproved heurstc that taes nto account the weghts of covered vertexes gves almost unformly better results, for both types of generated problem. The Greedy2 would gve results that would be better up to %. The mprovement was smallest for sparse graphs. ISSN: ISBN:

4 Table. Comparson of results for small and medum problems for Type problems 5* * * * * * * * * * * * * * * * * * * * * * * * * Table 3. Comparson of results for small and medum problems for Type2 problems 5* * * * * * * * * * * * * * * * * * * * * * * * Table 2. Comparson of results for large problems for Type problems 5* * * * * * * * * * * * * * Table 4. Comparson of results for large problems for Type2 problems 5* * * * * * * * * * * * * * ISSN: ISBN:

5 ACO used as ntal guess the result acqured by Greed2 so we shall compare ts results to t. Frst we pont out that ACO has mproved results n all the tested cases. The mprovement would vary for more and less dense graphs. In the case of graphs wth lowest densty, or n other words the average number of edges per vertex, the mprovement would be the smallest -2%. The advantages of ACO would be ncreased wth the ncrease of the densty and n the cases when the average number of edges per vertex s 2 t would be from 2% to even 7%. Ths shows that ACO s very effcent on ths problem, and that both greedy algorthms are not good for dense graphs. 5 Concluson In ths paper we have shown an mplementaton of ACO for MWDSP. We have presented a new heurstc functon for the greedy algorthm that taes nto account the weghts of vertexes beng covered and shown that t s an mprovement to the standard one. We used ths heurstc functon n our mplementaton of ACO. Tests for these methods have been done on a varety of graphs wth dfferent szes, edge denstes and weght generaton algorthms. Our results show that our heurstc functon mproves the performance of the greedy algorthm. ACO proved to be very effcent on the MWDSP and greatly mproved the qualty of results especally n dense graphs. In the future we wsh to extend wor to connected domnatng set problems due to the fact that they are more closely related to MANETs than the non-connected verson. Acnowledgment: Ths research s supported by Project 447, Mnstry of Scence, Republc of Serba. References: [] M. R. Garey, and D. S. Johnson, Computers and Intractablty. A Gude to the Theory of NPCompleteness., New Yor-San Francsco: W. H. Freeman and Company, 979. [2] J. Blum, M. Dng, A. Thaeler et al., "Connected Domnatng Set n Sensor Networs and MANETs," Handboo of Combnatoral Optmzaton, D. D.-Z. and P. P., eds., pp , Kluwer: Academc Publshers, 24. [3] A. K. Pareh, Analyss of a greedy heurstc for fndng small domnatng sets n graphs, Inf. Process. Lett., vol. 39, no. 5, pp , 99. [4] C. Ambühl, T. Erlebach, M. Mhalá et al., "Constant-factor approxmaton for mnmumweght (connected) domnatng sets n unt ds graphs." pp [5] X. Xu, Z. Tang, W. Sun et al., An Algorthm for the Mnmum Domnatng Set Problem Based on a New Energy Functon, n SICE Annual Conference, Sapporo, Japan, 24, pp [6] R. Msra, and C. Mandal, Mnmum Connected Domnatng Set Usng a Collaboratve Cover Heurstc for Ad Hoc Sensor Networs, IEEE Transactons on Parallel and Dstrbuted Systems, vol. 2, pp , 2. [7] S. Gutner, "Polynomal Kernels and Faster Algorthms for the Domnatng Set Problem on Graphs wth an Excluded Mnor," Parameterzed and Exact Computaton: 4th Internatonal Worshop, IWPEC 29, Copenhagen, Denmar, September -, 29, Revsed Selected Papers, pp : Sprnger- Verlag, 29. [8] C. K. Ho, Y. P. Sngh, and H. T. Ewe, An Enhanced Ant Colony Optmzaton Metaheurstc for the Mnmum Domnatng Set Problem, Appled Artfcal Intellgence, vol. 2, pp , 26. [9] S. S. Jan, Y. Peng-Yeng, and L. B. M.T., An Ant Colony Optmzaton Algorthm for the Mnmum Weght Vertex Cover Problem, Annals of Operatons Research, vol. 3, pp , 24. [] A. Vlachos, An Ant Colony Optmzaton (ACO) algorthm soluton to Economc Load Dspatch (ELD) problem, WSEAS Transactons On Systems, vol. 5, no. 8, pp , 26. [] []F. Kolahan, M. Abachzadeh, and S. Sohel, A comparson between Ant colony and Tabu search algorthms for job shop schedulng wth sequence-dependent setups, WSEAS Transactons on Systems, vol. 2, pp , 26. [2] N. E. Mastoras, and X. Zhuang, Image processng wth the artfcal swarm ntellgence, WSEAS Transactons on Computers, vol. 4, no. 4, pp , 25. [3] R. Jovanovc, M. Tuba, and D. Sman, An Object-Orented Framewor wth Correspondng Graphcal User Interface for Developng Ant Colony Optmzaton Based Algorthms, WSEAS Transactons on Computers, vol. 7, no. 2, 28 ISSN: ISBN: