An Analysis of Different Variations of Ant Colony Optimization to the Minimum Weight Vertex Cover Problem

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1 Mlan Tuba, Raa Jovanovc An Analyss of Dfferent Varatons of Ant Colony Optmzaton to the Mnmum Weght Vertex Cover Problem MILAN TUBA Faculty of Computer Scence Megatrend Unversty Belgrade Bulevar umetnost 29, N. Belgrade SERBIA RAKA JOVANOVIC Insttute of Physcs Belgrade Pregrevca 8, Zemun SERBIA Abstract: - Ant colony optmzaton (ACO) has prevously been appled to the Mnmum Weght Vertex Cover Problem wth very good results. The performance of the ACO algorthm can be mproved wth the use of dfferent varatons of the basc Ant Colony System algorthm, le the use of Eltsm, Ran based approach and the MnMax system. In ths paper, we have made an analyss of effectveness of these varatons appled to the Mnmum Weght Vertex Coverng Problem for dfferent problem cases. Ths analyss s done by the observaton of several propertes of acqured solutons by these algorthms le best found soluton, average soluton qualty, dsperson and dstrbuton of solutons. Key-Words: - Ant Colony, Mnmum Weght Vertex Cover, Optmzaton Problems, Populaton Based Algorthms Introducton One of the classcal graph theory problems s the Mnmal Vertex Cover Problem. The problem s defned for an undrected graph G = (V, E). V s the set of vertexes and E s a set of edges. A vertex coverng of a graph s set of vertexes V V that has the property that for every edge e(v,v 2 )E at least one of v,v 2 s an element of V. A mnmal vertex cover s a vertex coverng that has a mnmal number of vertexes. In ths paper we devote our attenton to an extenson of ths problem named the Mnmum Weght Vertex Cover Problem (MWVCP) n whch weghts are added to the vertexes. The soluton s not the vertex coverng wth a mnmal number of vertexes but wth a mnmal sum of weghts. A large number of real lfe problems could be converted to ths form. An example s the optmal postonng of garbage dsposal facltes. Every area needs to be covered by a garbage dsposal faclty, but not all potental postons are equal. It has been shown that ths problem s NP-complete even when t s restrcted to a unt-weghted planar graph wth the maxmum vertex degree of three []. The vertex cover s one of the core NP-complete problems that are frequently used for proof of NP-hardness of newly establshed ones. In the same way as for many other NP-complete problems, fndng the optmal solutons s very tme consumng and, n larger problem cases, even mpossble n realstc tme. Because of ths, varetes of dfferent methods have been presented for calculatng near optmal solutons. The frst method s a greedy heurstc approach of collectng the vertex wth the smallest rato between ts weght and degree [2], [3]. Degree of vertex s the number of edges that have t as a member. Ths problem has also been solved by the use of more complex method genetc algorthms [4]. Ant colony optmzaton s a meta-heurstc nspred by the behavor of ants used for solvng optmzaton problems. Ant colones are able to fnd the shortest possble path between ther nest and a food source, ths s done by the cooperaton of the ants n the colony. Each ant starts from the nest and wals toward food. It moves untl an ntersecton where t decdes whch path t wll tae. In the begnnng t seems as a random choce but after some tme the majorty of ants are usng the optmal path. Ths s acheved by usng pheromone. Each ant deposts pheromone whle walng whch mars the rout taen. The amount of pheromone ndcates the usage of a certan route. Pheromone tral evaporates as tme passes. Due to ths a shorter path wll have more of t because t wll have less tme to evaporate before t s deposted agan. The colony behaves ntellgently because each ant chooses paths that have more pheromone. Dorgo frst used the smulatons of ant colony for on the Travelng Salesman Problem and defned the new meta heurstc [5]. Ths method has been ISSN: Issue 6, Volume 6, June 2009

2 Mlan Tuba, Raa Jovanovc successfully used on a wde varety of problems snce then. The use of ant colony optmzaton (ACO) gave very good results when appled to the MWVCP. The results acqured by ACO where better that the ones calculated by genetc algorthms and local search methods le tabu search, and smulated annealng [6]. One of the methods of mprovng the effcency of ACO s the use of dfferent varatons of the basc algorthm. It has been shown that for some problems varatons of ACO gave results of dfferent qualty [7]. Because of ths, we have decded to analyze the mplementaton detals and effect of dfferent Ant colony optmzaton algorthms le Ant Colony system, the use of Eltsm, Ran Based Ant colony systems and the MnMax approach to MWVCP. In ths paper we wll compare several dfferent aspects of performance of these varatons. Ths paper s organzed as follows. In Secton 2 we gve an mathematcal defnton of the MWVCP. In Secton 3 we present the mplementaton of ACO for ths problem. In Secton 4 we show the mplementaton detals of ACO algorthm varatons. In Secton 5, we analyze and compare expermental results of the use of dfferent varatons of ACO on several problem nstances. 2 Mathematcal Defnton of MWVCP In ths secton we defne the MWVCP n strct mathematcal terms, through ts nteger lnear programmng form. Lnear programmng s a procedure for fndng the maxmum or mnmum of a lnear functon where the arguments are subject to lnear constrants. Ths type of formulaton s a very good gudelne for software mplementaton of problems. For the MVCP we accept the formulaton that was presented n artcle [8]: Let G = (V, E) be an undrected graph, where V{,2,.., n} E {(, j), j V} () V s the set of vertexes and E s the set of edges. Let m = E and n = V be the number of edges and vertexes respectvely. Let A = n x m be the vertexedge ncdence matrx of G. The cells of matrx A are defned n the followng way: a j,f edge j s ncdent to vertex (2) 0,otherwse In addton, we ntroduce a cost functon to the vertex set, w: V N, whch corresponds to the weghts of the ntal graph G. Wth the gven defntons, the MWVCP can now be descrbed by the followng nteger lnear programmng formulaton: Mnmze Subject to n n w() x a x, j,.., m j x {0,},,.., n 3 ACO for the MWVCP The use of ACO has proven to be effectve on varous types of problems from Economc Load Dspatch [9], Schedulng problems [0], even mage processng[], and ts use has been also very effcent on MWVCP. The MWVCP s n two man aspects dfferent than most of the problems solved usng ACO. To llustrate ths, for comparson we wll use the Travelng Salesman Problem (TSP). In the 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 ctes. In the case of MWVCP the soluton s a subset of the graph vertexes set, n whch the order s unmportant. In TSP, our heurstc functon s statc n the sense that t represents the dstance between ctes and does not change durng the calculaton of the path. Opposte to ths, for the MWVCP the heurstc functon s the rato between the weght and the degree of a vertex, whch s dynamc. The degree of a vertex changes as we add new vertexes to the soluton set because more edges become covered. We wsh to pont out that ACO wth a dynamc heurstc and a soluton that conssts of a subset nstead of a permutaton have been also used for solvng the set parttonng [2], maxmum ndependent set [3] and maxmum clque [4] problems. These two dfferences affect the basc algorthm n two drectons. Frst, ants leave the pheromone on vertexes nstead of on edges and second, we dynamcally update the graph, and wth t, the heurstc functon. The frst step n solvng these problems s representng the problem n a way that maes dynamc calculaton of the heurstc functon smple. Snce ants n ther search can move from a vertex to any other vertex, t s natural to use a fully connected graph G c (V,E c ) derved from G. In the artcles [6], [5] t s proposed to add weghts to ISSN: Issue 6, Volume 6, June 2009

3 Mlan Tuba, Raa Jovanovc edges n the new graph G c. If an edge exsts n G t s gven the weght, or 0 f t does not exst n the orgnal graph. We have adopted ths approach, whch s llustrated by Fg.. Fg. Expanson to the fully connected graph As we mentoned before we also have to update ths graph as we add new vertexes to the result set. Ths s done usng the followng rule: when we add vertex a, weghts of all edges n G c that are connected to a, are set to 0. Ths s llustrated by Fg. 2. Fg. 2 Addng a vertex to the soluton set 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 3: (, j) Value( E (, j)) (3) Ths update rule has two roles. Frst, we can dynamcally evaluate the preference of vertexes wth functon ψ κ. The second role s that t gves us the nformaton when all edges have been covered, or more precsely, f the total sum of edge weghts n G s 0 then all edges are covered. Now we can defne a dynamc heurstc (, ) (, ) j j Ec j (4) c In Equaton 4 w(j) s the weght of a vertex. Usng the heurstc defned wth η j n Equaton 4 we can setup the state transton rule for ants:, q q0 & j arg max A p 0, q q & j arg max j j, q q0 j j 0 A A (5) In Equaton 5 q 0 s the standard parameter that specfes the explotaton/exploraton rate of ndvdual ant searches. that appears n ACO. q s a random varable that decdes the type of selecton on each step. A s a lst of avalable vertexes. We pont out that opposte to the TSP transton rule t does not depend on the last selected vertex that s why we have τ nstead of τ j. To fully specfy an Ant Colony System we stll have to defne a 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 creatng better solutons to become more desrable, or n other words, t ntensfes explotaton., V (6) jv ( p) (7) Equaton 6 defnes the global update rule. In t Δτ ι s a qualty measure of soluton subset V that contans vertex, and wth t we defne an global update rule n Equaton 5. 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) 0 For the value of t 0 we tae the qualty measure of the soluton acqured wth the greedy algorthm when we select the vertex wth the best raton of vertex ISSN: Issue 6, Volume 6, June 2009

4 Mlan Tuba, Raa Jovanovc degree and weght. Parameter φ s used to specfy the strength of the local update rule. 4 Varatons of ACO for MWVCP There are dfferent methods of mprovng ACO le certan types of hybrdzatons. Standard hybrdzatons are the combnaton of the basc algorthm wth a local search [5] or some genetc algorthm [6]. These hybrdzatons are effectve n ncreasng the effcency of ACO but are often complcated for mplementaton. The complexty of ther mplementaton s due to the fact that to separate algorthms need to be developed one for ACO and another for the local search or for the genetc algorthm. The other method of mprovng the performance of ACO s the use of dfferent varatons of the basc algorthm. On TSP dfferent varatons of ACO gave dfferent qualty of results, and no varaton can be consdered the best [7]. That s why we have decded to performed a comparatve assessment of standard varatons of ACO on MWVCP. The varatons mostly dffer n the global update rule. We present several varatons of ACO by gvng ther global update rules. Ant System (AS), s the most basc mplementaton of ACO, n ths verson of the algorthm all ants are equal and leave pheromone. It s defned wth Equatons 9 and 7. In Equaton 7 AntS s the set of all the solutons created by ants n the current step of the algorthm. V Ants V AntS V jv (9) Renforced Ant System (RAS), whch s the same as Ant system, except that the global best soluton s renforced each teraton. V ' V gb jv V Ants gb V AntS jv (0) In some varatons of ths method the teraton best soluton s also renforced each teraton. Wth ths approach the basc AS s made to be slghtly greeder. It s defned wth Equatons 0 and 7. Eltst Ant System (EAS), In ths verson of the algorthm, ndvdual ants do not automatcally leave pheromone. In each teratons step of the colony, or n other words when all the ant complete ther solutons, only the global best soluton wll be used to update the pheromone tral. In ths way, the search s even more centralzed around the global best soluton. It s defned wth Equatons and 7., V gb () jv gb MnMax Ant System (MMAS) s same as the Eltst Ant colony System, but wth an extra constrant that all pheromone values are bounded, [ mn, max ]. We adopt the formulas presented n artcle [7] n whch τ max s calculated dynamcally as new best solutons are found by Equaton 2, and τ mn s calculated at the begnnng of calculatons wth Equaton 3. avg s the average number of vertexes that are possble to be chosen, p best s the possblty of the best overall soluton beng found and τ 0 s the ntal value of the pheromone tral gotten as the qualty measure of the greedy algorthm soluton. max mn gb (2) ( p) 0( n pbest ) ( avg ) n p best (3) Ths varaton has two effects that mprove the effectveness of EA. Frst, the pheromone tral wll not become very strong on some good vertexes and mang them a part of almost all newly created solutons. By gvng a lower bound to the pheromone tral the potental problem of certan parts of the soluton beng totally excluded from the search due to very wea values of pheromone s avoded. Ran Based Ant Colony System (RANKAS) [8] n whch except the qualty we also use the ran (R) of found solutons. Ran s defned by the qualty of the soluton compared to solutons found by other ants n the same teraton. It s defned wth Equatons 7 and 4: ISSN: Issue 6, Volume 6, June 2009

5 Mlan Tuba, Raa Jovanovc BRan { V ( R( V ) RK) V V gb ( V AntS)} V BRan jvgb ( RK R( V )) RK V BRan jv (4) In the mplementaton of ths algorthm, t s mportant how many best solutons wll be taen nto account when updatng the pheromone tral. In Equaton 4 parameter RK s used to defne the number of best raned ants who wll affect the tral. Ths parameter s user defned. Ths parameter s very mportant for the effectveness of ths algorthm. In ts extreme cases when RK s equal to 0 t s equvalent to EAS. The program for our experments was wrtten n C#, usng the framewor from artcle [9]. Ths framewor s dot net based and s desgned for creatng wndows applcatons. It s mplemented as a plug-n system so smlar research on the effect of dfferent ACO varatons can be conducted on other problems that could be solved by ths method just by creatng the basc ACO algorthm. We have created a plug-n for ths system and used exstng features to conduct our tests. The executable alpha verson of ths software (Fg. 3) and accompanyng Mcrosoft Vsual Studo project can be downloaded from All of our test have been performed on an Intel(R) Core(TM)2Duo CPU 3.6 GHz wth 4GB of RAM wth Mcrosoft Wndows Vsta Ultmate x64 Edton Verson 2007 Servce Pac. In the tables (,2,3,4,5,6,7,8,9,0) EAS (Eltst Ant Colony System) corresponds to the algorthm presented n artcle [6], n whch the effcency of usng ant colony optmzaton on ths problem was shown. 5 Applcaton and Results In ths secton, we present the comparatve assessment of dfferent varatons of ACO. In our test, the mplementaton of an teraton step has been done by the use of the followng pseudo code. Reset Graph Info Reset Soluton for all Ants whle (! AllAntsFnshed) for All Ants If(Ant Not Fnshed) begn add new vertex A to soluton based on probablty correct ants coverng graph data calculate new heurstc End If End for End whle local update rule for A Compute Δτ for varaton Compute τ Fg. 3. Graf-Ant software wth the plug-n for Mnmal Weghted Vertex Coverng Problem We have tested several graphs contanng dfferent number of edges and vertexes. In each test, we have used colones consstng of 0 ants. The exploraton rate was q 0 =0., and the nfluence factor of heurstcs was α=, evaporaton rates where φ=0., p=0.. In RANKAS we used RK = 5. For the ISSN: Issue 6, Volume 6, June 2009

6 Mlan Tuba, Raa Jovanovc ntal value of the pheromone tral, and τ 0 was calculated from the soluton ganed usng the greedy algorthm presented n artcle [3].In MMAS for the value of p best =0.05. For each varaton, we conducted 0 separate runs. In each test, we set a maxmum number of possble teratons and compared results obtaned up to that number of steps. The analyss s done by observng the best-found soluton, the average soluton value, dsperson and dstrbuton of solutons. The calculaton tme of each varaton of ACO s very smlar, so we excluded t from the analyss nstead we use the number of teratons. We generated random problem nstances. In whch weghts where randomly selected for vertexes from the nterval [20, 70]. We used graphs of 25, 50, 50, 250, 500 vertexes and for each of these szes, we tested two dfferent sets of edges. In the algorthm for edge set creaton, we would generate n edges from each vertex to random vertexes. n was a random number between [, 4] n Tables, 3, 5 and [, 0] n Tables 2, 4, 6, 7 and [0-20] n Tables 8, 0. We frst observe the behavor of these methods n small problem cases (Tables, 2). All ACO varatons gave the optmal soluton, except the two most basc AS and RAS. The optmal soluton was found usng a brute force method testng the whole soluton space. Table. Number of nodes 25, Number of edges 7, greedy algorthm soluton value 088, Maxmum number of teratons 250 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 2. Number of nodes 25, Number of edges 3, greedy algorthm soluton value 35, Maxmum number of teratons 250 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS For fndng the optmal soluton, we used a recursve method that mplements the followng pseudo code OptCover ( startindex, Connectons, Covered, Sum) begn f(startindex >= SzeOfGraph) return; f(sum > BestValue) return; f(covered = SzeOfGraph) begn f (Sum < BestValue) BestValue = Sum; return; end Connectons = Connectons; Covered = Covered; OptCover( startindex+, Connectons, Covered, Sum ); Connectons2 = Connectons; Covered2 = Covered; UpdateCoverng( Connectons2, startindex, Covered2); Sum = Sum + NodeValues[startIndex]; OptCover( startindex +,Connectons2, Covered2, Sum); end In larger problem cases, we dd not calculate the optmal soluton due to very long executon tme. The qualty of soluton acqured by AS and RAS varatons was bad n larger problem szes. In small problems n average RANKAS gave the best qualty of results, but the best soluton was found at a hgher number of teratons than MMAS and EAS. In the medum (Tables 3, 4) and large (Tables 5, 6, 7, 8, 9, 0) problem cases MMAS and EAS gave the best results, wth EAS beng slghtly better. Table 3. Number of nodes 50, Number of edges 72, greedy algorthm soluton value 2238, Maxmum number of teratons 2000 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS In medum problems RANK preformed slghtly worse than these varatons, but n large cases ths dfference would greatly ncrease. For problems of ths sze RANK performance was smlar to AS and ISSN: Issue 6, Volume 6, June 2009

7 Mlan Tuba, Raa Jovanovc RAS. Ths could be explaned by the very large soluton space and because of ths a need for a more focused search. EAS and MMAS have ths property and because of ths gve better results. We also wsh to pont out that t s hghly possble that f sgnfcantly longer calculaton tme were used RANKS would have mproved ts results. We beleve ths s because ths varatons has a slow convergence to optmal solutons but has a lower possblty of gettng trapped n local optma. Table 4. Number of nodes 50, Number of edges 374, greedy algorthm soluton value 2238, Maxmum number of teratons 2000 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 5. Number of nodes 50, Number of edges 562, greedy algorthm soluton value 6782, Maxmum number of teratons 2000 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 7. Number of nodes 250, Number of edges 970, greedy algorthm soluton value 0877, Maxmum number of teratons 500 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 8. Number of nodes 250, Number of edges 8399, greedy algorthm soluton value 239, Maxmum number of teratons 500 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 9. Number of nodes 500, Number of edges 858, greedy algorthm soluton value 22395, Maxmum number of teratons 500 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 6. Number of nodes 50, Number of edges 470, greedy algorthm soluton value 6834, Maxmum number of teratons 2000 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS Table 0. Number of nodes 500, Number of edges 50973, greedy algorthm soluton value 22626, Maxmum number of teratons 500 Varaton Best Best Value Average Value Iteraton AS EAS RAS RANKAS MMAS ISSN: Issue 6, Volume 6, June 2009

8 Mlan Tuba, Raa Jovanovc To better qualfy the performance of dfferent varatons of ACO we also observed the dsperson and dstrbuton of solutons obtaned by them. As a measure of dsperson, we used the standard devaton gven by the Equaton 5 s 2 ( W W ) (5) n The results for these propertes where very smlar for all problem cases, because of ths we only present the data for problems from tables 3 and 6. We show the standard devaton n table. Table. Standard devaton of solutons Varaton Stand. Devaton Stand. Devaton for Table 3 for Table 6 AS EAS RAS RANKAS MMAS We frst notce that AS and RAS have a small devance, whch can be explaned by the fact that they are trapped n local optma n early steps of the algorthm. The better performng varatons have notceable greater devances. The mportance of ths property s better understood f we observe the dstrbuton of the solutons n Fg. 4 and 5. Now we can see the dfference n behavor of AS and RAS. These two methods gve the average soluton and best solutons of smlar qualty, but the dstrbuton s very dfferent. AS gves us a much smaller varety of solutons, than RAS. It s surprsng that by renforcng of the best soluton the dsperson grows whch s opposte to the frst mpresson of the effect of focusng the search near good solutons. The effect of no or wea renforcement of best teraton solutons and global best soluton s that, the ants search a wder area near the best solutons but when a better soluton s found, the pheromone tral wll not change suffcently. Because of ths, the majorty of the ants wll not move to the area near the newly found best soluton. Ths explans the fact that RANKS converges to good solutons slower that EAS and MMAS but can fnd better ones. In RANKS the renforcement of the global best s weaer than n these two methods, because of ths areas around each soluton are more detaled searched, but t taes longer to move to regons around new best solutons. Fg. 4. Dstrbuton of solutons for problem from Table 3 AS-flled crcles, EAS empty crcles, MMAS flled trangles, RANKAS-empty trangles, RAS-flled squares Fg 5. Dstrbuton of solutons of problem from Table 6 AS-flled crcles, EAS empty crcles, MMAS flled trangles, RANKAS-empty trangles, RAS-flled squares ISSN: Issue 6, Volume 6, June 2009

9 Mlan Tuba, Raa Jovanovc The last concluson that we mae from the analyss of dsperson and dstrbutons s that MMAS s a more relable for acqurng good results than EAS. EAS has a greater dsperson than MMAS. Ths means that even f there s a hgher possblty of gettng the best overall soluton wth EAS t s more le to get a good soluton wth MMAS. In practcal use, ths means f we are gong to perform a larger number of separate runs to fnd the best soluton t s better to use EAS, but n the case of a small number of attempts MMAS s a better choce. 6 Concluson In ths paper, we have done an extensve analyss of the standard varatons of ACO, whch are Ant System, Renforced Ant system, Eltst Ant system, Mn Max Ant System and Ran based Ant System on the MWVC problem. To do ths we have transformed the standard varaton formulas to a form that could be appled on ths problem. We used our prevously developed framewor [9] to create software for conductng tests. Our frst observaton was that the dfference n calculaton tme for all the varatons was neglect able. We have analyzed dfferent aspects of solutons calculated for these varatons le the best-found soluton, average soluton qualty, dsperson and dstrbuton of solutons. From our analyss, we came to the followng conclusons. The two basc algorthms AS and RAS gave sgnfcantly worst results n all tested cases than the other methods and are not a good choce for ths problem. An overall best varaton dd not exst but t depended on the sze of the problem, and the avalable resources. In small problem cases EAS, MMAS and RANKS gave good results. RANKS gave the best qualty of results but t had found them n a hgher of number teratons than the other two methods. In our tests, t has been shown that EAS and MMAS gave the best results n large problem cases wth EAS havng slghtly better results when best solutons and average soluton qualty were compared. When we too nto account the dsperson and dstrbuton of solutons, we concluded that EAS could not be seen as the better method than MMAS. Ths s due to the fact that MMAS solutons where less dspersed and because of that the method can be consdered to be more relable. The choce of whch of these two methods we shall use depends on our resources f plan to conduct a large number of separate runs t s better to use EAS, and f a small number f tests s planed MMAS s the better choce. Although RANKS preformed worse than EAS and MMAS n our test t should not be dscarded. By analyzng the tendences n our experments, we have observed that RANKS converges slower than the other two methods but can come to better solutons f the calculaton tme s suffcently long. In further research, we wsh to adopt and mplement the suspcon hybrdzaton for ACO used on the TSP[9]. Parallelzaton of ACO proved to be very effectve and n some cases even gve super lnear mprovement, because of ths we wsh to mplement parallel verson of ACO on ths problem and compare the effectveness of dfferent parallel topologes. References: [] Karp, R.M.. Reducblty Among Combnatoral Problems. In R.E. Mller and J.W. Theater, Complexty of Computer Computatons, New Yor: Plenum Press, 972 [2] Chvatal, V., A Greedy-Heurstc for the Set Cover Problem. Mathematcs of Operatons Research, Vol.4, 979, pp [3] Clarson, K.L., A Modfcaton of the Greedy Algorthm for Vertex Cover, Informaton Processng Letters, Vol. 6, 983, pp [4] Asho Kumar Gupta, Alo Sngh, A Hybrd Heurstc for the Mnmum Weght Vertex Cover Problem, Asa-Pacfc Journal of Operatonal Research, 2006, vol. 23, No 2, pp [5] Dorgo M, Manezzo V: Ant Colony system: Optmzaton by a colony of cooperatng agents, IEEE Transactons on Systems, Man and Cybernetcs - Part B Vol. 26, No., 996, pp [6] Shyong Jan Shyu, Peng-Yeng Yn, Bertrand M.T. Ln, An Ant Colony Optmzaton Algorthm for the Mnmum Weght Vertex Cover Problem, Annals of Operatons Research, Vol.3, 2004, pp , [7] D.Asmar,A. Elshaml, S. Areb, A Comparatve Assessment of ACO Algorthms Wthn a TSP Envronment. In DCDIS: 4th Internatonal Conference on Engneerng Applcatons and Computatonal Algorthms, Guelph, Ontaro, Canada, July [8] R. Garey and D. Johnson, Computers and Intractablty, A Gude to the Theory of NP- Completeness W.H. Freeman and Company, San Francsco, 979. ISSN: Issue 6, Volume 6, June 2009

10 Mlan Tuba, Raa Jovanovc [9] Vlachos Arstds, An Ant Colony Optmzaton (ACO) algorthm soluton to Economc Load Dspatch (ELD) problem. WSEAS Transactons On Systems, Vol 5, No 8, pp , 2006 [0] Kolahan, F., Abachzadeh, M., Sohel, S, A comparson between Ant colony and Tabu search algorthms for job shop schedulng wth sequence-dependent setups, WSEAS Transactons on Systems, Vol. 2, 2006, pp [] Mastoras, N.E., Zhuang, X, Image processng wth the artfcal swarm ntellgence, WSEAS Transactons on Computers, Vol 4, No. 4, 2005, pp [2] Broderc Crawford, Carlos Castro, Ant Colones usng Arc Consstency Technques for the Set Parttonng Problem, Professonal Practce n Artfcal Intellgence, Sprnger Boston, 2006, pp [3] Youme L, Zongben Xul, An ant colony optmzaton heurstc for solvng maxmum ndependent set problems, Proceedngs of the 5th Internatonal Conference on Computatonal Intellgence and Multmeda Applcatons, 2003 pp: [4] Serge Fenet, Chrstne Solnon, Searchng for Maxmum Clques wth Ant Colony Optmzaton, Applcatons of Evolutonary Computng, Sprnger Berln / Hedelberg, 2003, pp [5] Y.J Feng, Z.R. Feng, Ant colony system hybrdzaton wth smulated annealng for flowshop schedulng problems. WSEAS Transacton on Busness and Economcs, Vol., No., 2004, p [6] Hong-hao Zuo, Fan-lun Xong, Tme Ant Colony Algorthm wth Genetc Algorthms, Informaton Acquston, IEEE Internatonal Conference on Volume, Issue, Aug. 2006, pp [7] T. Stützle et H.H. Hoos, MAX MIN Ant System, Future Generaton Computer Systems, Vol. 6, 2000, pp [8] B. Bullnhemer, R. F. Hartl, and C Strauss. A new ran-based verson of the ant system: a computatonal study, Central European Journal for Operatons Research and Economcs, Vol.7 No., 999, p , [9] Raa Jovanovc, Mlan Tuba, Dana 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, 2008, pp Acnowledgment: The research was supported by the Mnstry of Scence, Republc of Serba, Projects no and 403 ISSN: Issue 6, Volume 6, June 2009