An Efficient Algorithm for Minimum Vertex Cover Problem

Transcription

1 An Effcent Algorthm for Mnmum Vertex Cover Problem Rong Long Wang Zheng Tang Xn Shun Xu Member Non-member Non-member Paper An effcent parallel algorthm for solvng the mnmum vertex cover problem usng bnary neural network s presented. The proposed algorthm whch s desgned to fnd the smallest vertex cover of a graph, uses the bnary neural network to get a near-smallest vertex cover of the graph, and adjusts the balance between the constrant term and the cost term of the energy functon to help the network escape from the state of the near-smallest vertex cover to the state of the smallest vertex cover or better one. The proposed algorthm s tested on a large number of random graphs and benchmark graphs. The smulaton results show that the proposed algorthm s very satsfactory and better than prevous works for solvng the mnmum vertex cover problem. Keywords: Vertex cover, Bnary neural network, Local mnmum, NP-complete problem 1. Introducton The goal of the mnmum vertex cover problem s to fnd a smallest subset C of the vertces of an undrected graph G such that all edges of G are adjacent to at least one vertex n the set C. The mnmum vertex cover problem s of central mportance n computer scence. It s very ntractable (1). In 1972, Karp (2) showed that the vertex cover problem s NP-complete. Furthermore, the mnmum vertex cover problem remans NPcomplete even for certan restrcted graphs, for example, the bounded degree graphs (3). Because the mnmum vertex cover problem has many mportant practcal applcatons, especally n multple sequence algnments for computatonal bochemstry (4) (5), the problem has been wdely studed by many researchers. Garey and Johnson (6) presented a smple approxmaton algorthm based on maxmal matchng for the general graphs. Hochbaum (7) presented an algorthm of approxmaton rato 2 2/d for graphs of degree bounded by d. Monen and Speckenmeyer (8) mproved ths bound to 2 (loglogn)/(2logn). On the sparse graphs, Berman and Fujto (9) presented an approxmaton algorthm for graphs of degree bounded by 3, whose approxmaton rato s bounded by 7/6 +e. The parameterzed algorthms are well known methods. The frst fxedparameter tractable algorthm for k-vertex cover problem (gven a graph G and a parameter k, decdng f G has a vertex cover of k vertces), was done by Fellows (10). Buss (11) developed the fxed-parameter tractable algorthm of runnng tme O(kn +2 k k 2k+2 ) for ths problem. More recently, parameterzed algorthms for the k-vertex cover problem have further drawn researchers Faculty of Engneerng, Fuku Unversty, Fuku-sh, Japan Faculty of Engneerng, Toyama Unversty, Toyama-sh, Japan attenton, and contnuous mprovements on the problem have been developed. Buss s algorthm was mproved to O(kn+2 k k 2 ) by Downey and Fellows (12). Chen et al (13). presented an O(kn k k 2 ) tme algorthm for ths problem. Recently, Dehne et al (14). have reported that they used fxed parameter tractable algorthm to solve the mnmum vertex cover problem on coarse-graned parallel machnes successfully. Khur et al (1). presented an evolutonary heurstc for the mnmum vertex cover problem. Despte many algorthms were presented for the problem, the problem s a well-known NP-complete (2),no tractable algorthm s known for solvng t. Furthermore, few parallel algorthms have been proposed to solve the problem. A possble parallel algorthm for solvng such optmzaton problems was ntroduced by Hopfeld and Tank (15), whch found a good soluton to some optmzaton problems n a reasonable amount of tme. Usng the neural network technques, Yuan et al (16). presented a parallel algorthm based on the Hopfeld network for the mnmum vertex cover problem. In ther algorthm, n order to help the network fnd the global mnmum, they ntroduced a specal term whch had dfferent values accordng to the degree of vertces. Although t was effectve for small graphs, the rate to get the mnmum vertex cover was very low for large graphs, and performance of the algorthm becomes poorer wth problem scale becomng large. In ths paper, for effcently solvng the mnmum vertex cover problem, we propose a new neural network algorthm. The proposed algorthm uses the bnary neural network to get a near-smallest vertex cover of the graph, and adjusts the balance between the constrant term and the cost term of the energy functon to help the network escape from the state of the near-smallest vertex cover to the state of the smallest vertex cover or better one. We show ts effectveness by extensve computa IEEJ Trans. EIS, Vol.124, No.7, 2004

2 An Effcent Algorthm for Vertex Cover Problem where s the logcal OR, X means the complement of X and d j ( =1, 2,,N,j =1, 2,,N) s element of the adjacency matrx of graph G. d j has only two values (1 or 0) whch express f the connecton between vertex and j exsts or not. In other words, f (, j) exsts, then d j s 1 else d j s 0. For example, for the graph of Fgure 1, we have the followng adjacency matrx. D = Fg. 1. A graph wth 10 vertces and 18 edges. tonal experments on a large number of random graphs and benchmark graphs (14). The performance of the proposed algorthm s compared wth Yuan s algorthm (16). The smulaton results show that the proposed algorthm can fnd better solutons than Yuan s algorthm. 2. Formulaton Let G =(V,E) be an undrected graph, a set V V s a vertex cover of G, f for every edge (, j) E, ether V or j V or both, j V. A vertex cover s mnmal or optmal f t has mnmum sze,.e., f there s no vertex cover that has fewer vertexes. The goal of the mnmum vertex cover problem s to fnd a mnmum vertex cover. Fgure 1 shows a graph wth 10 vertces and 18 edges, where the black vertces ndcate the mnmum vertex cover of the graph. In general, an N-vertex M-edge mnmum vertex cover problem can be mapped onto a bnary neural network wth N 2 neurons. In ths work, we show a new bnary neural network representaton. In the new representaton, only N neurons are used for the N-vertex M-edge mnmum vertex cover problem. The output (v =1or v = 0) of the neuron ( =1,,N) expresses f the -vertex s n the cover or not, respectvely. { 1 f vertex s n the cover v = (1) 0 otherwse Thus the number of vertces n the cover can be expressed by: E 1 = v (2) Because the vertex cover must nclude at least one vertex of edge (, j), the constraned condton of the mnmum vertex cover problem can be wrtten as: E 2 = d j v v j (3) Thus, the problem can be mathematcally transformed nto the followng optmzaton problem. ( Mnmze E 1 = ) v (4) Subject E 2 = d j v v j =0 (5) j When we follow the mappng procedure by neural network, the energy functon for the mnmum vertex cover s gven by: E = A v + B d j v v j (6) where A, B are coeffcents. Because the relatons between Boolean representaton and the arthmetc representaton for three typcal functons are as followng: X =1 X X Y = X + Y XY X Y = XY (7) The energy functon can be expanded as: E = A v + B d j (v v j v v j ) +B d j (8) As n the energy functon E, the last term s a constant, the energy can be rewrtten as: E = A v + B d j (v v j v v j ) (9) The network can fnd the soluton of the problem by seekng the local mnmum of the energy functon E usng the followng equatons: u (t) = E = w j v j + I (10) j C

3 where w j s weght of a synaptc connecton from the j-th neuron to the -th one, I s external nput of -th neuron and s also called threshold. Each neuron updates ts nput value u based on the moton equaton. Specfcally, the value u (t +1)atteraton step (t + 1) s gven by: u (t +1)=u (t)+ u (t) (11) The output v s updated from u usng a non-lnear functon called neuron model. In the bnary neural network, the followng McCulloch-Ptts (17) bnary neuron model has been used as the nput/output functon: { 1 f u > 0 v = (12) 0 otherwse Each neuron updates ts nput potental accordng to the updatng rule (Eq.(11)) and sends ts output n response to the nput accordng to the nput/output functon (Eq.(12)). All neurons operate n parallel and each adjusts ts own state to the states of all the others; n consequence, the network converges to a fnal confguraton. In ths way, we can fnd the soluton to the mnmum vertex cover problem smply by observng the fnal confguraton that the network converged. Unfortunately, because the network wll attempt to take the best path to the nearest mnmum, whether global or local, the qualty of the soluton s not very good. 3. Method for Escapng Local Mnma In ths secton, we propose a method to help the network escape from the state of the near-smallest vertex cover to the state of the smallest vertex cover or better one. We rewrte the Eq.(9) as follow: and E = AE 1 + BE 2 (13) E 1 = E 2 = v (14) d j (v v j v v j ) (15) j We analyze the characterstcs of the bnary neural network n a mnmum. We use E to denote the varaton of energy of the network wth the state change of neuron. It s well known that a local mnmum always satsfes E 0 for =1, 2,..., N (16) The varaton of energy of the network wth the state change of any neuron can be wrtten as: E =(A E 1 + B E 2 ) v =(A + B E 2 ) v (17) From Eq.(15) we have: E 2 =2 j d j (v 1) (18) Because d j and v j are equal to 0 or 1, thus we have: 2k E 2 0 (19) where k s the number of vertces connected to vertex. To make the energy value of the neural network decrease wth the state change of at least one neuron, we can have the followng rule from Eq.(17): (1) Select one neuron that satsfes the followng equaton: E 2 < 0 (20) It s worth to note that a neuron wth Eq.(20) certanly exst because n a local mnmum, E 0 (Eq.(17)) and for every fred neuron, v s -1. (2) f v > 0(v = 0) and E 0, thus A B E 2, then we modfy parameter A accordng to the followng rule: A new = B E 2 δ<a (21) Otherwse( v < 0,v = 1), A new = B E 2 + δ<a (22) where δ s a small postve constant and usng Eq.(20) we can see that the modfed parameter A new wll be postve. After modfcaton, the parameter set becomes A new, B, and wth the state change of the neuron, the varaton of energy of the network can be descrbed by the followng formula by applyng Eq.(21),(22) nto Eq.(17). { δ v for v E = > 0 (23) δ v for v < 0 It s evdent that E (Eq.(23)) s smaller than zero. The dervatves of Eq.(23) show that the energy of the network decreases wth the state change of the neuron by the above rule (Eq.(21) and (22)). Thus, we can see that by adjustng the balance between the constrant term and the cost term of the energy functon, the local mnmum that the network falls nto s elmnated. 4. Algorthm The followng procedure descrbes the proposed algorthm for the general mnmum vertex cover problem n synchronous parallel mode. In order to reduce the computaton tme we set the maxmum number of the teraton step, when the teraton step exceeds the maxmum one, the network s consdered as fallng nto a convergence state. Ths rule s usually used n the bnary neural networks. If the targ cost s the target total cost set by the user as expected one, we have: 1496 IEEJ Trans. EIS, Vol.124, No.7, 2004

4 An Effcent Algorthm for Vertex Cover Problem Table 1. The smulaton results for the random graphs. Propose Algorthm Graph Algorthm of Yuan et al. wthout the modfcaton of A wth the modfcaton of A Ver. Edg. Den. Cover CPU (s) Cover Cover CPU (s) (1) Set the parameter δ to 0.01 and ntal parameters A, B to 1.0. (2) Intalze the state of the network. (3) Update the neural network n synchronous parallel mode wth current parameter set untl the network converges to a stable state or the teraton step exceeds the maxmum one. (4) Check the network, f targ cost s reached, then termnates ths procedure. (5) Select one neuron wth satsfyng Eq.(20) (6) Use rule (Eq. (21) and (22)) to compute the new parameter A. (7) Go to the step 3. It s worth to note that about the condton of termnaton, though t could requre more CPU tme for some problems, approprate tunng of targ cost s useful n the practcal smulaton. 5. Smulaton Results In order to assess the effectveness of the proposed algorthm, extensve smulatons were carred out on two types of graphs: the randomly generated graphs and the benchmark graphs for the vertex cover problem. The algorthm was mplemented on PC Staton (Pentum4 1.8GHz). The frst type of graph s random graph (18). The random graph s defned n terms of two parameters, n and d. The parameter n specfes the number of vertces n the graph; the parameter d, 0<d<1, s the edge densty. Wth parameters n and d, the number of edges of the graph wll be dn(n 1)/2. We generated 14 such graphs to test the proposed method. The method of Yuan et al (15). was also executed for comparson. For each of nstances, 100 smulaton runs were performed. Informaton on the test graphs as well as all results was shown n Table 1. The results that we recorded for each graph were the best solutons by the algorthm of Yuan et al., and by the proposed algorthm. For readers reference, the computaton tmes to obtan the best soluton and the solutons found by the proposed algorthm wthout the modfcaton of parameter A were also gven n Table 1. The computaton tme was the average of 100 Table 2. The ncrease rato of computaton tme aganst the problem sze. Graph Edge densty: 0.15 Edge densty: 0.25 Vertex Edges Increase rato edges Increase rato % % % % % % % % % % % % smulatons. In general, wth the ncrease of the sze of the problem, the computatonal cost wll ncrease. In Table 2, we showed the ncrease rato of computaton tme aganst the problem sze (numbers of edges) under the graphs of same edge densty. The ncrease rato s defned by: ncrement of CPU rato = 100 (24) ncrement of edges From the Table 2, we can see that although the computatonal cost ncrease wth the ncrease of the task sze, the ncrease rato s very small. The expermental results can be summarzed as follows: (1) The soluton qualty of the proposed algorthm s much better that of Yuan et al s method. (2) The soluton found by the proposed algorthm wth the modfcaton of A s much better than that wthout the modfcaton of A. In other words, the method for escapng local mnma n the proposed algorthm s very effcent. (3) The computaton tme of proposed algorthm s about or less than 4 second even for relatvely large random graphs up to 300 vertces. Besdes, although the computatonal cost ncrease wth the ncrease of the task sze, the ncrease rato s very small. The computatonal cost for each graph s reasonable. To see f the proposed algorthm can fnd optmal solutons, we test the proposed algorthm on some benchmark graphs wth known answers. The problem of generatng benchmark graphs for NP-hard combnatoral problems has been nvestgated n (19) (20). Procedures for C

5 Table 3. The smulaton results for benchmark graphs. Algorthm of Yuan et al. Proposed Algorthm Graph Ver. Edg. k Vertex Cover CPU (s) Vertex Cover CPU (s) RG RG G RG RG RG RG RG RG RG generatng benchmark graphs wth known answers were descrbed n (21).In (22) the generatng procedures for vertex cover were evaluated expermentally on several approxmaton algorthms. In (14), some graphs generated by (22) were used to test ther algorthm for the vertex cover problem. Here we used some of these graphs n (14) to test the proposed algorthm. The results were shown n Table 3 where k refers the sze of the optmal soluton. In Table 3 the solutons and the computaton tme of the proposed algorthm were gven. From Table 3, we can see that the proposed algorthm could fnd optmal solutons for each benchmark graph. 6. Concluson We have proposed an effcent parallel algorthm for solvng the mnmum vertex cover problem usng bnary neural network. The proposed algorthm has two phases. The frst phase uses the neural network to decrease the energy and fnd a near-mnmum vertex cover. The second phase adjusts the balance between terms of energy functon, thus makng the network escape from the local mnmum vertex cover. The proposed algorthm was tested on a large number of random graphs and benchmark graphs and compared wth Yuan s algorthm. The expermental results ndcated that the proposed algorthm s better than prevous works and can fnd optmal or near optmal solutons. (Manuscrpt receved June 19, 2003, revsed Nov. 17, 2003) References ( 1 ) S. Khur and T. Back: An evolutonary heurstc for the mnmum vertex cover problem, J. Kunze and H. Stoyan, edtors, KI-94 Workshops (Extended Abstracts), pp.83 84, Bonn (1994). ( 2 ) R. M. Karp: Reducblty among combnatoral problems, Complexty of Computer Computaton, Plenum Press, New York, pp (1972). ( 3 ) M. R. Garey, D. S. Johnson, and L. Stockmeyer: Some smplfed NP-complete graph problems, Theoretcal Computer Scence, Vol.1, pp (1976). ( 4 ) R. Nedermeer and P. Rossmanth: Upper bounds for vertex cover further mproved, Proceedngs of the 16th Symposum on Theoretcal Aspects n Computer Scence (STACS 99), Sprnger-Verlag LNCS 1563, pp (1999). ( 5 ) U. Stege: Resolvng Conflcts from Problems n Computatonal Bology. Ph.D. thess, No.13364, ETH Zurch (2000). ( 6 ) M.R. Garey and D.S. Johnson: Computers and Intractablty: A Gude to the Theory of NP-completeness. San Francsco: Freeman (1979). ( 7 ) D. S. Hochbaum: Effcent bounds for the stable set, vertex cover and set packng problems, Dscrete Appl. Mathematcs, Vol.6, pp (1983). ( 8 ) B. Monen and E. Speckenmeyer: Ramsey numbers and an approxmaton algorthm for the vertex cover problem, Acta Informatca, Vol.22, pp (1985). ( 9 ) P. Berman and T. Fujto: On approxmaton propertes of the ndependent set problem for low degree graphs, Theory of Computng Syst., Vol.32, pp (1999). (10) M. R. Fellows: On the complexty of vertex cover problems, Techncal report, Computer Scence Department, Unversty of New Mexco (1988). (11) J.F. Buss and J. Goldsmth: Nondetermnsm wthn P. SIAM Journal on Computng, Vol.22, pp (1993). (12) R.G. Downey and M.R. Fellows: Fxed parameter tractablty and completeness II: completeness for W[1], Theoretcal Computer Scence, Vol.141, pp (1995). (13) J. Chen, I. A. Kanj, and W. Ja: Vertex cover: further observatons and further mprovements, 25th nternatonal Workshop on Graph- Theoretcal Concepts n Computer Scence (WG 99), LNCS, pp (1999). (14) F. Dehne, et al.: Solvng large FPT problems on coarse graned parallel machnes, Avalable: carleton.ca/ fpt/papers/ndex.htm. (15) J.J. Hopfeld and D.W. Tank: Neural computaton of decsons n optmzaton problems, Bo. Cybern., No.52, pp (1985). (16) S. Y. Yuan and S. Y. Kuo: A new technque for optmzaton problems n graph theory, IEEE Trans. on Computers, Vol.47, pp (1998). (17) W. S. McCulloch and W. H. Ptts: A logcal calculus of deas mmanent n nervous actvty, Bull. Math. Bophys, Vol.5, pp (1943). (18) D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon: Optmzaton by smulated annealng: An expermental evaluaton; Part 1, graph parttonng, Operatons Research, Vol.37, No.6, pp (1989). (19) B. Krshnamurthy: Constructng test cases for parttonng heurstcs. IEEE Trans. on Computers, Vol.36, pp (1987). (20) B.N. Khoury, P.M. Pardalos, and D.-Z. Du: A test generator for the stener problem n graphs, ACM Trans. on Mathematcal Software, Vol.19, pp (1993). (21) L.A. Sanchs: Generatng hard and dverse test sets for NPhard graph problems, Dscrete Appl. Mathematcs, Vol.58, pp (1995). (22) L.A. Sanchs: Test case constructon for the vertex cover problem, DIMACS Seres n Dscrete Mathematcs and Theoretcal Computer Scence, Vol.15, pp (1994) IEEJ Trans. EIS, Vol.124, No.7, 2004

6 An Effcent Algorthm for Vertex Cover Problem Rong Long Wang (Member) receved a B.S. degree from Hangzhe teacher s college, Zhejang, Chna and an M.S. degree from Laonng Unversty, Laonng, Chna n 1987 and 1990, respectvely. He receved hs D.E. degree from Toyama Unversty, Toyama, Japan n From 1990 to 1998, he was an Instructor n Benx Unversty, Laonng, Chna. In 2003, he joned Unversty of Fuku, Fuku Japan, where he s currently an Instructor n Department of Electrcal and Electroncs Engneerng. Hs current research nterests nclude genetc algorthm, neural networks, and optmzaton problems. Xn Shun XU (Non-member) receved a B.S. degree from Shandong Normal Unversty, Shandong, Chna and a M.S. degree from Shandong Unversty, Shandong, Chna n 2000 and Now he s workng toward the Ph.D. degree at Toyama Unversty, Japan. Hs man research nterests are neural networks and optmzatons Zheng Tang (Non-member) receved a B.S. degree from Zhejang Unversty, Zhejang, Chna n 1982 and an M.S. degree and a D.E. degree from Tsnghua Unversty, Bejng, Chna n 1984 and 1988, respectvely. From 1988 to 1989, he was an Instructor n the Insttute of Mcroelectroncs at Tsnhua Unversty. From 1990 to 1999, he was an Assocate Professor n the Department of Electrcal and Electronc Engneerng, Myazak Unversty, Myazak, Japan. In 2000, he joned Toyama Unversty, Toyama, Japan, where he s currently a Professor n the Department of Intellectual Informaton Systems. Hs current research nterests nclude ntellectual nformaton technology, neural networks, and optmzatons. C