Some special clique problems

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1 Some special clique problems Reate Witer Istitut für Iformatik Marti-Luther-Uiversität Halle-Witteberg Vo-Seckedorff-Platz, D 0620 Halle Saale Germay Abstract: We cosider graphs with cliques of size k for k i special itervals to solve the clique-problem i efficiet time. Key-Words: Clique, Idepedet set, Vertex cover, Strog NP-completeess, Time complexity, Polyomial algorithm. Itroductio The strog NP-completeess ad the hopeless outlook for a good result by meas of approximatio show the hardess of fidig large cliques i graphs. We cosider possibilities to solve the clique problem i special cases of the clique umber k. 2. Fudametal defiitios ad coectios The clique decisio problem is give as follows: Defiitio 2.: CLIQUEG,k Istace: Graph G = [V, E] with a set V of vertices ad a set E V V of edges, V =, k N, k. Questio: Does G have a k-clique, i.e., does exist a set V V with V = k, so that for every pair v, v 2 V v, v 2 E?

2 The idepedet set problem ad the vertex cover problem are defied as follows: Defiitio 2.2: INDEPENDENT SET G,k Istace: Graph G = [V, E] with a set V of vertices ad a set E V V of edges, V =, k N, k. Questio: Does G cotai a idepedet set of size k, i.e., a subset V V with V k such that o two vertices i V are joied by a edge i E? Defiitio 2.3: VERTEX COVER G,k Istace: Graph G = [V, E] with a set V of vertices ad a set E V V of edges, V =, k N, k. Questio: Is there a vertex cover of size k, i.e., such that for each edge v, v 2 E at least oe of v ad v 2 belogs to V? The complemet of G is G = [V, E] with E = {v, v 2 ; v, v 2 V v, v 2 E}. Lemma 2.: For ay graph G = [V, E] ad subset V V, V = k the followig statemets are equivalet: V is a clique i G. V is a idepedet set for G. V \ V is a vertex cover for G.[] CLIQUE is a NP-complete problem [6]. It is ot kow ay polyomial time algorithm for solvig this problem. The complexity does t deped o the size of umbers. Extremely large umbers would be ulikely to occur. We ca use to for odes. A limitatio of the size of umbers is t profitable. CLIQUE is strog NP-complete. There do t exist pseudopolyomial algorithms uless P = N P [6]. The PCP-theory proves the clique problem as ot solvable by approximatio with reasoable results [2]. 2

3 3. The umber of subsets of V for k as a liear fuctio of All kow algorithms for decisio of CLIQUE i geeral ispect k subsets of vertices. I practice we ofte have limitatios to k, such that the expoetial time of algorithms is t so bad. Let s see how may subsets V we have to examie by differet k with k = c for a costat c > 0. Lemma 3.: Give ay graph G = [V, E] with V = ad a atural umber k we estimate the umber ν of all possible subsets V V with V = k as follows: a 2+ /2 < ν < + /2 for k = /2 b 27 /3 3 < ν < 27 /3 2 for k = /3 8 c 256 / 5 < ν < d e ν = k /6 < ν < / for k = / /6 for k = / /2 < ν < /2 for k = /2 a For the first relatio we give a iductive proof over k. = 2k. For k = there is k+ k = 5 < 2 = 2. With the iductive hypothesis k+ k < 2k k we get k+ k 2k+2k+2 < 2k 2k+2k+2 k+ 2 k = 2k+2 k+ 2 k+. Because of k+5 < 2k+2k+2 k+k+ 2 we get Let k+5 k+ < k+ k 2k+2k+2 < 2k+2 k+ 2 k+ with the first estimatio. So we have 2+ /2 < k = ν for k = /2. We cosider a k = k k = k k k k = 2k! k! 2 2 Because of 2i for i N we obtai a k < 2i < 2i k 2i k+ = k 2k k 3 = 2k+a k. = 2k k. k

4 a 2 k < 2k+, a k < 2k+, 2k k < k 2k+. It followes ν = k < + /2 for k = /2. b We prove by iductio over k. Let = 3k. For k =, = 3 is 27 9 = 9 < 3 = 3 = ν. Because of iductive hypothesis 3 27 k k < 3k k we get 3 27 k 33k+3k+2 3k 33k+3k+2 k 2k+2k+ k 2k+2k+ = 3k+3 k+. With = 3 27 k k 27 k k+ < 3 k 3 27 k+ k+ 27 k 27k 2 +27k+6 = k 2 +6k k 33k+3k+2 k 2k+2k+ /3 < k = ν for followes the iductive claim ad 27 3 k = /3. Now we prove the secod relatio. For k = is 3 < Because of the iductive hypothesis 3k k = 3k! 3k+3 k+ = 2 27 k!2k! k+. 33k+3k+2 2k+2k+ < 2 27 c - e Aalogous by iductio over k k 33k+3k+2 2k+2k+ < 2 = 3 < k we obtai 27 k 27k 2 +k+6/27 k 2 +3/2k+/2 Coclusio 3.2: a 27 /3 3 < ν < + /2 for k [/3, /2] [/2, 2/3] b 6656 /6 325 < ν < 27 /3 2 for k [/6, /3] [2/3, 5/6] c /2 < ν < / 27 for k [/2, /] [3/, /2] I frot of adaptig a CLIQUE problem for a give k we fid out upper ad lower bouds of time complexity with lemma 3.. So we coclude the CLIQUE problem as feasible i practice or ot.. Fast geeratio of big cliques If G with vertices has a clique of size k, we ca costruct a graph G 2 with 2 vertices ad with k 2 clique poits.

5 Lemma.: Let G = [V, E], k = V ad G 2 = [V 2, E 2 ] with V 2 = V V ad E 2 = {u, v, u, v ; u = u v, v E u, u E}. G has a k-clique if ad oly if G 2 has a k 2 -clique [5]. : If G has a clique C = {v, v 2,..., v k } with C = k the G 2 has a clique C 2 = {u, v; u, v C} with C 2 = k 2. : If G 2 has a clique C 2 with C 2 = k 2 the C = {u; v V with u, v C 2 } is a clique i G. Suppose C k. Oe of the C sets C u = {v; u, v E} for u C are cliques ad have at least k odes. Lemma.2: a G has a k -clique G 2 has a k-clique G has a k -clique. b G has a idepedet set of size k G 2 has a k 2 -clique G 2 has a idepedet set of size k 2. c G has a vertex cover of size k G 2 has a k 2 -clique. a is obvious due to Lemma. b G has a idepedet set of size k G has a k-clique Lemma 2. G 2 has a k 2 -clique Lemma.. G 2 = G 2 has a idepedet set of size k 2 E 2 = E 2 []. c G has a vertex cover of size k G has a k-clique Lemma 2. G 2 has a k 2 -clique Lemma.. Assume G = [V, E] with V = has a clique of size k. The G 2r has a clique of size k 2r. So we ca costruct graphs with 2r odes ad a k 2r -clique, r N. We implemeted a program with LEDA to build such graphs. 5

6 5. The cardiality of V for k as a logarithmic fuctio of Lemma 5.: Give ay graph G = [V, E] with V = ad k. For the umber ν of subsets V V with V = k is log log < ν < log + log for k = log ad k = log. log log < ν = log = log =... log + log! < log + log Let G = [V, E] with V =, k. The ruig time or time complexity of a algorithm A o iput w = {v, v 2 ; v, v 2 E}, k is the fuctio T, k that is the maximum of the ruig time of A o w over all iputs w with V =. Theorem 5.2: Let G = [V, E ] with V =. If there is a algorithm A for CLIQUEG, k 2 with time complexity T, k 2 = t, k 2 for k 2 < log, the a algorithm A 2 for CLIQUEG, k 2 exists with time complexity T, k 2 = 2k + 2 3k + t k, k for ay k 2 <. Give G = [V, E ], k 2 <, = V, we costruct a algorithm A 2 : a We costruct a graph G 2 = [V 2, E 2 ] with V 2 = {{v, v 2,..., v k }; v,..., v k V } ad E 2 = {{u, u 2,..., u k }, {v, v 2,..., v k }; {u,..., u k, v,..., v k } V ad 2k clique i G } b ad apply algorithm A to G 2, k i t k, k steps. If G 2 has a k-clique the the poits i every ode of G 2 are pairwise coected i V. The poits of two coected odes i G 2 are pairwise coected i E. So a k-clique i G 2 delivers a k 2 -clique i G. O the other had k 2 -clique odes i G we ca dissect i k sets of k odes. 6

7 Every set is a ode i the clique of G 2. Therefore: G 2 has a k-clique G has a k 2 -clique. V 2 = k < k + k < k. The umber of edge tests i G 2 is E 2 = k k k. is the time to prove whether 2k odes build a clique i G 2k 2 2 test for a edge i G 2. So G is calculated to G 2 i k 2k k k 2 2 < k + k 2k + 2 2k 2 2k + < 2k + 2 3k steps. Because of V 2 > k k ad log V2 > k log k k the coditio clique umber < log of vertex umber for A comes true. Theorem 5.3: Let G = [V, E] with V =. If there is a polyomial algorithm A for CLIQUEG, k for k < log, the a algorithm A 2 for CLIQUEG, k exists with time complexity T, k = k O k = 2 O k log for ay k. Let t, k = p, p costat, for k < log. t k, k = k p < k + p k, p costat. With theorem 5.2 follows T, k = 2 k k + t k, k 2 k k + k + p k = k O k. Theorem 5.: Let G = [V, E] with V =. CLIQUEG, k for k Olog is decidable i polyomial time. 7

8 G = [V, E] with V =. We costruct G. Accordig [3] exists a algorithm A for VERTEX COVERG, k with time complexity Ok k. G has a k-clique G has a vertex cover of size k. Algorithm A with G, k produces i O k k steps the result. Let k c log, c costat. The time complexity is Oc log c log = Oc log + c log Coclusio Before decidig a clique problem it s clever to look i which sector does k lie. I depedece o this sector we ca fid out the time complexity. For k ear /2 we have the worst case of time. For k ear the time is polyomial. If there exists a polyomial algorithm for k < log the CLIQUE problem has a complexity better as expoetial. Ackowledgemet I would like to thak Gregor Erz for creatig the LEDA source code. Refereces [] M. R. Garey, D. S. Johso. Computers ad Itractability. A Guide to the Theory of NP-completeess. Bell Telephoe Laboratories Icorporated 979 [2] E. W. Mayr, H. J. Prömel, A. Steger. Lectures o Proof Verificatio ad Approximatio Algorithms. Spriger 998 [3] R. Niedermeier, P. Rossmaith. Upper bouds for Vertex Cover Further Improved. Proceedigs of the 6th STACS, Spriger LNCS 999 [] C. H. Papadimitriou. Computatioal Complexity. Addiso-Wesley, Readig MA 99 [5] C. H. Papadimitriou, K. Steiglitz. Combiatorial Optimizatio. Pretice Hall 982 [6] I. Wegeer. Theoretische Iformatik. Teuber 993