Approximating the minimum independent dominating set in perturbed graphs

Transcription

1 Aoximating the minimum indeendent dominating set in etubed gahs Weitian Tong, Randy Goebel, Guohui Lin, Novembe 3, 013 Abstact We investigate the minimum indeendent dominating set in etubed gahs gg, ) of inut gah G = V, E), obtained by negating the existence of edges indeendently with a obability > 0. The minimum indeendent dominating set MIDS) oblem does not admit a olynomial unning time aoximation algoithm with wost-case efomance atio of n 1 ɛ fo any ɛ > 0. We ove that the size of the minimum indeendent dominating set in gg, ), denoted as igg, )), is asymtotically almost suely in Θlog V ). that the obability of igg, )) 4 V algoithm of oven wost-case efomance atio time. Futhemoe, we show is no moe than V, and esent a simle geedy 4 V and with olynomial exected unning Keywods: Indeendent set, indeendent dominating set, dominating set, aoximation algoithm, etubed gah, smooth analysis 1 Intoduction An indeendent set in a gah G = V, E) is a subset of vetices that ae ai-wise non-adacent to each othe. The indeendence numbe of G, denoted by αg), is the size of a maximum indeendent set in G. One close notion to indeendent set is the dominating set, which efes to a subset of Deatment of Comuting Science, Univesity of Albeta. Edmonton, Albeta T6G E8, Canada. weitian@ualbeta.ca goebel@ualbeta.ca Coesondence autho. guohui@ualbeta.ca 1

2 Tong et al. vetices such that evey vetex of the gah is eithe in the subset o is adacent to some vetex in the subset. In fact, an indeendent set becomes a dominating set if and only if it is maximal. The size of a minimum indeendent dominating set of G is denoted by ig), while the domination numbe of G, o the size of a minimum dominating set of G, is denoted by γg). It follows that γg) ig) αg). Anothe elated notion is the vetex) coloing of G, in which two adacent vetices must be coloed diffeently. Note that any subset of vetices coloed the same in a coloing of G is necessaily an indeendent set. The chomatic numbe χg) of G is the minimum numbe of colos in a coloing of G. Clealy, αg) χg) V. The indeendence numbe αg) and the domination numbe γg) and the chomatic numbe χg)) have eceived numeous studies due to thei cental oles in gah theoy and theoetical comute science. Thei exact values ae NP-had to comute [4], and had to aoximate. Raz and Safa showed that the domination numbe cannot be aoximated within 1 ɛ) log V fo any fixed ɛ > 0, unless NP DTIME V log log V ) [9, 3]; Zuckeman showed that neithe the indeendence numbe no the chomatic numbe can be aoximated within V 1 ɛ fo any fixed ɛ > 0, unless P = NP [14]; fo ig), Halldósson oved that it is also had to aoximate within V 1 ɛ fo any fixed ɛ > 0, unless NP DTIME o V ) ) [5]. The above inaoximability esults ae fo the wost case. Fo analyzing the aveage case efomance of aoximation algoithms, a obability distibution of the inut gahs must be assumed and the most widely used distibution of gahs on n vetices is the andom gah Gn, ), which is a gah on n vetices, and each edge is chosen to be an edge of G indeendently and with a obability, whee 0 = n) 1. A gah oety holds asymtotically almost suely a.a.s.) in Gn, ) if the obability that a gah dawn accoding to the distibution Gn, ) has the oety tends to 1 as n tends to infinity [1]. Let Ln = log 1/1 ) n. Bollobás [] and Luczak [7] showed that a.a.s. χgn, )) = 1 + o1))n/ln fo a constant and χgn, )) = 1 + o1))n/ lnn)) fo c/n n) o1) whee c is a constant. It follows fom these esults that a.a.s. αgn, )) = 1 o1))ln fo a constant and αgn, )) = 1 o1)) lnn)/ fo C/n o1). The geedy algoithm, which colos vetices of Gn, ) one by one and icks each time the fist available colo fo a cuent vetex, is known to oduce a.a.s. in Gn, ) with n ɛ 1 a coloing whose numbe of colos is lage than the χgn, )) by only a constant facto see Ch. 11 of the monogah of Bollobás [1]). Hence the

3 Minimum indeendent dominating set 3 lagest colo class oduced by the geedy algoithm is a.a.s. smalle than αgn, )) only by a constant facto. Fo the domination numbe γgn, )), Wieland and Godbole showed that a.a.s. it is equal to eithe Ln LLn)ln n)) + 1 o Ln LLn)ln n)) +, fo a constant o a suitable function = n) [13]. It follows that a.a.s. ign, )) Ln LLn)ln n)) + 1. Recently, Wang oved fo ign, )) an a.a.s. ue bound of Ln LLn)ln n)) + k + 1, whee k = max{1, L} [1]. Aveage case efomance analysis of an aoximation algoithm ove andom instances could be inconclusive, because the andom instances usually have vey secial oeties that distinguish them fom eal-wold instances. Fo instance, fo a constant, the andom gah Gn, ) is exected to be dense. On the othe hand, an aoximation algoithm efoms vey well on most andom instances can fail miseably on some had instances. Fo instance, it has been shown by Kučea [6] that fo any fixed ɛ > 0 thee exists a gah G on n vetices fo which, even afte a andom emutation of vetices, the geedy algoithm oduces a.a.s. a coloing using at least n/ log n colos, while χg) n ɛ. analysis. To ovecome this, Sielman and Teng [10] intoduced the smoothed This new analysis is a hybid of the wost-case and the aveage-case analyses, and it inheits the advantages of both, by measuing the exected efomance of the algoithm unde slight andom etubations of the wost-case inuts. If the smoothed comlexity of an algoithm is low, then it is unlikely that the algoithm will take long time to solve actical instances whose data ae subect to slight noises and imecision. Fomally, let A be an algoithm we want to analyze and Q be the quality measuement. Without loss of geneality, we assume the lage Q the wose the algoithm A is, such as Q being the unning time of algoithm A. Given x an inut instance, QA, x) denotes the efomance of algoithm A on x. Let U denote the set of instances and U n denote the subset of size-n instances. The smoothed efomance measue of algoithm A unde Q essentially is the wost of the exected efomances of algoithm A in a small neighbohood of an instance, ove all instances, and is defined ecisely as M smooth n, σ) = su x U n E [QA, x + σ x )], whee is the andom noise and σ is the etubation aamete measues the stength of noise. Intuitively, σ x is the etubation on instance x, in which the the magnitude of the etubation is elated to the magnitude of inut. We have the following obsevations. If the smoothed efomance measue of algoithm A unde Q is good with some elatively small σ and some ea-

4 4 Tong et al. sonable andom model fo, then it is unlikely algoithm A would efom vey bad in eal wold alications unde quality measue Q, because eal wold instances ae often subect to a slight amount of noise, esecially when they ae obtained fom measuements of eal-wold henomena. A classic examle is the Simlex method fo linea ogamming. Simlex method is a vey actical algoithm, but it has exonential unning time in the wost-case. Sielman and Teng [10] had shown that Simlex method has olynomial smoothed unning time, which exlains the above henomenon efectly. Though the smoothed analysis concet was oiginally intoduced fo the comlexity of algoithms, we extend its idea to deict the essential oeties of oblems. In this ae, we study the aoximability of the minimum indeendent dominating set MIDS) oblem unde the smoothed analysis, and we esent a simle deteministic geedy algoithm beating the stong inaoximability bound of n 1 ɛ, with olynomial exected unning time. The MIDS oblem, and the closely elated indeendent set and dominating set oblems, have imotant alications in wieless netwoks, and have been studied extensively in the liteatue. Ou obabilistic model is the smoothed extension of andom gah Gn, ) also called semi-andom gahs in [8]), oosed by Sielman and Teng [11]: given a gah G = V, E), we define its etubed gah gg, ) by negating the existence of edges indeendently with a obability of > 0. That is, gg, ) has the same vetex set V as G but it contains edge e with obability e, whee e = 1 if e E o othewise e =. Fo sufficiently lage, Manthey and Plociennik esented an algoithm aoximating the indeendence numbe αgg, )) with a wost-case efomance atio O n) and with olynomial exected unning time [8]. Re-define Ln = log 1/ n. We fist ove on γgg, )), and thus on igg, )) as well, an a.a.s. lowe bound of Ln LLn)ln n)) if > 1 n. We then ove on αgg, )), and thus on igg, )) as well, an a.a.s. ue bound of ln n/ if < 1 o ln n/1 ) othewise. Given that the a.a.s. values of αgn, )) and ign, )) in andom gah Gn, ), ou ue bound comes with no big suise; nevetheless, ou ue bound is deived by a diect counting ocess which might be inteesting by itself. Futhemoe, we extend ou counting techniques to ove on igg, )) a tail bound that, when 4 ln n/n < 1, P[igG, )) 4n/] n. We then esent a simle geedy algoithm to aoximate igg, )), and ove that its wost case efomance atio is 4n/ and its exected unning time is olynomial.

5 Minimum indeendent dominating set 5 A.a.s. bounds on the indeendent domination numbe We need the following seveal facts. Fact 1 e x 1+x 1 + x e x holds fo all x [ 1, 1]. Fact n n ) ne ) holds fo all = 0, 1,,..., n. Fact 3 Jensen s Inequality) Fo a eal convex function fx), numbes x 1, x,..., x n in its domain, ) and ositive weights a i, f ai x i ai ai fx i ) ; the inequality is evesed if fx) is concave. ai Given any gah G = V, E), let gg, ) denote its etubed gah, which has the same vetex set V as G and contains edge e with a obability of 1, if e E, e =, othewise..1 An a.a.s. lowe bound Recall that γgg, )) and igg, )) ae the domination numbe and the indeendent domination numbe of gg, ), esectively. Also, Ln = log 1/ n. Theoem 4 Fo any gah G = V, E) and 1 n < 1, a.a.s. γgg, )) Ln LLn)ln n)). Poof. Let S be the collection of all -subsets of vetices in gg, ), and these n ) sets of S ae odeed in some way. Define I of S, V, is a dominating set; set X = I. as a boolean vaiable to indicate whethe o not the -th -subset Clealy, γgg, )) < imlies that thee ae size- dominating sets. Theefoe, P[γgG, )) < ] P[X 1] EX ), whee EX ) is the exected value of X. We abuse the notation E a little, but its meaning should be clea at evey occuence.)

6 6 Tong et al. Fo the -th -subset V, let E be the subset of induced edges on V fom the oiginal gah G = V, E); let V c = V V, the comlement subset of vetices. Also, fo each vetex u V c, define Eu, V ) = {u, v) E v V }, and its size n u = Eu, V ). Using Fact 1, we can estimate EX ) as follows: n ) EX ) = EI ) = =1 = = = n ) =1 n ) =1 1 v V ex n ) ex =1 =1 v V v V 1 u,v) 1 u,v) 1 u,v) n ) ex n u 1 ) n u n ) ex =1 nu 1 ). 1 Since function fx) = 1 )x is convex in the domain [0, n], by Jensen s Inequality, the above becomes 1 n ) n u n EX ) ex u V c n )1 ) 1. =1 Since function gx) = e axb with a = 1 ) 1 n [0, n ], again by Jensen s Inequality, we futhe have 1 n EX ) ex n ) n 1 and b = n )1 ) is concave in the domain n ) =1 n u n )1 )..1) Recall that n u is numbe of edges in the oiginal gah G = V, E) between u and vetices n ) n of V. Each edge e E is thus counted towads the quantity n u exactly 1 =1

7 Minimum indeendent dominating set 7 times. That is, n ) =1 n n u = 1 ) E = Using Eq..), Fact and = Ln LLn)ln n)), Eq..1) becomes E n EX ) ex n ) n )1 ) 1 n ) ex n )1 ) 1 ne ) ex n ) Ln)ln n) ) ex ln n + ln n ) n n ) n ) E n..) ) = exln)ln n) LLn)ln n)) ln n + ln Ln)ln n) + Ln)ln n)/n) = ex LLn)ln n)) ln n ln Ln)ln n)/n 1)) ex LLn)ln n)) ln n ln ))..3) f The ight hand side in Eq..3) aoaches 0 when n +. Since > 1 n Ln LLn)ln n)) is an a.a.s. lowe bound on γgg, )). This oves the theoem. guaantees 1, Since P[igG, )) < ] P[γgG, )) < ], we have the following coollay: Coollay 1 Fo any gah G = V, E) and 1 n < 1, a.a.s. igg, )) Ln LLn)ln n)).. An a.a.s. ue bound Recall that αgg, )) is the indeendence numbe of gg, ). Theoem 5 Fo any gah G = V, E), a.a.s. αgg, )) Poof. ln n, if ln n n, 1 ], ln n 1, if [ 1, 1 ln n n ). Let S be the collection of all -subsets of vetices in gg, ), and these n ) sets of S ae odeed in some way. Define I as a boolean vaiable to indicate whethe o not the -th -subset

8 8 Tong et al. of S is an indeendent set; set X = I. Since αgg, )) > imlies that thee is at least one indeendent -subset, i.e. X > 0, the obability of the event αgg, )) > is less than o equal to the obability of the event X > 0, i.e. On the othe hand, let A denote the event I P[αgG, )) > ] P[X > 0]. follows that X = 0 is equivalent to the oint event A, i.e. = 0, i.e. the -th -subset is not indeendent. It P[X = 0] = P[ A ] P[A ] = 1 P[I = 1]). Theefoe, we have P[αgG, )) > ] 1 1 P[I = 1])..4) Let E denote the subset of edges of gg, ), each of which connects two vetices in the -th -subset of S. Note that E [0, ) ]. Among all the edges of E, assume thee ae n coming fom the oiginal edge set E of G. It follows that P[I = 1] = 1 e ) = e E Using this and Fact 1 in Eq..4) gives us 1 ) n 1 ) ). of them P[αgG, )) > ] 1 1 P[I = 1]) Conside the function fx) = and 0 x ). Since its deivative n ) 1 ex P[I = 1] ) 1 P[I =1 = 1] n ) P[I = 1 ex = 1] 1 P[I = 1] =1 n ) = 1 ex =1 1 ax b 1 a x b f x) = 1 1 in Eq..5), whee a = ax b ln a 1 a x b) < 0, if a < 1, = 0, if a = 1, > 0, if a > 1, ) n 1 ) ) n..5) 1 ) 1 > 0, b = 1 ) 0, 1),

9 Minimum indeendent dominating set 9 fx) is stictly deceasing if a < 1, o stictly inceasing if a > 1. Theefoe, the maximum value of function fx) is achieved at x = 0 if a 1, o at x = ) if a 1. When 1, that is a = 1 1, Eq..5) becomes n ) 1 ) P[αgG, )) > ] 1 ex =1 1 1 ) ) n 1 ) ) = 1 ex..6) 1 1 ) To ove P[αgG, )) > ] 0 as n +, we only need to ove that n 1 ) ) 1 1 ) 0 as n +. Using Fact, we have n 1 ) ) 1 1 ) = n ) 1 ) ne ) )..7) ) 1 Setting = ln n/. We see that + as n +. On the othe hand, when is lage enough, we have 1 ) 1 ) 1 = 1 o1))..8) 1 1

10 10 Tong et al. Using Eq..8) and Fact 1, when n is sufficiently lage, Eq..7) becomes The quantity n 1 ) ) 1 1 ) = ne ) 1 1 ) 1 + o1)) = ) ne ) 1 ne o1)) ) o1)) ne ) 1 + o1)) 1 ex ne = )) ex o1)) ne 1+ = 1 + o1)).9) = e e o1)) e o1))..10) e 4 5 in Eq..10) is less than 0.5 when n is sufficiently lage, the latte aoaches 0 when n +. This oves that when 1, P[αgG, )) > ] 0 as n +. That is, when 1, a.a.s. αgg, )) ln n/. When 1, that is a = 1 1, q = 1 1 and exactly the same agument as when 1 alies by elacing with 1 q, which shows that a.a.s. αgg, )) ln n/1 ). This oves the theoem. Since αgg, )) igg, )), P[igG, )) > ] P[αgG, )) > ] and thus we have the following coollay: Coollay Fo any gah G = V, E), a.a.s. igg, )) ln n, if ln n n, 1 ], ln n 1, if [ 1, 1 ln n n ).

11 Minimum indeendent dominating set 11 3 A tail bound on the indeendent domination numbe Theoem 6 Fo any gah G = V, E) and 4 ln n n, 1 ], Poof. P[igG, )) 4n 4n ] P[αgG, )) ] n. The oof of this theoem flows exactly the same as the oof of Theoem 5. In fact, with 1, we have both Eq..6) and Eq..7) hold. Diffeent fom the oof of Theoem 5 whee = ln n/, we have now = 4n ln n/ and theefoe Eq..8) holds as well. Again, using Eq..8) and Fact 1, when n is sufficiently lage, Eq..9) still holds. It then follows fom Fact 1 that Eq..6) becomes P[igG, )) ] P[αgG, )) ] ne 1+ 1 ex 1 + o1))). 3.1) e Using = 4n, we ove in the following that ne o1)) = o1). And consequently by Fact 1 again and = 4n 8n, Eq. 3.1) becomes P[igG, )) ] ne 1+ e e ) 1 + o1)) e ne 1+ e = e ex ln + 1 ln n 1 )) = e ex ln ln n 1 ) = e ex ln ln n 1 1 ) 4 ) ) n. 3.) The quantity ln ln n 1 ) in Eq. 3.) is non-negative when n, since ln ln n 1 1 ln8n) n ln n ln8n) + 1 4n 4 ln n ln n n 4 = 1 ln8n) 5 ) 0.

12 1 Tong et al. It follows that Eq. 3.) becomes P[igG, )) ] e ex ln ln n 1 ) ) n e e n < n. This oves the theoem. 4 Aoximating the indeendent domination numbe We esent next a simle algoithm, denoted as Aox-IDS, fo comuting an indeendent dominating set in gg, ). In the fist hase, algoithm Aox-IDS eeatedly icks a maximum degee vetex and udates the gah by deleting the icked vetex and all its neighbos; it teminates when thee is no moe vetex and etuns a subset I of V. If I 4n, algoithm Aox-IDS teminates and oututs I; othewise it moves into the second hase. In the second hase, algoithm Aox-IDS efoms an exhaustive seach ove all subsets of V, and etuns the minimum indeendent dominating set I. Theoem 7 Fo any gah G = V, E) and 4 ln n n, 1 ], algoithm Aox-IDS is a 4n - aoximation to the MIDS oblem on the etubed gah gg, ), and it has olynomial exected unning time. Poof. Note that igg, )) 1. The subset I of V comuted by algoithm Aox-IDS is a dominating set, since evey vetex of V is eithe in I, o is a neighbo of some vetex in I. Also, no two vetices of I can be adacent, since othewise one would be emoved in the iteation its neighbo was icked by the algoithm. Theefoe, I is an indeendent dominating set of gg, ). It follows that if algoithm Aox-IDS teminates afte the fist hase, I 4n igg, )). Also clealy the fist hase takes On 3 ) time. In the second hase, a maximum of n subsets of V ae examined by the algoithm. Since checking each of them to be an indeendent dominating set o not takes no moe than On ) time, the oveall unning time is O n n ). Note that this hase etuns I with I = igg, )). As αgg, )) I > 4n, Theoem 6 tells that the obability of executing this second hase is no

13 Minimum indeendent dominating set 13 moe than n. Theefoe, the exected unning time of the second hase is On ). This oves the theoem. 5 Conclusions We have efomed a smooth analysis fo aoximating the minimum indeendent dominating set oblem. The obabilistic model we used is the etubed gah gg, ) of the inut gah G = V, E) [11]. We have oved a.a.s. bounds and a tail bound on the indeendent domination numbe of gg, ), and esented an algoithm with the wost-case efomance atio of with olynomial exected unning time. 4 V and Acknowledgement This eseach was suoted in at by NSERC. Refeences [1] B. Bollobás. Random Gahs. Academic Pess, New Yok, [] B. Bollobás. The chomatic numbe of andom gahs. Combinatoica, 8:49 55, [3] U. Feige. A theshold of fo aoximating set cove. Jounal of the ACM, 45:634 65, [4] M. R. Gaey and D. S. Johnson. Comutes and Intactability: A Guide to the Theoy of NP-comleteness. W. H. Feeman and Comany, San Fancisco, [5] M. M. Halldósson. Aoximating the minimum maximal indeendence numbe. Infomation Pocessing Lettes, 46:169 17, [6] L. Kučea. The geedy coloing is a bad obabilistic algoithm. Jounal of Algoithms, 1: , [7] T. Luczak. The chomatic numbe of andom gahs. Combinatoica, 11:45 54, 1991.

14 14 Tong et al. [8] B. Manthey and K. Plociennik. Aoximating indeendent set in etubed gahs. Discete Alied Mathematics, 16: , 013. [9] R. Raz and S. Safa. A sub-constant eo-obability low-degee test, and sub-constant eoobability PCP chaacteization of NP. In Poceedings of the 9th Annual ACM Symosium on Theoy of Comuting STOC), ages , [10] D. A. Sielman and S.-H. Teng. Smoothed analysis of algoithms: Why the simlex algoithm usually takes olynomial time. Jounal of the ACM, 51: , 004. [11] D. A. Sielman and S.-H. Teng. Smoothed analysis: an attemt to exlain the behavio of algoithms in actice. Communications of the ACM, 5:76 84, 009. [1] C. Wang. The indeendent domination numbe of andom gah. Utilitas Mathematica, 8: , 010. [13] B. Wieland and A. P. Godbole. On the domination numbe of a andom gah. The Electonic Jounal of Combinatoics, 8:#R37, 001. [14] D. Zuckeman. Linea degee extactos and the inaoximability of max clique and chomatic numbe. Theoy of Comuting, 3:103 18, 007.