On the Approximability and Hardness of Minimum Topic Connected Overlay and Its Special Instances

On the Approximabil ity and Hardn ess of Minim um T opic Connected Overla y and Its Sp eci al Instances ✩ , ✩✩ Jun Hoso da c , Jura j Hromk ovi ˇ c a , T aisuke Izumi c , Hirotak a Ono b , Monik a Steinov´ a a , Koichi W ada c a Dep artment of Computer Scie nc e, ETH Zurich, Switzerland b Dep artment of Ec onomic Engine ering, Kyushu Unive rsity, Jap an c Gr aduate Scho ol of Engine ering, N agoya Institute of T e chnolo gy, Jap an Abstract In the context of designing a scalable ov erlay net work to supp or t decentralized topic-based pub/sub communication, the Minimum T opic-Connected Ov erlay problem ( Min-TCO in short) has be e n inv estigated: Given a set of t topics a nd a collection of n users to gether with the lists o f topics they a re interested in, the aim is to connect thes e users to a netw ork by a minim um num ber of edges such that ev ery graph induced by us ers in terested in a common topic is connected. It is kno wn that Min-TCO is N P -hard and approximable within O (log t ) in po lynomial time. In this pap er, we further inv estigate the problem and some of its spe cial instances. W e giv e v arious hardness results for instances where the n um ber of topics in whic h an user is in terested in is b ounded by a cons tant , and also for the instances wher e the num ber of users in terested in a common topic is constan t. F or the latter case, w e present a first c o nstant approximation algor ithm. W e also present s ome polynomia l-time algorithms for very r estricted instances of Min-TCO . Keywor ds: topic-connected ov erlay , approximation alg orithm, APX, hardness 1. In trod uctio n Recently , the spr eading of so cial netw orks and other s e rvices based on shar- ing co nt ent allo w ed the dev elopmen t of man y-to-many comm unication, often suppo rted by these services . Publishers publish information through a lo gical ✩ This research is partly supp orted by the Ja pan So ciet y for the Promotion of Science, Gran t-in-Aid for Scient ific Research, 21500 013, 21680001, 22650004, 22700010, 23104511, 23310104 , F oundation for the F usion of Science T echnology (FOST) and INAMORI FOUN- DA TION. The researc h is also partiall y funded b y SNF grant 200021-132510 /1. ✩✩ Some of the results of this pap er w ere presen ted at MFCS 2011 and PODC 2011. Email addr esses: juraj.hrom kovic@inf .ethz.ch (J ura j Hromko vi ˇ c), t-izumi@ nitech.ac .jp (T aisuke Izumi), hirotaka@e n.kyushu- u.ac.jp (Hirotak a Ono), monika.s teinova@i nf.ethz.ch (Monik a Steinov´ a), wada@nit ech.ac.jp (Koichi W ada) Pr e print submitte d to Elsevier Octob er 29, 2018 channel that is consumed by subscrib ed users. This environment is often mo d- eled by publish/ subscrib e (pub/sub) systems that can b e classified int o tw o categorie s. When the channels are a sso ciated with a collection of attributes and the messages are deliv ered t o a subscrib er only if their attributes matc h user-defined constr a int s, we sp ea k ab out c ontent-b ase d pub/sub systems. E ach channel in topic-b ase d pub/sub s ystems is a sso ciated with a sing le topic and the messag es are distributed to the users v ia channels by his/her topic se- lection. There a re numerous implementations of pub/sub sy s tems, for details see [1, 4, 6, 7, 22, 23, 2 5]. In our paper, we fo cus on topic-based p eer- to-p eer pub/sub sy s tems. In such a system, subscr ib e rs interested in a pa rticular topic hav e to be connected without the use of in termediate ag ent s (such as servers). Many as pects o f s uch a system can b e studied (see [9, 20]). Minimizing the dia meter of the ov erlay net work can minimize the ov erall time in which a mes sage is distributed to all the subscr ib er s. When minimizing the (av erage) degree of no des in the net work, the subscr ibe rs need to maintain a smaller num ber of connectio ns. In this pape r , we study the minimization o f the overall n um ber of connections in the system. A small num ber of connections may b e necessar y due to maintenance requirements o r may b e helpful since thus information ag grega ted int o a single message can b e broa dcasted to the netw ork a nd thus amor tize the head count of otherwise small messa ges. W e study her e the har dnes s of Minimum T opic-Conne cte d Overlay ( Min- TCO ) which was studied in differe nt scenar ios in [2, 9, 17, 18]. In Min-TCO , we are given a collection of user s, a s e t of topics, and a us e r-interest assig nment , we w an t to connect users in an o verla y net w ork G s uch that all users inter- ested in a commo n topic are connected and the o verall num ber of edges in G is minimal. The hardness of the pro blem was studied in [9] and [2]. In [9], the inapproximabilit y by a co nstant was proved and a logar ithmic-factor a ppr oxi- mation algo r ithm was pres ent ed. In [2], the low er bo und on the approximability of Min-TCO was improved to Ω(log ( n )), where n is the num ber of use r s. Moreov er, we fo cus here o n sp ecial instances of Mi n-TCO . W e study the case where, for each topic, there is a c onstant num ber of users interested in it. W e also consider the case where the num ber of topics in which any use r is interested is bounded b y a constant. W e believe that such restrictio ns on the instances hav e wide practica l a pplications such a s when a publisher has a limited num ber of s lo ts for users or the user’s application limits the num ber of topics that he/s he can follow. In the study of the gener al M in-TCO , we extend the metho d prese n ted in [9] and des ign a n appro ximation-prese r ving re duction fro m instances of the min- im um hitting set pr oblem to instances of Min-TCO . This reduction do es not only prov e a similar lower b ound as in [2], but also shows that Min-TCO is LOG AP X -complete and thus, concerning approximability , equiv alent with such a famous pro ble m as the minimu m set co v er. As our r eduction is not blo w- ing up the n um ber o f user s interested in a co mmon topic, the reduction is also a n appro ximation-prese r ving r e ductio n for the c a se w he r e the num ber of users interested in a co mmon topic is limited to a constan t. F urthermor e, we 2 design a o ne-to-one reductio n of these instances to sp ecial instances of the hit- ting set problem. As these s p ecia l instances o f the hitting set pr oblem are constantly a pproximable, we immediately obtain the firs t appr oximation algo - rithm for our sp ecial instances. This, together with our approximation preserv- ing reduction, shows that the restriction of Min- TCO to such sp ecial ins tances is AP X -complete. Finally , due to the one- to-one reduction and the proper - ties o f the s p ecia l insta nce s of hitting s et pro blem, we show the existence o f a po lynomial-size kernel and a non-trivial exact algo rithm, all fo r the instances of Min-TCO where the num ber of us e rs interested in a commo n topic is b ounded by a constant. F or the case, where the num ber of to pics of Min-TCO is b ounded fr om ab ove by (1 + ε ( n )) − 1 · log log n , for ε ( n ) ≥ 3 / 2 log log log n log log n − 3 / 2 log l og log n ( n is the n um ber of user s), w e pr esent a polynomia l-time alg orithm that computes the optimal solution. In the s tudy of instances where the num ber of topics an y user is interested in is restricted to a cons ta n t, w e show that, if this num ber is at most 6, Min-T CO cannot b e a pproximated within a factor of 694 / 693 in p olynomial time, unless P = N P , even if any pair of t wo user s is interested in at most three commo n topics. The pa per is or ganized as follows. Section 2 is devoted to the prelimina ries and a s umma r y of known results. The har dness, a pproximation results, kernel- ization and an exact algorithm for instanc e s of Min- TCO , wher e w e limit the nu mber of users in terested in a commo n topic by a constant, a r e discus sed in Sec - tion 3. This section also provides the discussio n ab out LO G AP X -completeness of the genera l Min-TCO . The res ults rela ted to the instances o f Min-TCO , where the num b er of to pics that each user is interested in is co nstant, are presented in Section 4. Section 5 contains a p olynomial- time a lgorithm that solves Min-TCO when the num ber o f to pic s is small. The co nclusion is pr ovided in Section 6. 2. Preliminaries In this section, w e define ba sic notions us ed througho ut the pap er. W e assume that the reader is familiar with notions of gra ph theory . Let G = ( V , E ) be an undirected gr aph, where V is the set of vertices and E is the set of edges. Let V ( G ) and E ( G ) denote the set o f vertices and the set of edge s of G , resp ectively . W e denote by E [ S ] the s et of edge s of G in the subgra ph induced by the v ertices from S ⊆ V , i. e., E [ S ] = { { u, v } ∈ E | u , v ∈ S } . The g r aph induced b y S ⊆ V is denoted a s G [ S ] = ( S, E [ S ]). By N [ v ] w e denote the close d neighb orho o d of vertex v , i. e., N [ v ] = { u ∈ V | { u, v } ∈ E } ∪ { v } . A graph G is called c onne ct e d , if, for any u 1 , u ℓ ∈ V , there ex is ts a pa th ( u 1 , u 2 , . . . , v ℓ ) such that { u i , u i +1 } ∈ E , for all 1 ≤ i < ℓ . Let x b e a n instance of an optimization problem (in this pap er, Min- TCO , Min-VC or Min-HS ), then b y | x | we denote the size of this ins ta nce, i. e., the nu mber of vertices and topics of an instance of Min-TCO and the n umber of elements and sets of an insta nce of Min-H S . F or a set S , | S | denotes the size of the set, i. e., the num ber of its elements. 3 The set of users or no des of our netw ork is deno ted b y U = { u 1 , u 2 , . . . , u n } . The topics ar e T = { t 1 , t 2 , . . . , t m } . Each user subscrib es to several topics. This relation is expressed by the user interest function INT : U → 2 T . The s e t of all vertices o f U interested in a topic t is denoted by U t . F or instance, if user u ∈ U is interested in topics t 1 , t 3 and t 4 , then we hav e INT( u ) = { t 1 , t 3 , t 4 } and u ∈ U t 1 , U t 3 , U t 4 . F or a given set of users U , a set of to pic s T , and an interest function INT, w e say that a gr aph G = ( U, E ) with E ⊆ {{ u, v } | u, v ∈ U ∧ u 6 = v } is t -topic-c onne cte d , for t ∈ T , if the subgraph G [ U t ] is connected. W e call the graph topic-c onne cte d if it is t -topic-co nnected for each topic t ∈ T . Note that the topic-c onnectedness prop erty implies that a messa ge published for topic t is transmitted to all user s interested in this topic without using non- in terested users as intermediate no des. The mos t gener al problem that we study in this paper is called Minimum T opic Conne ct e d O verlay : Problem 1. Min- TCO is the following optimiza tio n problem: Input: A set of users U , a se t of topics T , and an user interest function INT : U → 2 T . F easi ble sol utions: Any set o f edges E ⊆ {{ u , v } | u, v ∈ U ∧ u 6 = v } suc h that the gr aph ( U, E ) is topic-co nnec ted. Costs: Size of E . Goal: Minimization. In this pa per we study als o s o me of its sp ecial instances. W e restrict the nu mber of users that are interested in a co mmon topic, i. e., the size of U t , to a constan t. W e also study the instance s where each user is interested in a constant num ber of topics. The definitions necessary for these sp ecial instances are summarized in the b eg inning of the cor resp onding section. W e refer here to the famous m inimum hitting set pr oblem ( Min-HS ) and minimum set c ov er pr oblem ( Min-SC ). In Min-HS , we ar e given a system of s ets S = { S 1 , . . . , S m } on n elements X = { x 1 , . . . , x n } (i. e., S j ⊆ X ). A fea s ible solution of this pr oblem is a s et H ⊆ X , such that S j ∩ H 6 = ∅ fo r all j . Our goal is to minimize the size of H . The M in-SC is the dual problem to Min-H S . In this pro blem, w e are g iven a system of sets S = { S 1 , . . . , S m } on n elements X = { x 1 , . . . , x n } , a feasible solution is a set S ⊆ S of sets such that for all i there exists j such that x i ∈ S j ∈ S and the go al is the minimization o f the size of S . There ar e many mo difications and subpro blems of the hitting set pr oblem that are in tensively studied. In our pap er, we refer to the d -HS problem – a restriction o f Min -HS to instances where | S i | ≤ d for all i . The Min -HS is equiv a lent to the Mi n-SC ([3]), thus all the pr op erties of Min- SC ca rry over to Min-HS . F ollowing from these prop erties, we have LO G AP X - completeness of Mi n-HS ([10]) and AP X -completeness of d -HS ([21]). There is a well known d - approximation algorithm for d -HS ([5]), it can b e appr oximated 4 with ratio d − ( d − 1) ln ln n ln n ([12]), it is N P -hard to approximate it within a factor ( d − 1 − ε ) ([1 1]) and d -H S is not approximable within a factor b etter than d , unless the unique g ames c o njecture fails ([15]). W e use the sta ndard definitions from complexity theory (for deta ils see [13]): • F or N P O pro blems in the class P T AS , there exists an algorithm that, for arbitrar y ε > 0, pro duces a solution in time p olynomial in the input size (but po ssibly exp onential in 1 / ε that is within a factor (1 + ε ) from optimal. • The N P O pro blems in the class AP X are a pproximable by so me constant- factor approximation algorithm in p olynomial time. • F or N P O problems in the class LO G AP X , there exists a p olynomial- time logarithmic-fa ctor approximation algorithm. Thu s P T AS ⊆ AP X ⊆ LO G AP X . Definition 1. Let A and B be tw o N P O minimization pro blems. Let I A and I B be the sets of the instances of A a nd B , r esp ectively . Let S A ( x ) and S B ( y ) b e the sets of the feasible solutions and let cost A ( x ) and cost B ( y ) b e p olynomia lly computable meas ur es of the instances x ∈ I A and y ∈ I B , r esp ectively . W e say that A is AP-r e ducible to B , if ther e exis t functions f a nd g and a cons ta n t α > 0 such that: 1. F or any x ∈ I A and any ε > 0, f ( x, ε ) ∈ I B . 2. F or any x ∈ I A , for any ε > 0, and any y ∈ S B ( f ( x, ε )), g ( x, y , ε ) ∈ S A ( x ). 3. The functions f and g ar e computable in po lynomial time with resp ect to the sizes of instances x and y , for any fixed ε . 4. The time complexity of computing f and g is nonincreasing with ε for all fixed instances o f size | x | and | y | . 5. F or any x ∈ I A , for any ε > 0, and for any y ∈ S B ( f ( x, ε )) cost B ( y ) min { cost B ( z ) | z ∈ S B ( f ( x, ε )) } ≤ 1 + ε impl ies cost A ( g ( x, y , ε )) min { cost A ( z ) | z ∈ S A ( x ) } ≤ 1 + α · ε. 3. Results for Min-TCO When The N um b er o f Users Interested i n a Common T opic is a Constant In this whole sec tion, we denote by a tr iple ( U, T , INT) an instance o f Min - TCO . W e fo cus her e on the case wher e the num ber of user s tha t share a topic t , i. e., max t ∈ T | U t | , is b ounded. W e present her e a low er b ound on the approximability , a cons tant approxi- mation algo r ithm and an AP X -completeness pr o of for these restricted ins tances of Min-TCO . 5 3.1. Har dness r esults It is easy to see that, if max t ∈ T | U t | ≤ 2, then Min-TCO can b e solved in linear time, b ecause t wo user s sharing a topic t should b e directly co nnected by an edge, which is the unique minimum s olution. Theorem 1. If max t ∈ T | U t | ≤ 2 , then Min-T CO c an b e solve d in line ar time. W e ex tend the metho ds fro m [9] and design an AP-reduction from d -HS to Min-TCO , where max t ∈ T | U t | ≤ d + 1. Theorem 2. F or arbi tr ary d ≥ 2 , ther e exists an AP-r e duction f r om d -H S to Min-TCO , wher e max t ∈ T | U t | ≤ d + 1 . Proof. Let I HS = ( X, S ) be a n instance of d -HS and let ε > 0 b e arbitrary . W e o mit the subs c ript in the functions cost d − HS and cost Min − TCO as they are unambiguous. F or the instance I HS , we create an instance I TCO = ( U, T , INT) of Min-TCO with max t ∈ T | U t | ≤ d + 1 with | X | + k users, wher e k = | X | 2 ·  1+ ε ε  , as follows (the function f in the definition o f AP-r eduction). U = X ∪ { p i | p i / ∈ X ∧ 1 ≤ i ≤ k } , T = { t i S j | S j ∈ S ∧ 1 ≤ i ≤ k } , INT( x ) = ( { t i S j | x ∈ S j ∧ S j ∈ S ∧ 1 ≤ i ≤ k } for x ∈ X { t i S j | S j ∈ S } for x = p i Observe that the instance co n tains k · |S | topics a nd its size is polynomial in the size of I HS . The users interested in a topic t i S j ( S j ∈ S ) are exactly the elements that are mem b ers of set S j in d -HS plus a sp e cial user p i (1 ≤ i ≤ k ). Let S ol TCO be a feasible solution of Min-TCO on instance I TCO . W e par tition the solution into le vels. Level i is a se t L i of the edg es o f S ol TCO that are inc ident with the sp ecial user p i . In addition, w e denote by L 0 the set of edg es of S ol TCO that are not incident with any sp ecial user. Therefor e , S ol TCO = S k i =0 L i and L i ∩ L j = ∅ (0 ≤ i < j ≤ k ). W e claim tha t, for any L i (1 ≤ i ≤ k ), the set of the non-sp ecial users incident with edges o f L i is a feasible so lution of the instance I HS of d -HS . T his is tr ue since, if a set S j ∈ S is no t hit, none of the edges { x, p i } ( x ∈ S j ) is in L i . But then the user s interested in topic t i S j are not interconnected as user p i is disconnected. Let j b e c hosen such that L j is the smallest of all sets L i , for 1 ≤ i ≤ k . W e construct S ol HS by pic king all the non-sp ecia l users that are incident to some edge from L j (the function g in the definition of AP-reduction). Denote an optimal solution of d -HS and Min-TCO for I HS and I TCO by O pt HS and Opt TCO , resp ectively . If we knew Opt HS , w e would be able to construct a feasible solution of Min- TCO o n I TCO as follo ws. First, w e pick the edges { x, p i } , x ∈ Opt HS , for a ll sp ecial users p i , and include them in the solution. This wa y , for any topic 6 t ∈ INT( p i ), w e connect p i to so me element of X that is in terested in t , to o. T o have a fea sible solution, w e could miss some edges b etw een s o me ele ments of X . So, we pick all the edges b etw een elements from X . The feasible s olution of Min-TCO on I TCO that we obtain has roug hly cos t k · cost ( O pt HS ) + | X | 2 ≥ cost ( O pt TCO ) . On the other hand, if we replace all lev els L i (1 ≤ i ≤ k ) by lev el L j in S ol TCO , w e still hav e a feasible solution o f Min-TCO on I TCO , with c o st p os sibly smaller. Thus k · cost ( S ol HS ) ≤ cost ( S ol TCO ) . W e use these tw o ineq ualities to b ound the cost of S ol HS : k · cost ( S ol HS ) ≤ cost ( S ol TCO ) cost ( Opt TCO ) ·  k · cost ( O pt HS ) + | X | 2  and th us cost ( S ol HS ) cost ( Opt HS ) ≤ cost ( S ol TCO ) cost ( Opt TCO ) ·  1 + | X | 2 k  . If cost ( S ol TCO ) /cost ( Op t TCO ) ≤ 1 + ε and α := 2, then we hav e cost ( S ol HS ) cost ( Opt HS ) ≤ (1 + ε ) ·  1 + | X | 2 k  ≤ (1 + ε ) ·  1 + ε 1 + ε  = 1 + 2 ε. It is easy to see that the five conditions of Definition 1 are satisfied and th us we hav e an AP -reduction. ✷ Corollary 1 . F or any δ > 0 and p olynomial-time α -appr oximatio n algorithm of Min-T CO with max t ∈ T | U t | ≤ d + 1 , ther e exists a p olynomi al-time ( α + δ ) - appr oximation algorithm of d -HS . Proof. The appr oximation algo rithm for d -HS w ould use Theo rem 2 with k := | X | 2 · ⌈ α δ ⌉ . ✷ Our theorem also implies the following negative re s ults on approximabil- it y . One o f them holds if unique games c onje ctur e is true. This conjecture is discussed, for example, in [24] and was in tro duced by Kho t in [14]. Corollary 2 . Min-TCO with max t ∈ T | U t | ≤ d ( d ≥ 3 ) is N P -har d to ap pr ox- imate within a factor of ( d − 1 − ε ) , for any ε > 0 , and, if the un ique ga mes c onje ctur e hold s, ther e is no p olynomial-time ( d − ε ) -appr oximatio n algorithm for it. Proof. Otherwise, the reductio n descr ibed in the proof of Theorem 2 would imply an a pproximation a lgorithm for d -HS with a ratio b etter than d − 1 and d resp ectively . This would direc tly contradict theorems prov en in [11] and [15]. ✷ 7 The following corolla ry is an improvemen t of the already known results of [9] where a n O (log | T | )-a pproximation algo rithm is pr esented, and of [2] w he r e a low er b ound of Ω(log ( n )) o n the approximabilit y is shown. W e clo se the gap by desig ning a reductio n that can reduce any problem fro m class LO G AP X to Min-TCO preserving the a pproximation ratio up to a constant. Corollary 3 . Min-TCO is LO G AP X -c omple te. Proof. Min-TCO is in the cla ss LOG AP X since it admits a logar ithmic a pprox- imation algorithm as pres ent ed in [9]. Our reduction from the pr o of o f Theo- rem 2 is indep e nden t of d and thus an AP- reduction fro m LO G AP X -complete Min-HS to Min- TCO . ✷ 3.2. A Constant Appr oximation Algorithm In this subsection, w e pre sent a r eduction from Min-TCO with max t ∈ T | U t | ≤ d to O ( d 2 ) -HS thus sho wing that there ex is ts a constant approximation alg o- rithm for Min-T CO with max t ∈ T | U t | ≤ d as d -HS is constantly approximable. Moreov er, the constant approximation algor ithm classifies this problem to b e a mem b e r of the class AP X and thus, since the AP X -hardness was prov en in Sub- section 3.1, we conclude that Min-T CO with max t ∈ T | U t | ≤ d is AP X -complete. Recall that a partition of vertices V in graph G is a tuple ( A, B ), such that A ⊆ V , B ⊆ V , A ∩ B = ∅ , and A ∪ B = V . Definition 2. Let V = { v 1 , . . . , v n } b e a s et of vertices and for every partition ( A i , B i ) of V , let E i = { { u, v } | u ∈ A i ∧ v ∈ B i } . Then w e call the system S = { E 1 , . . . , E m } of a ll sets of e dges b etw een vertices of all the pa rtitions of V a char acteristic syst em of edges on V . In other words, S contains a ll sets o f edges that form a ma ximum bipartite gr aph o n V . In the fo llowing lemma, w e show the basic prop erties of characteristic sys- tems of edges. Lemma 1 . L et S = { E 1 , . . . , E m } b e a cha r ac teristic system of e dges on the set V of n vertic es. The n 1. m = 2 n − 1 − 1 . 2. | E j | ≤ ⌊ n/ 2 ⌋ · ⌈ n/ 2 ⌉ , for al l j , 1 ≤ j ≤ m . 3. Any t wo sets E i and E j differ in at le ast n − 1 elements ( 1 ≤ i < j ≤ m ). 4. H ⊆ {{ u , v } | u, v ∈ V ∧ u 6 = v } is a hitting set o f ( {{ u, v } | u , v ∈ V ∧ u 6 = v } , S ) if and only if ( V , H ) is c onne cte d. 5. The size of S is minimal su ch that p art 4 holds. Proof. Observe that the complementary gra ph ( V , F j ) ( F j = {{ u, v } | u, v ∈ V ∧ u 6 = v } \ E j ) c ontains tw o complete gr aphs – o ne on the v ertices of A j and other on the vertices of B j , and it is a maximal graph (in the nu mber o f e dges) that is not connected. W e use this o bserv ation to pro ve the las t tw o par ts of our lemma. 8 Part 1 : W e co un t the different partitions ( A j , B j ) of the vertices V as each such par tition determines a differ en t set E j of edge s . There are 2 n wa ys how to distribute vertices from V in to pa r titions. W e ha v e to s ubtract 2 p os sibilities for the cases where one o f A j or B j is empty . Ea ch of the o ther p os sibilities is counted twice – o nce when the vertices a re pres en t in A j and o nce when they are present in B j . Part 2: Let the tw o sets of vertices A j and B j of a pa rtition contain k > 0 and n − k vertices. Then the size of E j is k · ( n − k ). This function reaches its maximum for k = n/ 2 and th us we can conclude that, for a ll j , 1 ≤ j ≤ m , we hav e | E j | ≤ ⌊ n/ 2 ⌋ · ( n − ⌊ n/ 2 ⌋ ) = ⌊ n/ 2 ⌋ · ⌈ n/ 2 ⌉ . Part 3: Let us consider tw o different pa r titions ( A i , B i ) and ( A j , B j ) of the vertices V . The sets A i and A j m ust differ b y at least one v ertex. W.l.o.g., let the vertex v ∈ A i and v / ∈ A j . Then, due to the transition of the vertex v from A i to B j , there a re | B i | edges that are in E i but canno t b e in E j , a nd there are | A i | − 1 edges that are not in E i , but are in E j . Thus, the ov erall difference in the n um ber of elemen ts b etw een the sets E i and E j is a t least | A i | + | B i | − 1 = n − 1. Part 4 : Fir st, we prov e the if case. Suppose that H is a hitting s et, but ( V , H ) is not connected. Since S contains co mplemen ts o f all maximal sets of edges that indu ce a disconnected gr aph, ther e ex is ts j (1 ≤ j ≤ m ) such that H ⊆ F j . But then, since E j is complementary to F j , it follows tha t E j ∩ H = ∅ . Thu s, H cannot b e a hitting set as E j is not hit. F or the only-if case, suppo se that ( V , H ) is co nnected, but H is not a hitting set of ( {{ u, v } | u, v ∈ V ∧ u 6 = v } , S ). Then there exists j suc h that E j is no t hit by H and thus H ⊆ F j . Y et in such a case, b y our ass umption, ( V , F j ) is not connected and thus ( V , H ) ca nnot be connected as well. Part 5: Let S ′ = S \ E j , ( {{ u, v } | u, v ∈ V ∧ u 6 = v } , S ′ ) b e an insta nce of Min-HS . The n we claim that F j is a hitting set of ( {{ u, v } | u , v ∈ V ∧ u 6 = v } , S ′ ). First, observe that F j 6 = ∅ since E j cannot co nt ain a ll the edges. Moreover, there exists e ∈ E i ( E i ∈ S ′ ) such that e / ∈ E j . Then e ∈ {{ u, v } | u, v ∈ V ∧ u 6 = v } \ E j = F j and thus F j is a hitting set. Howev er, by the definition of F j , ( V , F j ) cannot be co nnected and thus, the if cas e of part 4 do es not ho ld. ✷ Now we ar e rea dy to present a simple one-to-one r eduction of Min-TCO with max t ∈ T | U t | ≤ d to O ( d 2 ) -HS . The core co nc e pt is to construct a system of sets that has to b e hit in O ( d 2 ) -HS as a union ov er all the topics of the character is tic systems of edges on the vertices interested in the topic. Theorem 3. Ther e exists a one-to-one r e duction of instanc es of Min-TCO with max t ∈ T | U t | ≤ d to instanc e of O ( d 2 ) -HS . Proof. Let I TCO = ( U, T , INT) b e an instance of Min- TCO with max t ∈ T | U t | ≤ d . F or ea ch topic t ∈ T w e define S t to be the characteristic system o f e dges on vertices in U t . Note tha t Lemma 1 holds for ea ch S t with n := d . W e co nstruct 9 an O ( d 2 ) -HS instance I HS = ( X , S ) as follows: X = {{ u , v } | u, v ∈ U ∧ u 6 = v } S = [ t ∈ T S t . The system c o nt ains  | U | 2  elements and at mo s t | T | ·  2 d − 1 − 1  sets in S and th us has a size p olyno mial in | I TCO | . O b viously , the constructio n of I HS takes time p olynomial in | I TCO | , to o. W e now show that a feasible solutio n of I TCO corres p o nds to a feasible so lution of I HS and vice versa. First, consider a feasible solution S ol HS of I HS and a to pic t ∈ T . Due to our c onstruction, the system S co nt ains the characteristic system S t on vertices U t . Therefor e, by Lemma 1 part 4 and the fact that S ol HS is a hitting set, we know that the gra ph induced by the edges in S ol HS on vertices U t is connected. Now, co nsider a feasible so lution S ol TCO of I TCO . By th e following argu- men t, we can ea sily see that S ol TCO hits a ll the sets in S . Let P ∈ S be a s et that is not hit by S ol TCO . Then there exists t such that P ∈ S t and thus a set o f the characteristic system was not hit and S ol TCO is not a hitting set of S t . Y et in such a ca se, consider ing Lemma 1 par t 4, the s ubg raph induced on vertices U t by edges from S ol TCO cannot be connected and that is in contradiction with the definition of Min -TCO with max t ∈ T | U t | ≤ d . ✷ Theorem 4. Ther e exists a p olynomial -time ( ⌊ d/ 2 ⌋ · ⌈ d/ 2 ⌉ ) -appr oximatio n al- gorithm for Min-T CO with max t ∈ T | U t | ≤ d . Proof. W e employ the reduction from Theore m 3 together with the well-kno wn d -approximation algo rithm fo r d -HS . Since the size of each set in S is at most ⌊ d/ 2 ⌋ · ⌈ d/ 2 ⌉ (Lemma 1 part 2), b y application o f this appr oximation algor ithm on O ( d 2 ) -HS instance ( X , S ) we o btain a ⌊ d/ 2 ⌋ · ⌈ d/ 2 ⌉ approximate solution of our Min-T CO instance with ma x t ∈ T | U t | ≤ d . Note tha t our r eduction is tight in the size of S as it is minimal (Lemma 1 part 5), thus to achiev e an improv emen t in the approximation a lgorithm, a different metho d ha s to be developed. ✷ Corollary 4 . Min-TCO with max t ∈ T | U t | ≤ 3 inherits the appr oximation har d- ness of Min-VC . Corollary 5 . Min-TCO with max t ∈ T | U t | ≤ d is AP X -c ompl ete, for arbitr ary d ≥ 3 . Proof. The AP X - hardness follows from the AP X -hardness o f d -HS ([21]). Due to our reduction the pr oblem be longs to the class AP X . ✷ 3.3. Min- TCO and Par ametrize d Complexity The ory In this subsec tio n, we shortly summarize the consequences of our reduction from Theore m 3 for the field of pa r ametrized complexity , na mely we present an 10 exact a lgorithm and a kernelization fo r Min-TCO with max t ∈ T | U t | bo unded by a constant. In the resear ch ar ea of ex act algo rithm design, one searches for an exact solution in exp onential time. The main goal is to make the base of the exp o- nent iation as small as p os sible. A k ernelization is a pro cess in which an instanc e is r educed to a smaller instance in p o lynomial time. Then, instead of solv ing the or iginal instance, it is sufficient to solve the problem on the smaller one and then, in p oly no mial time, transform its solutio n ba ck to the initial insta nce. Problem 2. Min- d -TCO ( k ) is the following pa r ametrized problem: Input: Instance of Min -TCO with max t ∈ T | U t | ≤ d and a para meter k . Goal: A feasible so lution of the Min-TCO instanc e o f size at most k . Problem 3. d -HS( k ) is the following par ametrized problem: Input: Instance of d -HS and a para meter k . Goal: A feasible so lution of the d -HS instance o f size at mos t k . W e first transform the given instance o f M in-TCO with max t ∈ T | U t | ≤ d into an instance of O ( d 2 ) -HS as in Theore m 3 and then we apply the kernelization from [16] to obtain a kernel of Min- d -TCO( k ) or the ex act algo rithm from [19] to obtain the first nontrivial exact alg orithm for solving Min- d -TCO( k ). Theorem 5. Min- d -TCO( k ) has a kernel of size (2 c − 1) · k c − 1 + k with c = ⌊ d/ 2 ⌋ · ⌈ d/ 2 ⌉ . Theorem 6. Min- d -TCO( k ) on n vertic es c an b e solve d in time O ( c k + n 2 ) with c = ⌊ d/ 2 ⌋ · ⌈ d/ 2 ⌉ − 1 + O ( d − 2 ) . 4. Hardness of M in-TCO W he n the Num ber of Connections of a User is Constan t It is natural to co ns ider Min-TCO with b ounded num ber o f connections p er user, i. e., to bo und max u ∈ U | INT( u ) | , since the num ber of topics in which o ne user is interested in is usually no t to o large . W e show that, sadly , Min-TCO is AP X -hard even if max u ∈ U | INT( u ) | ≤ 6. T o show this, we desig n a reduction from minimum vertex c ove r ( M in-VC ) to Min- TCO . The minimum vertex cover problem is just a different name for d -HS with d = 2. F or a b etter prese ntation, in this section, we refer to Min-VC instea d of 2-HS . Given is a graph G = ( V ′ , E ′ ) a nd a po sitive integer k as an instance of Min- V C , where the goal is to decide whether the given g raph ha s a solutio n of size at most k . W e construct a n instance of Min- TCO as follows. Let V = V (1) ∪ V (2) be the set of users, where V (1) = { v (1) | v ∈ V ′ } a nd V (2) = { v (2) | v ∈ V ′ } . F or 11 each edg e e ∈ E ′ , w e prepar e three topics, t (0) e , t (1) e and t (2) e . The s et of topics is the union of all these topics, i. e ., T = S e ∈ E ′ { t (0) e , t (1) e , t (2) e } . The user interest function INT is defined a s INT( u (1) ) = [ e ∈ E ′ [ N [ u ]] { t (0) e , t (1) e } INT( u (2) ) = [ e ∈ E ′ [ N [ u ]] { t (0) e , t (2) e } . The following lemma s hows the r elation b etw een the s olutions o f the tw o pr o b- lems. Lemma 2 . The instanc e ( V , T , INT) of Min- TCO define d as ab o ve has an opti- mal solution of c ost k + 2 | E ′ | if and only if t he instanc e ( V ′ , E ′ ) of Min -V C has an optimal solution of c ost k . Mor e over, any fe asible solut ion H of ( V , T , INT) c an b e tr ansforme d into a fe a- sible solut ion of ( V ′ , E ′ ) of c ost at m ost | H | − 2 | E ′ | . Proof. It is obvious that any feas ible so lution H of the instance o f Min -TCO contains the edge { u ( i ) , v ( i ) } ( i ∈ { 1 , 2 } ), for every edg e e = { u, v } , b ecause only u (1) and v (1) (resp., u (2) and v (2) ) are interested in topic t (1) e (resp., t (2) e ). Since each fea sible s olution H of ( V , T , INT) must co nt ain the edg e s { u (1) , v (1) } and { u (2) , v (2) } , for every edge e = { u, v } ∈ E ′ , it is sufficient to consider only the to pics t (0) e . The num ber o f edges in H connecting a vertex fr om V (1) with a vertex from V (2) is at most | H | − 2 | E ′ | . F or a n edge e = { u, v } , the vertices that a re interested in t (0) e are u (1) , v (2) , v (1) and u (2) . Since these four vertices hav e to b e connected, H contains at le a st one edge of { u (1) , u (2) } , { v (1) , v (2) } , { u (1) , v (2) } and { v (1) , u (2) } . The optimal solution of ( V , T , INT) contains a t most tw o of these four e dges, namely the edges { u (1) , u (2) } and { v (1) , v (2) } . Observe tha t, for e ach edge f that is incident with vertex u in G , the edg e { u (1) , u (2) } connects the solution to b e t (0) f -connected. The only to pic that the other tw o edg es connect is t (0) { u,v } and th us they can b e r e placed by { u (1) , u (2) } or { v (1) , v (2) } . In any non-o ptimal solution, mor e than tw o o f the four edges may b e present and the replacement of edg e s { u (1) , v (2) } and { v (1) , u (2) } by { u (1) , u (2) } and { v (1) , v (2) } , resp ectively , may lead to a decr ease of the cost o f the solution. W e assume that the so lution of ( V , T , INT) has be e n tra nsformed so that it do es not contain cr o ss edges b etw een u ( i ) and v (3 − i ) ( i ∈ { 1 , 2 } ). The vertices that co rresp ond to the edges b et ween the tw o lay ers V (1) and V (2) form a feasible so lutio n of Min- V C . As discuss e d ab ov e, its s ize is a t most | H | − 2 | E ′ | for a feasible solution H and exactly | H | − 2 | E ′ | for a n optimal solution H . This prov es one implicatio n o f the first cla im a nd the seco nd cla im. W e show that, if Min-VC has an optimal solution of size k , then the instanc e of Min-T CO ha s an optimal so lution of s iz e k + 2 | E ′ | . F rom an optimal solution W ⊆ V ′ of Min-V C , we cons truct the optimal solution o f Min- TCO a s H = 12 {{ u (1) , v (1) } , { u (2) , v (2) } | { u, v } ∈ E ′ } ∪ {{ u (1) , u (2) } | u ∈ W } . Cle a rly , the size of H is e x actly k + 2 | E ′ | . As W is the smallest s et of vertices that cov ers all the edges o f E ′ , its cor resp onding edges of Min-T CO pro duce the minimal set o f edges that co nnect every topic with sup erscr ipt 0. Th us, H sa tisfies the connectivity requir ement for every topic t ∈ T and is optimal. ✷ W e use the Min-VC on degr ee-b ounded gra phs, w hich is AP X - hard, to show low er b ounds for our restricted Min-TCO . By the above reduction and the lemma, we prove the following theo rem. Theorem 7. Min-T CO with max v ∈ U | INT( v ) | ≤ 6 c annot b e appr oximate d within a factor of 694 / 69 3 in p oly nomial time, u n less P = N P , even if | INT( v ) ∩ INT( u ) | ≤ 3 holds for every p air of differ ent users u, v ∈ U . Proof. W e pro ve the statemen t b y contradiction. Supp ose that there exists an approximation algo rithm A for Mi n-TCO with the ab ov e stated restrictions that has the r atio (1 + δ ). Let G = ( V ′ , E ′ ) b e an insta nce of Mi n-VC and let G b e cubic and regular (i. e., ea ch vertex is incident with exa ctly three edg es). W e constr uct an insta nce I TCO of M in-TCO a s stated ab ov e and we apply our a lgorithm A to it to obtain a feasible solution S ol TCO . F rom such a so lution, by L e mma 2, we cr eate a feasible solution of the or iginal Min-V C instance S ol VC . W e denote b y Opt TCO and Opt VC the optimal solutions o f I TCO and G , r e s pec tiv ely . Let d b e a consta nt such that d · cost ( O pt VC ) = 3 | V ′ | . Since G is cubic a nd regular , cost ( O pt VC ) ≥ | E ′ | / 3 = | V ′ | / 2 and thus d ≤ 6. Observe that, due to Lemma 2, cost ( Opt TCO ) = cost ( Opt VC ) + 2 | E ′ | = cost ( Opt VC ) + d · cost ( O pt VC ) and cost ( S ol TCO ) ≥ cost ( S ol VC ) + 2 | E ′ | = cost ( S ol VC ) + d · cost ( O pt VC ). These tw o estimations give us the following bo und cost ( S ol VC ) + d · cost ( O pt VC ) cost ( Opt VC ) + d · cost ( O pt VC ) ≤ cost ( S ol TCO ) cost ( Opt TCO ) ≤ 1 + δ. The abov e inequality allows us to b ound the ratio of our Min-VC solution S ol VC and the optimal s olution Op t VC : cost ( S ol VC ) cost ( Opt VC ) ≤ (1 + δ ) · ( d + 1) − d = 1 + δ ( d + 1) ≤ 1 + 7 δ. F or δ := 1 693 , w e obtain a 100 99 -approximation algorithm for Min-VC on 3- regular gra phs which is directly in co nt radiction with a theo rem prov en in [8]. ✷ Corollary 6 . Min-TCO with max v ∈ U | INT( v ) | ≤ 6 is AP X -har d. Corollary 7 . Min-TCO with | INT( v ) ∩ INT( u ) | ≤ 3 , for al l u s ers u, v ∈ U , is AP X -har d. 13 This res ult is almost tight, the case when | INT( v ) ∩ INT( u ) | ≤ 2 is still ope n. The follo wing theor em sho ws that Min-T CO with | INT( v ) ∩ INT( u ) | ≤ 1, for every pair of distinct users u, v ∈ U , can b e solved in linear time. Theorem 8. Min-T CO c an b e solve d in line ar time, if | INT( v ) ∩ INT( u ) | ≤ 1 holds for every p air of u sers u, v ∈ U , u 6 = v . Proof. W e execute the following simple a lgorithm. First set the s o lution E := ∅ . Then sequen tially , for each topic t , choos e its represen tative v ∗ ∈ U ( t ∈ INT( v ∗ )) and add edges {{ v ∗ , u } | u ∈ U t \ { v ∗ }} to the solution E . W e show that, if | INT( v ) ∩ INT( u ) | ≤ 1, for all distinct u , v ∈ U , then the solution E is optimal. Observe tha t, in our cas e, any edge in any fea s ible so lution is present beca use of a unique topic. W e cannot find an edge e = { u, v } of the s o lution tha t b elo ng s to the subgraphs for tw o different topic s . (Otherwise | INT( v ) ∩ INT ( u ) | > 1 and our assumption w ould be wrong for the tw o endp oints of the edg e e .) T hus, any solution c o nsisting of spa nning trees for every topic is feasible and optimal. Note that its size is | T | · ( | U | − 1 ). ✷ Corollary 8 . Min-TCO with ma x u ∈ U | INT( u ) | ≤ 2 c an b e solve d in line ar time. 5. A P olynomial-Time Algorithm for Min- TCO wi th Bounded Number of T opics In this section, we present a simple br ute-force a lgorithm tha t achiev es a p oly nomial running time when the num ber of topics is b ounded by | T | ≤ log log | U | − 3 2 log log log | U | . Theorem 9. The optimal solution of Min-TCO c an b e c omput e d in p oly nomial time if | T | ≤ (1 + ε ( | U | )) − 1 · log log | U | , for a function ε ( n ) ≥ 3 / 2 lo g log log n log log n − 3 / 2 lo g log log n . Proof. Let ( U, T , INT) b e an instance of Min-T CO such that | T | ≤ (1 + ε ( | U | )) − 1 · log log | U | . Moreover, | T | > 2, otherwise the problem is solv a ble in p olynomial time. W e shorten the nota tion b y setting t = | T | a nd n = | U | . First observe that, if u, v ∈ U and INT ( u ) ⊆ INT( v ), ins tead of solving instance ( U, T , INT), we can s o lve Min-TCO on instance ( U \ { u } , T , INT) a nd add to such solution the direct edg e { u, v } . Note that u ha s to be incident with at least one edg e in any solution. Th us, the addition of the edge { u, v } cannot increase th e cost. Mo reov er, a n y other user that w ould b e connected to u in some solution ca n be also connected to v . Thus, w e can r emov e u , solve the smaller insta nce and then add u b y a single edge. Such a solution is feasible and its size is unchanged. W e say that vertex u is dominate d by the vertex v if INT( u ) ⊆ INT( v ). 14 Therefore, b efore applying our simple a lgorithm, we remove from the in- stance all the user s that a re dominated by some o ther user . W e denote the set of remaining users (i. e., tho se with incompara ble sets of interesting topics) by M . The larges t sys tem of incomparable sets o n n elements is called a Sp erner system and it is a well known fact that its size is at most  n ⌊ n/ 2 ⌋  . Since every user in M must ha v e differen t set of interesting topics and these sets are all incomparable, we hav e m = | M | ≤  t ⌊ t/ 2 ⌋  ≤ 2 t √ t . (T o verify the b o und, consider t to b e o dd or even and use  2 n n  ≤ 4 n √ 3 n +1 , n ≥ 1 .) Our simple algorithm exhaus tiv ely sear ch es over all the po ssible solutions on instance ( M , T , INT) and then rec onnects each of the remov ed us e rs U \ M by a single edge. The transfor mation to s et M and the connectio n o f the removed users is clearly p olyno mial. Thus, w e o nly need to show that our exhaustive search is p olynomial. Observe tha t the size of the optimal solution is at mo st t ( m − 1), a s merged spanning trees , for all the topics, form a feasible solution. Our algorithm ex- haustively searches ov er all p ossible solutions, i. e., it tries every p oss ible set of i edges fo r 1 ≤ i ≤ t ( m − 1) and verifies the topic-c o nnectivity re q uirements for such sets of edges. The verification of e a ch set can be done in p oly no mial time. The n um b e r of sets it chec ks can b e b ounded as follows: t ( m − 1) X i =1   m 2  i  ≤ tm X i =1  m 2 i  ≤ tm ·  m 2 tm  ≤ tm · m tm ≤ m tm · O (log 2 n ) (Note that tm ≤ m 2 / 2 and th us the binomial co efficient is ma x imal in tm . Oth- erwise the num ber of all p oss ible choices of edges int o a so lution is p olynomial in n .) T o c hec k a po lynomial n um ber of sets, it is sufficient to b ound the factor m tm by a po lynomial, i. e., b y at most n c for so me c > 0 . (In all our calculations, log stands for the binar y loga rithm, howev er any o ther lo garithm can b e used as the change will effect the exp onent by a consta nt.) W e consider tw o c a ses: A: First a ssume that t ≤ log log n 1+2 ε ( n ) , then m ≤ 2 t ≤ (log n ) (1+2 ε ( n )) − 1 . W e use the upp er bo unds on t a nd m to estimate the num ber of sets our exhaustive sea rch has to chec k: m tm ≤ (log n ) (1+2 ε ( n )) − 2 · log log n · (log n ) (1+2 ε ( n )) − 1 ≤ n c . 15 Then we take the logar ithm o f the inequality , leading to (1 + 2 ε ( n )) − 2 · log log 2 n ≤ c · (log n ) 2 ε ( n ) 1+2 ε ( n ) . After another lo g arithm o pe ration, we obtain the following inequa lit y: − 2 log(1 + 2 ε ( n )) + 2 lo g log log n ≤ 2 ε ( n ) 1 + 2 ε ( n ) · log log n + log c. W e prov e inequality (1) instead. In the end, we will see that the function ε ( n ) is p ositive, except for the first few v a lues. Th us, for large inputs, 2 log(1 + 2 ε ( n )) is p ositive a nd thus the ab ove inequalities will ho ld, too . 2 log lo g log n ≤ 2 ε ( n ) 1 + 2 ε ( n ) · log log n (1) W e are now able to estimate the function ε ( n ): ε ( n ) ≥ log log log n log log n − 2 lo g log log n . (2) Due to the following case , we use ε ( n ) ≥ 3 / 2 log log log n log log n − 3 / 2 log l og log n that als o satisfies (2) a nd is p ositive for n ≥ 16. B: T o conclude the pro o f, assume that log log n 1+2 ε ( n ) < t ≤ log log n 1+ ε ( n ) . Since we hav e bo th an upp er and a lower b ound on t , we can refine the estimation o f m : m ≤ 2 t √ t ≤ (log n ) (1+ ε ( n )) − 1 · (1 + 2 ε ( n )) 1 / 2 · (log log n ) − 1 / 2 . W e show that m tm is po lynomial in n similar ly as in the pr evious case: (log n ) (1+ ε ( n )) − 2 · (log log n ) 1 / 2 · (log n ) (1+ ε ( n )) − 1 · (1+2 ε ( n )) 1 / 2 ≤ m tm ≤ n c . Then we take the logar ithm o f the inequality , leading to (1 + ε ( n )) − 2 · (log log n ) 3 / 2 · (1 + 2 ε ( n )) 1 / 2 ≤ c · (log n ) ε ( n ) 1+ ε ( n ) . Assume that (1 + 2 ε ( n )) 1 / 2 ≤ 1 + ε ( n ), except for the first few v a lues, then it is sufficient to prove a simpler inequality: (1 + ε ( n )) − 1 · (log log n ) 3 / 2 ≤ c · (log n ) ε ( n ) 1+ ε ( n ) . After another lo g arithm o pe ration, we obtain the following inequa lit y: − log(1 + ε ( n )) + 3 / 2 · log log log n ≤ ε ( n ) 1 + ε ( n ) · log log n + log c. 16 Again, assuming tha t log (1 + ε ( n )) > 0 if n tends to infinity , to prove the ab ov e inequa lit y , it is sufficient to show that 3 / 2 lo g log log n ≤ ε ( n ) 1 + ε ( n ) · log log n. Thu s we a re a ble to b ound the function ε ( n ) as ε ( n ) ≥ 3 / 2 lo g log log n log log n − 3 / 2 lo g log log n . Observe that ε ( n ) > 0 for n ≥ 16, thus b oth assumptions that we made hold for | U | ≥ 1 6 which co ncludes the pro of. ✷ 6. Conclusion In this pap er , we hav e closed the gap in the approximation hardness o f Min-TCO by showing its LO G AP X -completeness. W e studied a subproblem of Min-TCO where the num ber of us e rs interested in a commo n topic is b ounded by a consta n t d . W e show ed that, if d ≤ 2, the restricted Min-TCO is in P and, if d ≥ 3, it is AP X -co mplete. The latter result, tog ether with the constant approximation algorithm we pre s ent ed, allows us to prov e low er b ounds on approximabilit y o f these sp ecial instances that match any low er b ound kno wn for any pro blem fr om the class AP X . F urthermore, we studied instances o f Min- TCO where the num ber of topics in which a single user is interested in is b ounded by a consta n t d . W e prese n ted a reduction that shows that such instances a r e AP X -hard for d = 6 . In this reduction, a n y tw o user s hav e a t most three common topics, thus the r eduction shows also that Min-TCO res tricted in this wa y is AP X -hard. 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