Tight Approximation Bounds for Vertex Cover on Dense k-partite Hypergraphs

Transcription

1 Tight Approximatio Bouds for Vertex Cover o Dese -Partite Hypergraphs Mare Karpisi Richard Schmied Claus Viehma arxiv: v [cs.ds] Jul 20 Abstract We establish almost tight upper ad lower approximatio bouds for the Vertex Cover problem o dese -partite hypergraphs. Itroductio A hypergraph H (V,E) cosists of a vertex set V ad a collectio of hyperedges E where a hyperedge is a subset of V. H is called -uiform if every edge i E cotais exactly vertices. A subset C of V is a vertex cover of H if every edge e E cotais at least a vertex of C. The Vertex Cover problem i a -uiform hypergraph H is the problem of computig a miimum cardiality vertex cover ih. It is well ow that the problem isnp -hard eve for 2 (cf. [3]). O the other had, the simple greedy heuristic which chooses a maximal set of oitersectig edges, ad the outputs all vertices i those edges, gives a -approximatio algorithm for the Vertex Cover problem restricted to -uiform hypergraphs. The best ow approximatio algorithm achieves a slightly better approximatio ratio of ( o()) ad is due to Halperi []. O the itractability side, Trevisa [22] provided oe of the first iapproximability results for the -uiform vertex cover problem ad obtaied a iapproximability factor of 9 assumigp NP. I 2002, Holmeri [] improved the factor to ǫ. Diur et al. [7, 8] gave cosecutively two lower bouds, first ( 3 ǫ) ad later o ( ǫ). Moreover, assumig Khot s Uique Games Cojecture (UGC) [7], Khot ad Regev [8] proved a iapproximability factor of ǫ for the Vertex Cover problem o -uiform hypergraphs. Therefore, it implies that the curretly achieved ratios are the best possible. Dept. of Computer Sciece ad the Hausdorff Ceter for Mathematics, Uiversity of Bo. Supported i part by DFG grats ad the Hausdorff Ceter grat EXC mare@cs.ui-bo.de Dept. of Computer Sciece, Uiversity of Bo. Wor supported by Hausdorff Doctoral Fellowship. schmied@cs.ui-bo.de Dept. of Computer Sciece, Uiversity of Bo. Wor partially supported by Hausdorff Ceter for Mathematics, Bo. viehma@cs.ui-bo.de

2 The Vertex Cover problem restricted to -partite -uiform hypergraphs, whe the uderlyig partitio is give, was studied by Lovász [20] who achieved a - 2 approximatio. This approximatio upper boud is obtaied by roudig the atural LP relaxatio of the problem. The above boud o the itegrality gap was show to be tight i []. As for the lower bouds, Guruswami ad Saet [0] proved that it is NP-hard to approximate the Vertex Cover problem o -partite -uiform hypergraphs to withi a factor of ǫ for 5. Assumig the Uique 4 Games Cojecture, they also provided a iapproximability factor of ǫ for 2 3. More recetly, Sachdeva ad Saet [2] claimed a early optimal NP - hardess factor. To gai better isights o lower bouds, dese istaces of may optimizatio problems has bee itesively studied [2, 5, 6, 4]. The Vertex Cover problem has bee ivestigated i the case of dese graphs, where the umber of edges is withi a costat factor of 2, by Karpisi ad Zeliovsy [6], Eremeev [9], Clemeti ad Trevisa [6], later by Bar-Yehuda ad Kehat [4] as well as Imamura ad Iwama [2]. The Vertex Cover problem restricted to dese balaced -partite -uiform hypergraphs was itroduced ad studied i [5], where it was proved that this restricted versio of the problem admits a approximatio ratio better tha if the 2 give hypergraph is dese eough. I this paper, we give a ew approximatio algorithm for the Vertex Cover problem restricted to dese -partite -uiform hypergraphs ad prove that the achieved approximatio ratio is almost tight assumig the Uique Games Cojecture. 2 Defiitios ad Notatios Give a atural umber i N, we itroduce for otatioal simplicity the set [i] {,..,i} ad set [0]. Let S be a fiite set with cardiality s ad [s]. We will use the abbreviatio ( S ) {S S S }. A-uiform hypergraphh (V(H),E(H)) cosists of a set of verticesv ad a collectio E ( V ) of edges. For a -uiform hypergraph ( H ad a vertex v V(H), ) we defie the eighborhood N H (v) of v by e {e E v e} e \ {v} ad the degree d H (v) of v to be {e E v e}. We exted this otio to subsets ofv(h), where S V(H) obtais the degree d H (S) by {e E S e}. A -partite -uiform hypergraph H (V,..,V,E(H)) is a -uiform hypergraph such that V is a disjoit uio of V,..,V with V i e for every e E ad i []. I the remaider, we assume that V i V i+ for all i [ ] ad O(). A balaced -partite -uiform hypergraph H (V,..,V,E(H)) is a -partite - uiform hypergraph with V i V for all i []. We set V ad m E as usual. For a -partite -uiform hypergraph H (V,..,V,E(H)) ad v V, we itroduce the v-iduced hypergraph H(v), where the edge set of H(v) is defied by 2

3 {e\{v} v e E(H)} ad the vertex set of H(v) is partitioed ito V i N H (v) with i [ ]. A vertex cover of a -uiform hypergraph H (V(H),E(H)) is a subset C of V(H) with the property that e C holds for all e E(H). The Vertex Cover problem cosists of fidig a vertex cover of miimum size i a give -uiform hypergraph. The Vertex Cover problem i -partite -uiform hypergraphs is the restricted problem, where a -partite -uiform hypergraph ad its vertex partitio is give as a part of the iput. We defie a -partite -uiform hypergraph H (V,..,V,E(H)) as ǫ-dese for a ǫ [0,] if the followig coditio holds: E(H) ǫ i [] V i For l [ ], we itroduce the otio of l-wise ǫ-dese -partite -uiform hypergraphs. Give a -partite-uiform hypergraphh, if there exists ai ( ) [] l ad a ǫ [0,] such that for all S with the property V i S for all i I the coditio d H (S) ǫ V i holds, we defie H to be l-wise ǫ-dese. i []\I 3 Our Results I this paper, we give a improved approximatio upper boud for the Vertex Cover problem restricted to ǫ-dese -partite -uiform hypergraphs. The approximatio algorithm i [5] yields a approximatio ratio of ( 2)( ǫ) l for l-wise ǫ-dese balaced -partite -uiform hypergraphs. Here, we desig a algorithm with a approximatio factor of 2+( 2)ǫ for the ǫ-dese case which also improves o the l-wise ǫ-dese balaced case for alll [ 2] ad matches their boud whel. A further advatage of this algorithm is that it applies to a larger class of hypergraphs sice the cosidered hypergraph is ot ecessarily required to be balaced. As a byproduct, we obtai a costructive proof that a vertex cover of a ǫ-dese -partite -uiform hypergraph H (V,..,V,E(H)) is bouded from below by ǫ V, which is show to be sharp by costructig a family of tight examples. O the other had, we provide iapproximability results for the Vertex Cover problem restricted to l-wise ǫ-dese balaced -partite -uiform hypergraphs uder the Uique Games Cojecture. We also prove that this reductio yields a 3

4 matchig lower boud if we use a cojecture o the Uique Games hardess of the Vertex Cover problem restricted to balaced -partite -uiform hypergraphs. This meas that further restrictios such as l-wise desity caot lead to improved approximatio ratios ad our proposed approximatio algorithm is best possible assumig this cojecture. I additio, we are able to prove a iapproximability factor uder P NP. 4 Approximatio Algorithm I this sectio, we give a polyomial time approximatio algorithm with improved approximatio factor for the Vertex Cover problem restricted to ǫ-dese -partite -uiform hypergraphs. We state ow our mai result. Theorem. There exists a polyomial time approximatio algorithm with approximatio ratio 2+( 2)ǫ for the Vertex Cover problem i ǫ-dese -partite -uiform hypergraphs. A crucial igrediet of the proof of Theorem is Lemma, i which we show that we ca extract efficietly a large part of a optimal vertex cover of a give ǫ-dese -partite -uiform hypergraph H (V,..,V,E(H)). More precisely, we obtai i this way a costructive proof that the size of a vertex cover of H is bouded from below by ǫ V. The procedure for the extractio of a part of a optimal vertex cover is give i Figure. We ow formulate Lemma : Lemma. Let H (V,..,V,E(H)) be a ǫ-dese -partite -uiform hypergraph with. The, the procedure Extract( ) computes i polyomial time a collectio R of subsets of V(H) such that the size of R is polyomial i V(H) ad R cotais a sets, which is a subset of a optimal vertex cover ofh ad its cardiality is at least ǫ V. As a cosequece, we obtai directly: Corollary. Give aǫ-dese-partite-uiform hypergraphh (V,..,V,E(H)) with, the cardiality of a optimal vertex cover of H is bouded from below by ǫ V. Before we prove Lemma, we describe the mai idea of the proof. Let OPT deote a optimal vertex cover of H. The procedure Extract( ) tests for the set R {v,..,v p } of the p heaviest vertices of V, if {v,..,v u } OPT ad v u OPT for every u [p]. Clearly, either R OPT or there exists a v u such that v u OPT. If the procedure already possesses a part of OPT deoted by R u, the, 4

5 Procedure Extract( ) Iput: ǫ-dese -partite -uiform hypergraph H (V,..,V,E) with. IF THEN (a) RETURN { e E e} 2. ELSE: (a) Let (v,..,v p ) be the vector cosistig of the first p heaviest vertices of V with d H (v i ) d H (v i+ ) (b) R {{v,..,v p }} (c) FOR i,..,p DO: 3. RETURN R i. R i {v [i ]} ii. Ivoe Extract(H(v i )) with output O iii. R R {R i S S O} E l [ ] V l Figure : Procedure Extract Extract( ) tries to obtai a large part of a optimal vertex cover of the v u -iduced hypergraphh(v u ). Hece, we have to show thath(v u ) must still be dese eough. We ow give the proof of Lemma. Proof. The proof of Lemma will be split i several parts. I particular, we show that give a ǫ-dese -partite -uiform hypergraph H (V,..,V,E(H)), the procedure Extract( ) ad its output R possess the followig properties:. Extract( ) costructs R i polyomial time ad the cardiality ofr is O( ). 2. There is a S R such that S is a subset of a optimal vertex cover of H. 3. For every S R, the cardiality of S is at least S ǫ V. (.) Clearly, R is upper bouded by V O( ) ad therefore, the ruig time of Extract( ) is O( ). (2.) ad (3.) We prove the remaiig properties by iductio. If we have, the set e E(H) e is by defiitio a optimal vertex cover of H (V,E(H)). Sice H is ǫ-dese, the cardiality of E(H) is lower bouded by ǫ V. We assume that >. Let H (V,..,V,E(H)) be a ǫ-dese -partite -uiform hypergraph ad OPT V(H) a optimal vertex cover of H. Let (v,..,v p ) be the vector cosistig of the first p E(H) l [ ] V l heaviest vertices of V with d H (v i ) d H (v i+ ). If {v,..,v p } is cotaied i OPT, we have costructed a subset of a 5

6 optimal vertex cover with cardiality p E(H) V l l [ ] ǫ V l l [] l [ ] V l ǫ V. Otherwise, there is a u [p] such that R u OPT ad v u OPT. But this meas that a optimal vertex cover of H cotais a optimal vertex cover of the v u -iduced ( )-partite ( )-uiform hypergraph H(v u ) i order to cover the edges e {e E v u e}. The situatio is depicted i Figure 2. v u -iduced Hypergraph H(v u ) v v 2 v u v u V V V 2 V Figure 2: The v u -iduced ( )-partite ( )-uiform hypergraph H(v u ) By our iductio hypothesis,extract(h(v u )) cotais a sets u which is a subset of a miimum vertex cover of H(v u ) ad of OPT. The oly claim, which remais to be prove, is that the cardiality ofs u is large eough. More precisely, we show that S u ca be lower bouded by ǫ V R u. Therefore, we eed to aalyze the desity of the v u -iduced hypergraph H(v u ). The edge set of H(v u ) is give by {e\{v u } v u e E}. Thus, we have to obtai a lower boud o the degree of v u. Sice {e E e R u } is upper bouded by R u l [ ] V l, the vertices i V \R u possess the average degree of at least deg H (v) ǫ V l {e E e R u } v V \R u l [] () V \R u V \R u ǫ V l R u V l l [] V \R u (ǫ V R u ) 6 V \R u l [ ] l [ ] V l (2) (3)

7 Sice the heaviest vertex i V \ R u must have a degree of at least (ǫ V R u ) l [ ] V l V \R u, we deduce that the edge set of H(v u ) deoted by E u ca be lower bouded by (ǫ V R u ) V l E u V \R u l [ ] Let H(v u ) be defied by (V u,..,v u,e u) with Vi u V i for all i [ ]. By our iductio hypothesis, the size of every set cotaied i Extract( ) is at least (ǫ V R u ) V l E u l [ ] V u l V V \R u (ǫ V R u ) V \R u l [ ] l [ ] V u l l [ ] l [ ] V l V l V (4) V (5) (ǫ V R u ) V (6) V \R u (ǫ V R u ) V ǫ V R u (7) V I (4), we used the fact that V u i V i for all i [ ]. Whereas i (5), we used our assumptio V V. All i all, we obtai R u S u R u +( ǫ V R u ) ǫ V. (8) Clearly, this argumetatio o the size of R u S u holds for every u [p] ad the proof of Lemma follows. Before we state our approximatio algorithm ad prove Theorem, we show that the boud i Lemma is tight. I particular, we defie a family of ǫ-dese -partite -uiform hypergraphs H(,l,ǫ) (V,..,V,E(H l )) with V i V for all i [],, ǫ { u u [l]} ad l such that Extract( ) returs a subset l of a optimal vertex cover with cardiality of exactly ǫ V. Lemma 2. The boud of Lemma is tight. Proof. Let us defie H(,p,ǫ) (V,..,V,E). For a fixed p ad, every partitio V i with i [] cosists of a set of l vertices. Let us fix a ǫ u l with u [l]. The, H(,l,ǫ) cotais the set V u V of u vertices such that E {{v,v 2,..,v } v V u,v 2 V 2,..,v V }. A example of such a hypergraph is depicted i Figure 3. Notice that H(,l,ǫ) (V,..,V,E) is ǫ-dese, sice E V j V u V u l ǫ. j [] 7

8 V u V V V 2 V Figure 3: A example of a hypergraph H(,l,ǫ) The procedureextract( ) returs a setr, i whichv u u is cotaied, sice V is the set of the p heaviest vertices of V. Hece, we obtai V u V u V V ǫ V. O the other had, the remaiig hypergraph H (V,..,V \V u,e(h )) with edge set E(H ) {e E e V u } is already covered, sice E(H ) is by defiitio of H(,p,ǫ) the empty set. Therefore, V u is a vertex cover of H(,p,ǫ) ad sice, accordig to Corollary, every vertex cover is bouded from below by ǫ V, V u must be a optimal vertex cover. Next, we state our approximatio algorithm for the Vertex Cover problem i ǫ-dese -partite -uiform hypergraphs defied i Figure 4. The approximatio algorithm combies the procedure Extract( ) to geerate a large eough subset of a optimal vertex cover together with the -approximatio algorithm due to 2 Lovász [20] applied to the remaiig istace. Algorithm Approx( ) Iput: ǫ-dese -partite -uiform hypergraph H (V,..,V,E) with 3. T {V } 2. ivoe procedure Extract(H) with output R 3. for all S R do : (a) H S (V(H)\S,{e E(H) e S }) (b) obtai a ( 2 )-approximate solutio S for H S (c) T T {S S} 4. Retur the smallest set i T Figure 4: Algorithm Approx( ) 8

9 We ow prove Theorem. Proof. Let H (V,..,V,E) be a ǫ-dese -partite -uiform hypergraph. From Lemma, we ow that the procedure Extract( ) returs i polyomial time a collectio C of subsets of V(H) such that there is a set S i C, which is cotaied i a optimal vertex cover of H. Moreover, we ow that the size of S is lower bouded by ǫ V. Next, we aalyze the approximatio ratio of our approximatio algorithm Approx( ). Clearly, the size of a optimal vertex cover of H is upper bouded by V. Let us deote by OPT the size of a optimal vertex cover of the remaiig hypergraph H defied by removig all edges e of H with e S. Furthermore, let S be the solutio of the 2 -approximatio algorithm applied to H. The approximatio ratio of Approx( ) is bouded by S + S S + OPT S + 2 OPT S + OPT S + OPT S + 2 OPT 2 S +( 2) S + OPT S + 2 OPT 2+( 2) 2+( 2) S V 2+( 2) ǫ V V 2+( 2)ǫ S S + 2 OPT (9) (0) () (2) (3) (4) I (), we used the fact that the size of the output of Approx( ) is upper bouded by V. Therefore, we have S + 2 OPT V. I (2), we ow from Lemma that S ǫ V. 5 Iapproximability Results I this sectio, we prove hardess results for the Vertex Cover problem restricted to l-wise ǫ-dese balaced -uiform -partite hypergraphs uder the Uique Games Cojecture [7] as well as uder the assumptio P NP. 5. UGC-Hardess The Uique Games-hardess result of [0] was obtaied by applyig the result of Kumar et al. [9], with a modificatio to the LP itegrality gap due to Ahorai et al. []. More precisely, they proved the followig iapproximability result: 9

10 Theorem 2. [0] For every δ > 0 ad 3, there exist a δ such that give H (V,..,V,E(H)) as a istace of the Vertex Cover problem i balaced -partite -uiform hypergraphs with V(H) δ, the followig is UGC-hard to decide: ( The size of a vertex cover of H is at least V ). δ 2( ) ( The size of a optimal vertex cover of H is at most V ). +δ ( ) As the startig poit of our reductio, we use Theorem 2 ad prove the followig: Theorem 3. For every δ > 0, ǫ (0,), l [ ], ad 3, there exists o polyomial time approximatio algorithm with a approximatio ratio 2+ 2( )( 2)ǫ +( 2)ǫ δ for the Vertex Cover problem i l-wise ǫ-dese -partite -uiform hypergraphs assumig the Uique Games Cojecture. Proof. First, we cocetrate o the ǫ-dese case ad afterwards, we exted the rage of l. As a startig poit of the reductio, we use the -partite -uiform hypergraph H (V,..,V,E(H)) from Theorem 2 ad costruct a ǫ-dese - partite -uiform hypergraph H (V,..,V,E ). Let us start with the descriptio of H ǫ. First, we joi the set C i of vertices ǫ to V i for every i [] ad add all possible edges e of H to E with the restrictio C e. Thus, we obtai V i + ǫ for all i []. ǫ Now, let us aalyze how the size of the optimal solutio of H trasforms. We deote by OPT a optimal vertex cover of H. The UGC-hard decisio questio from Theorem 2 trasforms ito the followig: ( ) 2( ) δ + ǫ ǫ OPT or OPT ( ) ( ) +δ + ǫ ǫ Assumig the UGC, this implies the hardess of approximatig the Vertex Cover problem i ǫ-dese hypergraphs for every δ > 0 to withi: ( ) δ + ǫ ǫ 2( ) ǫ ( ) δ( ǫ)+ ǫ 2( ) +δ + ǫ ǫ +δ( ǫ)+ ǫ (5) ( ) ǫ ( ) ( ǫ) + 2ǫ( ) 2( ) 2( ) ǫ + ǫ( ) ( ) ( ) δ (6) 0

11 ( ǫ) + 2ǫ( ) 2( ) 2( ) ǫ + ǫ( ) ( ) ( ) δ ǫ+2ǫ 2ǫ 2( ) ǫ+ǫ ǫ ( ) δ (7) +( 2)ǫ 2(+( 2)ǫ) δ (8) 2(+( 2)ǫ) +( 2)ǫ 2+2( 2)ǫ+(2 2)( 2)ǫ +( 2)ǫ 2+ (2 2)( 2)ǫ +( 2)ǫ 2+ 2( )( 2)ǫ +( 2)ǫ δ (9) δ (20) δ (2) δ (22) Fially, we have to verify that the costructed hypergraph H is ideed ǫ-dese. Notice that H ca have at most ( V ) ( + ǫ ǫ ) edges. Therefore, we obtai the followig: ( ǫ ǫ )( + ) ǫ ǫ ( + ) ǫ ǫ ǫ ǫ ( + ǫ ǫ ) ǫ ǫ +ǫ ǫ ǫ Notice that the costructed hypergraph is also l-wise ǫ-dese balaced. Hece, we obtai the same iapproximability factor i this case as well. Next, we combie the former costructio with a cojecture about Uique Games hardess of the Vertex Cover problem i balaced -partite -uiform hypergraphs. I particular, we postulate the followig: Cojecture. Give a balaced -partite -uiform hypergraph H (V,..,V,E(H)) with 3, let OPT deote a optimal vertex cover of H. For every δ > 0, the followig is UGC-hard to decide: ( ) ( ) 2 V δ OPT or OPT V +δ 2 Combiig Cojecture with the costructio i Theorem 3, it yields the followig iapproximability result which matches precisely the approximatio upper boud achieved by our approximatio algorithm described i Sectio 4: Theorem 4. For every δ > 0, ǫ (0,), l [ ], ad 3, there exists o polyomial time approximatio algorithm with a approximatio ratio 2+( 2)ǫ δ ǫ

12 for the Vertex Cover problem i l-wise ǫ-dese -partite -uiform hypergraphs assumig Cojecture. Proof. The UGC-hard decisio questio from Cojecture trasforms ito the followig: ( ) δ + ǫ ( ) 2 ǫ OPT or OPT +δ + ǫ 2 ǫ Assumig the UGC, this implies the hardess of approximatig the Vertex Cover problem i ǫ-dese -partite -uiform hypergraphs for every δ > 0 to withi: ( δ) + ǫ ǫ ( 2 2 +δ ) + ǫ ǫ ( δ) ( ǫ)+ ǫ ( 2 2 +δ ) ( ǫ)+ ǫ ( 2 2 ) ( ǫ)+ ǫ (23) 2 δ (24) 2( ǫ)+ǫ δ (25) 2+( 2)ǫ δ (26) 5.2 NP-Hardess Recetly, Sachdeva ad Saet proved i [2] a early optimal NP-hardess of the Vertex Cover problem o balaced -uiform -partite hypergraphs. More precisely, they obtaied the followig iapproximability result: Theorem 5. [2] Give a balaced -partite -uiform hypergraph H (V, E) with 4, let OP T deote a optimal vertex cover of H. For every δ > 0, the followig is NP-hard to decide: ( V 2( +)(2( +)+) δ ( V (2( +)+) +δ ) or ) OPT OPT Combiig our reductio from Theorem 2 with Theorem 5, we prove the followig iapproximability result uder the assumptio P NP : Theorem 6. For everyδ > 0,ǫ (0,),l [ ], ad 4, there is o polyomial time approximatio algorithm with a approximatio ratio 2 ( ǫ)+ǫ2( +)(2( +)+) 2( +)[ ǫ+ǫ(2( +)+)] 2 δ

13 for the Vertex Cover problem i l-wise ǫ-dese -partite -uiform hypergraphs assumig P NP. Proof. The NP-hard decisio questio from Theorem 5 trasforms ito the followig: ( ) 2( +)(2(+)+) δ + ǫ ǫ OPT ( ) or (2( +)+) +δ + ǫ ǫ OPT Assumig NP P, this implies the hardess of approximatig the Vertex Cover problem i ǫ-dese hypergraphs for every δ > 0 to withi: ( ǫ) 2(+)(2(+)+) + ǫ ǫ (2(+)+) + ǫ δ 2 ( ǫ)+ǫ2(+)(2(+)+) 2(+)(2(+)+) ǫ+ǫ(2(+)+) (2(+)+) δ (27) 2 ( ǫ)+ǫ2( +)(2(+)+) 2( +)[ ǫ+ǫ(2( +)+)] δ (28) 6 Further Research A iterestig questio remais about eve tighter lower approximatio bouds for our problem, perhaps coectig it more closely to the itegrality gap issue of the LP of Lovász [20]. Acowledgmet We tha Jea Cardial for may stimulatig discussios. Refereces [] R. Aharoi, R. Holzma, ad M. Krivelevich, O a Theorem of Lovász o Covers i r-partite Hypergraphs, Combiatorica 6, pp , 996. [2] S. Arora, D. Karger, ad M. Karpisi, Polyomial Time Approximatio Schemes for Dese Istaces of NP-Hard Problems, Proc. 27th ACM STOC (995), pp ; see also i J. Comput. Syst. Sci. 58, pp , 999. [3] N. Basal ad S. Khot, Iapproximability of Hypergraph Vertex Cover ad Applicatios to Schedulig Problems, Proc. 37th ICALP (200), LNCS 698, pp ,

14 [4] R. Bar-Yehuda ad Z. Kehat, Approximatig the Dese Set-Cover Problem, J. Comput. Syst. Sci. 69, pp , [5] J. Cardial, M. Karpisi, R. Schmied, ad C. Viehma, Approximability of Vertex Cover i Dese Bipartite Hypergraphs, Research Report CS-8539, Uiversity of Bo, 20. [6] A. Clemeti ad L. Trevisa, Improved No-Approximability Results for Miimum Vertex Cover with Desity Costraits, Theor. Comput. Sci. 225, pp. 3 28, 999. [7] I. Diur, V. Guruswami, ad S. Khot Vertex Cover o -Uiform Hypergraphs is Hard to Approximate withi Factor ( 3 ǫ), ECCC TR02-027, [8] I. Diur, V. Guruswami, S. Khot, ad O. Regev, A New Multilayered PCP ad the Hardess of Hypergraph Vertex Cover, SIAM J. Comput. 34, pp , [9] A. Eremeev, O some Approximatio Algorithms for Dese Vertex Cover Problem, Proc. Symp. o Operatios Research (999), pp [0] V. Guruswami ad R. Saet, O the Iapproximability of Vertex Cover o -Partite -Uiform Hypergraphs, Proc. 37th ICALP (200), LNCS 698, pp , 200. [] E. Halperi, Improved Approximatio Algorithms for the Vertex Cover Problem i Graphs ad Hypergraphs SIAM J. Comput. 3, pp , [2] T. Imamura ad K. Iwama, Approximatig Vertex Cover o Dese Graphs, Proc. 6th ACM-SIAM SODA (2005), pp [3] R. Karp, O the Computatioal Complexity of Combiatorial Problems, Networs 5, pp , 975. [4] M. Karpisi, Polyomial Time Approximatio Schemes for Some Dese Istaces of NP-Hard Optimizatio Problems, Algorithmica 30, pp , 200. [5] M. Karpisi, A. Rucisi, ad E. Szymasa, Computatioal Complexity of the Perfect Matchig Problem i Hypergraphs with Subcritical Desity, It. J. Foud. Comput. Sci. 2, pp , 200. [6] M. Karpisi ad A. Zeliovsy, Approximatig Dese Cases of Coverig Problems, Proc. DIMACS Worshop o Networ Desig: Coectivity ad Facilities Locatio (997), pp ; also published i ECCC TR97-004,

15 [7] S. Khot, O the Power of Uique 2-Prover -Roud Games, Proc. 34th ACM STOC (2002), pp [8] S. Khot ad O. Regev, Vertex Cover Might be Hard to Approximate to withi 2 ǫ, J. Comput. Syst. Sci. 74, pp , [9] A. Kumar, R. Maoara, M. Tulsiai, ad N. Vishoi, O LP-based approximability for strict CSPs, Proc. 22d ACM-SIAM SODA (20), pp [20] L. Lovász, O Miimax Theorems of Combiatorics, Doctoral Thesis, Mathematii Lapo 26, pp , 975. [2] S. Sachdeva ad R. Saet, Nearly Optimal NP-Hardess of Vertex Cover o -Uiform -Partite Hypergraphs, CoRR abs/05.475, 20. [22] L. Trevisa, No-Approximability Results for Optimizatio Problems o Bouded Degree Istaces, Proc. 33rd ACM STOC (200), pp