Hardness of Rainbow Coloring Hypergraphs

Transcription

1 Electronc Colloquum on Computatonal Complexty, Report No 47 07) Hardness of Ranbow Colorng Hypergraphs Venkatesan Guruswam Rsh Saket Abstract A hypergraph s k-ranbow colorable f there exsts a vertex colorng usng k colors such that each hyperedge has all the k colors Unlke usual hypergraph colorng, ranbow colorng becomes harder as the number of colors ncreases Ths work studes the ranbow colorablty of hypergraphs whch are guaranteed to be nearly balanced ranbow colorable Specfcally, we show that for any, k and l k/, gven a k-unform hypergraph whch admts a k-ranbow colorng satsfyng: n each hyperedge e, for some l e l all but l e colors occur exactly tmes and the rest ± ) tmes, t s NP-hard to compute an ndependent set of l+ k +ε)-fracton of vertces, for any constant ε > 0 In partcular, ths mples the hardness of even k/l)-ranbow colorng such hypergraphs The result s based on a novel long code PCP test that ensures the strong balancedness property desred of the k-ranbow colorng n the completeness case The soundness analyss reles on a mxng bound based on unform reverse hypercontractvty due to Mossel, Oleszkewcz, and Sen, whch was also used n earler proofs of the hardness of ω)-colorng -colorable 4-unform hypergraphs due to Saket, and k-ranbow colorable k-unform hypergraphs due to Guruswam and Lee Introducton A hypergraph s a collecton of vertces and subsets of the set of vertces called hyperedges It s q-unform f each hyperedge has exactly q vertces, n partcular a -unform hypergraph s the usual graph An ndependent set of a hypergraph s a subset of vertces that does not contan all the vertces of any hyperedge A fundamental property of a hypergraph s ts colorablty: t s sad to be k-colorable f the set of vertces can be colored usng k colors so that no hyperedge s monochromatc These color classes partton the vertces nto k dsjont ndependent sets The computatonal aspects of colorng graphs and hypergraphs have been the focus of a substantal body of research In bref, t s known that 3-colorable graphs can effcently be colored usng n O) -colors [4] see also [38, 6, 7, 3, 9, ]) Smlar n O) -approxmatons are known for -colorable 3-unform and 4-unform hypergraphs [, 5, 3] From the ntractablty perspectve, on graphs the best lower bound only rules out effcently colorng a 3-colorable graph Computer Scence Department, Carnege Mellon Unversty, Pttsburgh, PA, USA Emal: venkatg@cscmuedu Research supported n part by NSF grant CCF-5609 Ths work was ntated durng a vst by the author to Mcrosoft Research, Bangalore IBM Research, Bangalore, Inda Emal: rssaket@nbmcom ISSN

2 usng 4-colors [6, 6] For hypergraphs much better lower bounds rule out effcently colorng - colorable 8-unform hypergraphs usng explog n) Ω) ) colors [30,, 37] buldng upon the prevous expexpω log log n)) lower bound [, 4] For -colorable 4-unform hypergraphs a correspondng log n) c lower bound s known [36] These ntractablty results proceed by rulng out computng ndependent sets of densty e, fracton of vertces) αn) whch mples the same for /αn))- colorng the hypergraph For -colorable 3-unform hypergraphs however, [3] drectly showed the hardness of colorng usng constantly many colors Another noton of colorng n hypergraphs s that of ranbow colorablty a hypergraph can be k- ranbow colored f there exsts a colorng of the vertces usng k colors ensurng that each hyperedge contans vertces of all the k colors Any k ) of the k color classes therefore consttute an ndependent set, and thus such hypergraphs have at least one ndependent set of densty /k) Note that unlke usual hypergraph colorng, ranbow colorng becomes more restrctve as the number of colors ncreases, and the problem s to determne the largest k for whch there exsts a k-ranbow colorng Ths has been studed by Bollobás et al [8] who gave structural upper and lower bounds for several classes of hypergraphs In some scenaros such as modellng far resource allocatons) t s desrable that the colorng also be balanced, e, the colors should occur roughly the same number of tmes n any hyperedge Ths s related to mnmzng the dscrepancy of hypergraph -colorngs, for whch notable recent works [5, 3] have gven constructve algorthms Subsequent works have shown tght hardness results for zero dscrepancy case n hypergraphs wth unbounded hyperedges [0] and for bounded hypergraphs wth nearly zero dscrepancy [4] A perfectly balanced ranbow colorng s one n whch every color appears exactly the same number of tmes n each hyperedge Such hypergraphs are easy to effcently -color usng semdefnte programmng see [8]) In partcular, a k-unform k-ranbow colorable hypergraph aka, a k-unform k-partte hypergraph) can effcently be -colored On the other hand, effcent -colorng, or even fndng an ndependent set of densty / does not seem possble f the guaranteed colorng devates from beng perfectly balanced Indeed, Guruswam and Lee [8] proved the hardness of approxmately colorng a class of such hypergraphs They proved that t s NP-hard to compute an ndependent set of densty ε n a k-unform hypergraph, k ) whch s guaranteed to be k-ranbow colorable such that each color appears at least ) tmes n every hyperedge Ths mples the hardness of colorng such hypergraphs usng constantly many colors as well as that of non-trvally e, usng at least colors) ranbow colorng them The results of [8] do not, however, say anythng about colorng k-ranbow colorable q-unform hypergraphs for q < k Brakensek and Guruswam [9], under a conjectured ntractablty of a problem called V Label Cover that they formulate, proved hardness of fndng an ndependent set of densty ε n a k +)-unform k-ranbow colorable hypergraph Ths generalzed the case of -colorable 3-unform hypergraphs for whch the same hardness was shown by Khot and Saket [9] under the d-to- Games Conjecture of Khot [8] Unlke these conjecture based works however, our focus s on uncondtonal results Whle the structural guarantee consdered n [8] captures balanced ranbow colorngs, t also allows those n whch a partcular color may appear up to k + ) tmes n a hyperedge Ths s qute far off from beng balanced n the regme where k s comparable or larger than, for example when = whch corresponds to the smallest unformty relatve to k for whch the hardness apples The focus of ths work s the case of ranbow colorable hypergraphs wth a stronger balancedness condton on the colorng: n each hyperedge the occurrences of some of the colors s each off by at most from, and the rest of the colors have precsely occurrences

3 In our man result we show that t s hard to ranbow color such hypergraphs even wth far fewer colors Formally we show the followng: Theorem For any, k, l k/ and an arbtrarly small constant ε > 0, gven a k-unform hypergraph of sze n whch s guaranteed to be k-ranbow colorable such that: n each hyperedge e, for some l e l, there are l e colors that occur exactly ) tmes, l e colors that occur exactly + ) tmes and the rest of the colors occur exactly tmes, ) t s NP-hard to compute an ndependent set of densty l+) k + ε Ths mples, n partcular, that t s NP-hard to k/l)-ranbow color such hypergraphs Under DTIMEN Olog log N) ) reductons from 3SAT one can choose ε = log n) c where c s some postve constant dependng on, k and l Our result yelds wth = k = and l = ) the result of Saket [36] who showed the currently best ) hardness of log n) c -colorng -colorable 4-unform hypergraphs, mprovng on the Ω lower bound by Guruswam, Håstad, and Sudan [5], and Holmern [] For l = log log n log log log n and, k, Guruswam and Lee [8] studed ths problem and clamed a weaker hardness n an auxlary result of ther ntal work rulng out ndependent sets of densty /k) Our Technques The hardness reducton presented n ths paper follows the template of long code based Probablstcally Checkable Proofs PCPs) for the Label Cover problem The long code encodngs of the supposed labels of the Label Cover varables consttute the proof, whose locatons are the vertces of the resultng hypergraph nstance The PCP verfer queres a few locatons of the proof n each of ts random tests defnng the set of hyperedges The test accepts f the [k] := {,, k}-valued labels at the quered locatons descrbe a ranbow colorng satsfyng the desred balancedness crteron Let us now llustrate what we consder the novelty of our reducton: the PCP test for provng Theorem wth constants, k and l k/ The test queres k locatons from long codes correspondng to vertces of the Label Cover nstance wth constrants projectng on a common neghbor Its man buldng block s a dstrbuton P over [k]d ) k whch gves the query locatons restrcted to the pre-mages of a label of the common neghbor To construct P we defne, for t k, µ t to be the unform dstrbuton over all x ),, x t) ) [k] d ) t such that for every [d], x ),, x t) are dstnct Fgure a llustrates a sample from µ k Correspondng to the jth long code j []), let x,j,, x k,j ) be an d sample from µ k Now, for any choce of labels havng the fxed common projecton) gven by,, [d], x,),, x k,),, x,j) j,, x k,j) j,, x,),, x k,) ) [k] k s perfectly balanced e, each color n [k] occurs exactly tmes For the PCP test to work, t requres some perturbaton whle ensurng that the resultant colorng above remans nearly balanced However, Guruswam and Lee have snce wthdrawn ths clam Theorem 4 and Appendx D n the ECCC verson [7]) from later versons of ther paper [8] 3

4 a) A sample from µ k b) A sample from µ l µ k l ) Fgure : Samples from µ k and µ l µ k l ) k = 5, l = and d = 4 For ths purpose, we choose j [] at random and sample x,j,, x k,j ) from µ l µ k l llustrated n Fgure b) Formally, we defne: P := j tmes {}}{ µ k µ k µ }{{} l µ k l ) µ k µ k j = j tmes Notce that the margnal of P for any j [] s P := /)µ k + /)µ l µ k l ) As desred, for any choce of,, [d], for any x,j),, x k,j) ) n the support of P, x,),, x k,),, x,j) j,, x k,j) j,, x,),, x k,) ) s nearly balanced: for some l l, l of the colors n [k] occur exactly ) tmes, l of them occur exactly + ) tmes, and the rest of the colors exactly tmes To extend the test dstrbuton to the pre-mages of L labels on smaller sde of the Label Cover, we sample x,),, x k,),, x,j),, x k,j),, x,),, x k,) ) [k] Ld ) k by samplng ndependently for each [L], the restrcton of the above locatons to the coordnates {d ) +,, d} from P Let f j : [k] Ld {0, } denote the restrcton to the jth long code of a suffcently dense ndependent set Our soundness analyss shows that wth sgnfcant probablty over the choce of the long codes of the PCP test, two of the functons {f j } j [] are ntersectng juntas Otherwse, the expectatons nsde each long code would be uncorrelated yeldng a hyperedge nsde the supposed ndependent set These juntas are then decoded nto a good labelng for the Label Cover Ths [ motvates the frst step of the analyss: n a sngle long code lower boundng the expectaton k ] E s= f jx s,j) ) for some suffcently heavy f j We show that when E[f j ] l+ k + ε), [ k ] E f j x s,j) ) Ωε c ) s= for some c > 0 dependng on, k and l The proof of ths lower bound proceeds by representng the product nsde the expecton as AX)BY ) where X, Y ) = x,j),, x l,j) ), x l+,j),, x k,j) )) 4

5 It can be seen that X and Y are at most /)-correlated Further, usng the structure of the PCP test we show that E[A], E[B] Ωε) Usng ths coupled wth a lower bound on the mxng of Markov chans that was shown by Mossel, Oleszkewcz, and Sen [34] based on ther generalzed reverse hypercontractvty, yelds the desred lower bounds A smlar use for a smpler PCP test) of ths technque was made by Saket [36] Subsequently Guruswam and Lee [8] used reverse hypercontractvty to analyze a PCP test whch sampled unformly from [k] d ) k nstead of µ l µ k l for the j th long code, and used t to show ther NP-hardness result for O)-colorng k-ranbow colorable hypergraphs mentoned earler In ther work, Guruswam and Lee [8] leveraged a smoothness property frst defned by Khot [7]) of the Label Cover nstance for ther analyss whch used Gaussan nvarance theorems and decoded a labelng usng nfluental coordnates Unfortunately, achevng smoothness generates a sgnfcant blowup n the sze of the Label Cover whch renders the reducton somewhat neffcent In contrast, our analyss as also n [36]) s based on standard Fourer analyss and uses a projecton sze preservaton property of the vanlla Label Cover shown by Håstad [0] Ths lmts the sze blowup enablng us to upper bound the error ε n the NO case to /polylog n) under quas-polynomal tme reductons Prelmnares Consder for = to n, the product space Ω ) Ω ), µ ) where the margnals of µ are µ ), µ ) Let Ω s), µ s) ) = n = Ωs), n = µs) ), for s =, We say that X, Y ) Ω ) Ω ) s ρ- correlated 3 f ndependently for each [n], X, Y ) s sampled from µ wth probablty ρ, and from µ ) µ ) wth probablty ρ) The followng theorem s a straghtforward generalzaton of the specal case of Ω ) = Ω ) and µ = d proved n [34] The dervaton s provded n [8] and we ncorporate the explct parameters from [34] Theorem In above setup, let A Ω ) and B Ω ) be two sets such that µ ) {A}, µ ) {B} δ 0 Let X, Y ) Ω ) Ω ) ) be ρ-correlated Then, Pr [X A, Y B] δ ρ ρ We shall also use the Efron-Sten decompostons of functons over product spaces see [33] for reference) Proposton Let Ω = n = Ω, µ = n = µ ) be a product space Then, any f L Ω, µ) can be decomposed unquely as: fx) = f S x), S [n] where f S depends only on the coordnates n S and for S S, E [f S x S ] = 0 In partcular {f S } S [n] are orthogonal e, E[f S f S ] = 0 for S S The startng pont of the reducton s the LabelCover problem whch s defned as follows Ths s analogous to the noton of ρ-correlaton used n [34] and was also used n the reverse hypercontractvty based mxng bounds of [8] 3 See footnote 5

6 Defnton 3 An nstance L of LabelCover conssts of a bpartte graph G L U, V, E) along wth label sets [L] and [M] where M = dl For each edge e between u U and v V, there s a projecton π vu : [M] [L], such that πvu j) = d for each j [L] A labelng l u [L] to u and l v [M] to v satsfes the edge f π vu l v ) = l u The goal s to fnd a labelng of U and V to satsfy the maxmum number of edges The napproxmablty of LabelCover stated below follows from the PCP Theorem [3, ], Raz s Parallel Repetton Theorem [35] We also leverage a structural property proved by Håstad [0] showng that for any vertex n V the mage of a large subset of ts labels remans large under most of the projectons ncdent on v Theorem 4 For every postve nteger r, there s a determnstc N Or) tme reducton from a 3SAT nstance of sze N to an nstance LG L U, V, E), {π vu } {v,u} E, [L], [M]) of LabelCover wth the followng propertes: a U, V N Or) L, M, d 3r G s b-regular wth left and rght degrees bounded by Or) b There s a unversal constant c 0 > 0 such that [ for any v V and S [M], takng an expectaton over a random neghbor u of v, E π vu S) ] S c 0 Ths mples that over the choce of a random neghbor u of v, Pr [ π vu S) < S c 0 ] S c 0 c There s a unversal constant γ 0 > 0 such that, YES Case: If the 3SAT nstance s satsfable then there s a labelng to U and V that satsfes all edges of L NO Case: If the 3SAT nstance s unsatsfable then any labelng to U and V satsfes at most γ 0r fracton of the edges 3 Proof of Theorem In ths secton we prove the followng hardness reducton whch mples Theorem Theorem 3 For any constant ntegers, k and k/ l, and constant ε > 0, there s a polynomal tme reducton from 3SAT to a k-unform hypergraph G of sze n such that: YES Case If the 3SAT nstance s satsfable then there s a k-colorng of the vertces of G such that for n each hyperedge e for some l e l there exactly l e colors that appear ) tmes each and l e colors that appear + ) tmes each, and the other colors appear exactly tmes each In partcular, the hypergraph s k-ranbow colorable NO Case If the 3SAT nstance s not satsfable then there s no ndependent set n G of sze l+) k + ε fracton of vertces, mplyng that G s not k/l)-ranbow colorable In the above, ε can be chosen to be log n) c for some c dependng on, k and l f N Olog log N) - tme reducton from 3SAT s allowed 6

7 The nput s an nstance L of LabelCover from Theorem 4 consstng of a bpartte graph G L U, V, E), label sets [M] and [L] wth M = dl, and projectons {π vu : [M] [L] {u, v} E, u U, v V } such that π vu j) = d for any j [L] For each vertex v V, there s a unformly weghted long code H v whch s a copy of [k] M The set of vertces n the output G s v Hv and the k-unform hyperedges correspond to the PCP test that s descrbed n the next few paragraphs The parameters, k and l n Theorem 3 are fxed n the rest of ths secton 3 Dstrbutons Frst we defne Dt) for t k to be the unform dstrbuton over the set Γt) := {z [k] t z z j, < j t} ) Gven ths, let µ t be the dstrbuton over [k] d ) t where x ),, x t) ) [k] d ) t s sampled by ndependently for each [d] samplng x ),, x t) ) from Dt) Usng ths we defne the dstrbuton P over [k]d ) k by the followng samplng procedure Choose j [] unformly at random For each j [] \ {j } sample x,j),, x k,j) ) from µ k 3 For j sample x,j ),, x k,j ) ) from µ l µ k l ) 4 Ouput x,),, x k,),, x,j),, x k,j),, x,),, x k,) ) The margnal of P restrcted to any j [] s the same dstrbuton P where P := ) ) µ k + µ l µ k l ) ) Wth the above n place the PCP test of the verfer s gven below Test of PCP Verfer The verfer expects a colorng C v : H v [k] for all v V and executes the followng steps The verfer chooses a random vertex u U and of ts neghbors v,, v wth projectons π j := π vj u and long codes H j := H v j Let the C j := C vj be the correspondng colorngs The verfer samples: x,),, x k,),, x,j),, x k,j),, x,),, x k,)) [k] Ld ) k 7

8 from the dstrbuton whch s descrbed by samplng ndependently for each [L], ) x,) π ),, xk,) π ),, x,) π ),, xk,) π ) from the dstrbuton P defned above Let the margnal of restrcted to j [] be j notng that j s dentcal to P L up to permutaton of coordnates 3 The verfer accepts f the colorng C j x s,j) ) ) s [k],j [] has for some l l, l colors that appear exactly ) tmes each, l colors that appear exactly + ) tmes each and the rest of the colors appearng exactly tmes each The rest of ths secton s devoted to provng the YES and NO cases of Theorem 3 3 YES Case In ths case there s a labelng l v for v V such that for any u U and ts neghbors v, w, π vu l u ) = π wu l w ) Lettng C v x) = x lv for x H v and v V yelds a colorng of G that makes the verfer accept wth probablty In partcular, ths colorng satsfes the YES case of Theorem 3 33 NO Case ) Suppose that G contans an ndependent set I of l+) k + 4ε fracton of vertces By standard averagng and usng the b-regularty of the LabelCover nstance L we obtan that for at least ε fracton of good vertces u U, at least ε fracton of ts neghbors are heavy vertces v V whch satsfy, E x [k] M [f v x)] where f v : [k] M {0, } s the ndcator of I H v l + ) + ε, 3) k 33 Lower bound n each Long Code Fx a choce of a good u and ts heavy neghbors v,, v n the verfers test For convenence we let f j := f vj Fx some j [], and consder the expectaton E x,j),,x k,j) ) j [ k s= f j x s,j) ) ] 4) By rearrangng the coordnates and omttng the subscrpt j, the above s equvalent to the followng expectaton: [ k ] E x ),,x k) ) P L fx s) ), 5) s= 8

9 where E x [k] d ) L [k] M [fx)] l + ) + ε 6) k Usng the defnton of Γ) n ) and by abusng notaton slghtly let and defne functons X = x,, x l ) Γl)) M, and Y = x l+,, x k ) Γk l)) M, 7) AX) := l fx s ), and BY ) := s= Thus, the expectaton n 5) s equvalent to k s=l+ fx s ) 8) E X,Y [AX)BY )], 9) where X, Y ) s sampled from P L [ /)µ k + /)µ l µ k l )] L Note that, the margnal dstrbuton of X s µ L l and that of Y s µ L k l Thus, X and Y are /)-correlated Applyng Theorem we obtan, E X,Y [AX)BY )] mn{e[ax)], E[BY )]}) 3 0) The followng argument lower bounds the RHS of the above Lemma 3 For A and B defned as above, E[A], E[B] δ 0 := ε/ k l) Proof Consder a k-unform hypergraph H on vertex set [k] M and hyperedge set {x,, x k ) x ),, x k) ) Γk), [M]} Observe that H s a regular hypergraph e each vertex appears n the same number of hyperedges Usng the bound n 6) along wth an averagng, we obtan that at least ε fracton of the hyperedges x ),, x k) ) are dense satsfyng {s fx s) ) = } k l ) Consder a random choce of Y = x l+),, x k) ) sampled from µ L k l Dk l) M Ths s equvalent to a uar choce of a hyperedge n H and[ a uar subset of k l) of ts vertces From k ] ) we obtan E Y [BY )] = E x l+),,x k) ) Dk l) M s=l+ fxs) ) δ 0 Further, snce l k/ t s easy to see that E X [AX)] E Y [BY )] Usng Lemma 3 along wth 0) we obtan E X,Y [AX)BY )] δ 3 0, whch s rewrtten as: ] E X j),y j) ) j [A j X j) )B j Y j) ) δ := δ 3 0 ) 9

10 33 Analyzng over Long Codes Let us fx a choce of u and of ts neghbors v,, v Usng the notaton ntroduced n Secton 33: we have functons A j : Γl) M {0, } and B j : Γk l) M {0, } for j =,, From Proposton, ther Efron-Sten decomposton s gven as follows A j = A j,s, and B j = B j,s 3) S [M] S [M] Let R be a parameter to be decded later Defne the followng subsets of [M] ) : { } S 0 := S,, S ) [M] ) π j S j ) π j S j ) =, j < j, 4) { } S := S,, S ) [M] ) \ S 0 max S j R, 5) j { } S := S,, S ) [M] ) max π j S j ) > R c 0, 6) j S 3 := { S,, S ) [M] ) j st S j > R, π j S j ) R c 0 }, 7) where c 0 > 0 s the constant from Theorem 4 Note that 3 p=0 S p [M] ) Let us defne δ := Snce I s an ndependent set we also have, E X ),Y ) ),,X ),Y ) )) ] E X j),y j) ) j [A j X j) )B j Y j) ) 8) A j X j) )B j Y j) ) = 0 9) Subtractng 9) from 8), expandng the Efron-Sten decomposton and usng standard Fourer analyss we obtan δ, 0) where, 0 = and for p =,, 3, p = S,,S ) S 0 ] E X j),y j) ) j [ X j) ) Y j) ) E X ),Y ) ),,X ),Y ) )) X j) ) Y j) ) ) S,,S ) S p + E X j),y j) ) j [ X j) ) Y )] j) E X ),Y ) ),,X ),Y ) )) X j) ) Y j) ) ) 0

11 The defnton of S 0 and the propertes of Efron-Sten decompostons n Proposton mply that each term of the sum n the RHS of ) s zero Thus, 0 = 0 3) The goal of the rest of the analyss s to show that for an approprate choce of r n Theorem 4 and the parameter R, the expectaton over the choce of u and v,, v of each p p =,, 3) s small Specfcally, we shall show that a large E[ ] would yeld a good labelng to L contradctng ts NO case Further, s bounded by the dampenng nduced by the presence of subsets wth large projectons n ts sum, and E[ 3 ] s bounded by property b) of Theorem 4 On the other hand, E[δ] s sgnfcant due to ) thereby yeldng for us a contradcton n 0) We begn wth the followng upper bound on Lemma 33 Proof Usng E[fg] f g observe that S,,S ) S S,,S ) S + S,,S ) S S,,S ) S A j,sj B j,sj + S,,S ) S S,,S ) S,,S ), 4) where the last nequalty uses the standard Cauchy-Schwartz nequalty Observe that A j,s j snce {Xj) } k are ndependent Smlarly, B j,s = j Thus, 4) bols down to, S,,S ) S S,,S ) S S,,S ) S S,,S ) B j Aj,Sj 5) =

12 For analyzng and 3, we use the followng lemma whch s proved n Appendx A Lemma 34 E X ),Y ) ),,X ),Y ) )) X j) ) Y j) ) ) maxj π j S j ) Corollary 35 ] E X j),y j) ) j [ X j) ) Y j) ) A j,sj B j,sj ) πj S j ) Proof Use Lemma 34 wth S j = for all j j Usng the above we have the followng bounds for the two sums n Clam 36 Clam 37 S,,S ) S S,,S ) S E X j) ) Y j) ) ) R c0 6) ] E [ X j) ) Y j) ) ) R c0 7) Clams 36 and 37 are proved n Appendx B Usng them along wth p = n ) drectly yelds the followng lemma upper boundng Lemma 38 + ) ) R c0 Smlarly, we have the followng bounds for the two sums n 3 Clam 39 Clam 30 S,,S ) S 3 S,,S ) S 3 E X j) ) Y j) ) ] E [ X j) ) Y j) ) S j : S j >R π j S j ) <R c 0 S j : S j >R π j S j ) <R c 0 8) 9)

13 Clams 39 and 30 are proved n Appendx C Agan, wth p = 3 n ) Clams 39 and 30 drectly mply the followng lemma Lemma 3 3 S j : S j >R π j S j ) <R c 0 Pluggng n Lemmas 33, 38 and 3 nto 0) we obtan that for such a fxed choce of u and v,, v δ ) + S,,S ) S + S j : S j >R π j S j ) <R c 0 + ) R c0 30) For a good choce of u, and of ts heavy neghbors v,, v, as defned n 8), δ δ due to the lower bound n ) Takng an expectaton of 30) over the verfers choces and notng that wth probablty at least ε + u s good and v,, v are heavy, we obtan, ) δ ε+ + E u,{vj } + ) R c 0 E u,{vj } + E vj E u,{vj } S,,S ) S S,,S ) S S j : S j >R S,,S ) S Aj,Sj + S j : S j >R π j S j ) <R c 0 Pr [ π js j ) < R c 0 ] u + ) R c R c0 3) R c 0 ) where the last nequalty uses property b) of Theorem 4 Consder a labelng to the LabelCover nstance L gven by assgnng each vertex v V label l v randomly chosen from a subset S [M] sampled wth probablty A v,s A vertex u U s labeled by unformly at random choosng 3

14 ) of ts neghbors v,, v, random subsets S j wth probablty A vj,s j ndependently for j =,,, and assgnng a random label from j= π js j ) From the defnton of S n 5) the expected number of edges satsfed by ths strategy s at least R R E u,{v j A } j,sj, S,,S ) S whch by 3) s at least ) ) [δ ε+ R + R c 0 ] ) R c0 For any constant ε > 0, choosng the parameter R = poly/ε) and r = Θlog/ε)) n Theorem 4 yelds a contradcton to the NO Case of Theorem 4 Rulng out ε = log n) c Choosng r = log log N)/4 n Theorem 4 we get that the reducton s of sze n = N Or) 3r N Olog log N) The soundness of L s Ωlog log N) = Ωlog log n) Combnng ths wth the above analyss n the NO Case, choosng ε = log n) c and R = ε c for some postve constants c, c > 0 dependng on c 0,, l, k and γ 0 ) we obtan a contradcton to the NO Case of Theorem 4 4 Concluson Our work shows that n k-unform k-ranbow colorable hypergraphs, k ) such that n each hyperedge at most l of the colors appear ± tmes and the rest exactly tmes, t s NPhard to fnd ndependent sets of densty > l + )/k) It s an open challengng) queston to prove the NP-hardness of fndng ndependent sets of arbtrarly small constant densty n such hypergraphs The queston of computng ndependent sets of densty > 05 n perfectly balanced ranbow colorable hypergraphs s also smlarly open References [] S Arora, E Chlamtac, and M Charkar New approxmaton guarantee for chromatc number In Proceedngs of the ACM Symposum on the Theory of Computng, pages 5 4, 006 [] S Arora, C Lund, R Motwan, M Sudan, and M Szegedy Proof verfcaton and the hardness of approxmaton problems Journal of the ACM, 453):50 555, 998 [3] S Arora and S Safra Probablstc checkng of proofs: A new characterzaton of NP Journal of the ACM, 45):70, 998 [4] P Austrn, V Guruswam, and J Håstad +ε)-sat s NP-hard In Proceedngs of the Annual Symposum on Foundatons of Computer Scence, pages 0, 04 4

15 [5] N Bansal Constructve algorthms for dscrepancy mnmzaton In Proceedngs of the Annual Symposum on Foundatons of Computer Scence, pages 3 0, 00 [6] A Blum New approxmaton algorthms for graph colorng Journal of the ACM, 43):470 56, 994 [7] A Blum and D R Karger An Õn3/4 )-colorng algorthm for 3-colorable graphs Informaton Processng Letters, 6):49 53, 997 [8] B Bollobás, D Prtchard, T Rothvoß, and A D Scott Cover-decomposton and polychromatc numbers SIAM Journal on Dscrete Mathematcs, 7):40 56, 03 [9] J Brakensek and V Guruswam The quest for strong napproxmablty results wth perfect completeness In Proc APPROX-RANDOM, pages 4: 4:0, 07 URL: wezmannacl/report/07/080 [0] M Charkar, A Newman, and A Nkolov Tght hardness results for mnmzng dscrepancy In Proceedngs of the Annual ACM-SIAM Symposum on Dscrete Algorthms, pages , 0 [] H Chen and A M Freze Colorng bpartte hypergraphs In Proc IPCO, pages , 996 [] I Dnur and V Guruswam PCPs va low-degree long code and hardness for constraned hypergraph colorng In Proceedngs of the Annual Symposum on Foundatons of Computer Scence, 03 [3] I Dnur, O Regev, and C D Smyth The hardness of 3-unform hypergraph colorng Combnatorca, 55):59 535, 005 [4] V Guruswam, J Håstad, P Harsha, S Srnvasan, and G Varma Super-polylogarthmc hypergraph colorng hardness va low-degree long codes SIAM Journal of Computng, 46):3 59, 07 [5] V Guruswam, J Håstad, and M Sudan Hardness of approxmate hypergraph colorng SIAM Journal of Computng, 36): , 00 [6] V Guruswam and S Khanna On the hardness of 4-colorng a 3-colorable graph SIAM Journal of Dscrete Mathematcs, 8):30 40, 004 [7] V Guruswam and E Lee Strong napproxmablty results on balanced ranbow-colorable hypergraphs Electronc Colloquum on Computatonal Complexty ECCC), :43, 04 URL: [8] V Guruswam and E Lee Strong napproxmablty results on balanced ranbow-colorable hypergraphs In Proceedngs of the Annual ACM-SIAM Symposum on Dscrete Algorthms, pages 8 836, 05 [9] E Halpern, R Nathanel, and U Zwck Colorng k-colorable graphs usng relatvely small palettes J Algorthms, 45):7 90, 00 5

16 [0] J Håstad Some optmal napproxmablty results Journal of the ACM, 484): , 00 [] J Holmern Vertex cover on 4-regular hyper-graphs s hard to approxmate wthn - ε In Proceedngs of the Annual IEEE Conference on Computatonal Complexty, 00 [] S Huang log n)/0 o) hardness for hypergraph colorng CoRR, abs/ , 05 [3] D R Karger, R Motwan, and M Sudan Approxmate graph colorng by semdefnte programmng Journal of the ACM, 45):46 65, 998 [4] K Kawarabayash and M Thorup Combnatoral colorng of 3-colorable graphs In Proceedngs of the Annual Symposum on Foundatons of Computer Scence, pages 68 75, 0 [5] P Kelsen, S Mahajan, and R Harharan Approxmate hypergraph colorng In Proc SWAT, pages 4 5, 996 [6] S Khanna, N Lnal, and S Safra On the hardness of approxmatng the chromatc number Combnatorca, 03):393 45, 000 [7] S Khot Hardness results for colorng 3-colorable 3-unform hypergraphs In Proceedngs of the Annual Symposum on Foundatons of Computer Scence, pages 3 3, 00 [8] S Khot On the power of unque -prover -round games In Proceedngs of the ACM Symposum on the Theory of Computng, pages , 00 [9] S Khot and R Saket Hardness of fndng ndependent sets n -colorable and almost - colorable hypergraphs In Proceedngs of the Annual ACM-SIAM Symposum on Dscrete Algorthms, pages , 04 [30] S Khot and R Saket Hardness of colorng -colorable -unform hypergraphs wth log n)ω) colors SIAM Journal of Computng, 46):35 7, 07 [3] M Krvelevch, R Nathanel, and B Sudakov Approxmatng colorng and maxmum ndependent sets n 3-unform hypergraphs Journal of Algorthms, 4):99 3, 00 [3] S Lovett and R Meka Constructve dscrepancy mnmzaton by walkng on the edges SIAM Journal of Computng, 445):573 58, 05 [33] E Mossel Gaussan bounds for nose correlaton of functons GAFA, 9:73 756, 00 [34] E Mossel, K Oleszkewcz, and A Sen On reverse hypercontractvty Geometrc and Functonal Analyss, 33):06 097, 03 [35] R Raz A parallel repetton theorem SIAM Journal of Computng, 73): , 998 [36] R Saket Hardness of fndng ndependent sets n -colorable hypergraphs and of satsfable CSPs In Proceedngs of the Annual IEEE Conference on Computatonal Complexty, pages 78 89, 04 [37] G Varma Reducng unformty n Khot-Saket hypergraph colorng hardness reductons Chcago J Theor Comput Sc, 06, 06 [38] A Wgderson Improvng the performance guarantee for approxmate graph colorng Journal of the ACM, 304):79 735, 983 6

17 A Proof of Lemma 34 For r =,, L, j =,,, let us defne the operator T r) j on the space of functons gx),, X ) ) where T r) j g s the expectaton of g over the rerandomzaton of Xj ) π r) µ l Further, let Clearly, T r) j X j) ) = Moreover, from Proposton, T r) j A j,s j X j ) ) = T r) g := E j [] j []\{j } j [ ] T r) j g 3) X j) ) T r) j A j,s j X j ) ) 33) { 0 f r π j S j ), A j,s j X j ) ) otherwse Gven S = S,, S ) and r [L] let qs, r) := {j [] r π j S j )} From 3), 33) and 34) we obtan that for a fxed X ),, X ) ), and therefore, Now, T r) T ) T L) E X ),Y ) ),,X ),Y ) )) = E {X j),y j) ) µ L k } = E {X j),y j) ) µ L k } X j) ) = X j) ) = L r= 34) ) qs, r) X j) ), 35) X j) ) Y j) ) T ) T L) ) ) qs, r) X j) ) 36) X j) ) Y j) ) L X j) ) Y j) ) whch completes the proof L r= ) maxj π j S j ), r= ) ) qs, r) usng E[fg] f g ) ) ) qs, r) usng 36)) 7

18 B Proofs of Clams 36 and 37 Proof of Clam 36 Consder the term nsde the sum on the LHS of 6): E X j) ) Y j) ) 37) Fx S s such that π S ) > R c 0 The sum over all S,, S ) [M] ) ) of 37) s upper bounded usng Lemma 34) by, ) R c 0 Now note that, = = = = S,,S ) S,,S ) S,,S ) E E S,,S ) E S,,S ) S,,S ) E A,S S,,S ) j= E A,S A j j= S,,S ) S,,S ) E E E B,S B j j= B,S S,,S ) j= E [ A,S ]) E [ B,S ]) = A,S B,S, 4) where we appled Cauchy-Schwartz n 38) and used that fact that A j and B j are {0, } valued 38) 39) 40) 4) 8

19 functons n 4) To see how 39) equals 40) observe that, E E A,S S,,S ) j= = E [ A,S ] S,,S ) j= S,,S ) ] E [ A j,s j S,,S ) = 0, usng the ndependence of {X j) } and the orthogonalty of {A j,s} S [M] for each j [] The same analyss holds for the second product n 39) Thus, the sum of 37) over all S,, S ) such that π S ) > R c 0 s at most ) R c 0 A,S B,S ) R c 0 S : π S ) >R c 0 S : π S ) >R c 0 A,S S : π S ) >R c 0 B,S 43) ) R c0 44) Summng the above bound for all j [] for whch π j S j ) > R c 0 yelds the clam Proof of Clam 37 From Corollary 35 the LHS of 7) s upper bounded by S,,S ) S,,S ) Aj,Sj Bj,Sj ) R c 0 S,,S ) = B j,sj Sj Sj = A j B j ) R c0 ) R c 0 ) R c 0 ) R c0 45) C Proofs of Clams 39 and 30 Proof of Clam 39 The proof s a slght varaton to that of Clam 36 n Appendx B and proceeds n a smlar manner tll the bound n 43) except we don t have the /) Rc0 factor outsde and 9

20 the nner sum s over S st S > R and π S ) < R c 0 yeldng an upper bound of S : S >R π S ) <R c 0 A,S for j = Summng ths over all j [] completes the proof of the clam Proof of Clam 30 The proof s smlar to that of Clam 37 n Appendx B except that that t s done separately for each S st S > R and π S ) < R c 0, and upon summng over all such S ths yelds an upper bound of S : S >R π S ) <R c 0 A,S B,S, S : S >R π S ) <R c 0 A,S for j = Agan, summng ths over all j [] completes the proof of the clam, 0 ECCC ISSN