Approximability Distance in the Space of H-Colourability Problems

Appro ximabilit y Distane in the Spae of H -Colourabilit y Problems T omm y Färnqvist, P eter Jonsson, and Johan Thapp er Departmen t of Computer and Information Siene Link öpings univ ersitet SE-581 83 Link öping, Sw eden {tomfa, petej, johth}ida.liu.se Abstrat. A graph homomorphism is a v ertex map whi h arries edges from a soure graph to edges in a target graph. W e study the appro x- imabilit y prop erties of the W eighte d Maximum H -Colour able Sub gr aph problem ( Max H -Col ). The instanes of this problem are edge-w eigh ted graphs G and the ob jetiv e is to nd a subgraph of G that has maximal total edge w eigh t, under the ondition that the subgraph has a homo- morphism to H ; note that for H = K k this problem is equiv alen t to Max k -ut . T o this end, w e in tro due a metri struture on the spae of graphs whi h allo ws us to extend previously kno wn appro ximabilit y results to larger lasses of graphs. Sp eially , the appro ximation algo- rithms for Max ut b y Go emans and Williamson and Max k -ut b y F rieze and Jerrum an b e used to yield non-trivial appro ximation results for Max H -Col . F or a v ariet y of graphs, w e sho w near-optimalit y re- sults under the Unique Games Conjeture. W e also use our metho d for omparing the p erformane of F rieze & Jerrum's algorithm with Hås- tad's appro ximation algorithm for general Max 2-Csp . This omparison is, in most ases, fa v ourable to F rieze & Jerrum. Keyw ords : optimisation, appro ximabilit y , graph homomorphism, graph H -ol- ouring, omputational omplexit y 1 In tro dution Let G b e a simple, undireted and nite graph. Giv en a subset S ⊆ V ( G ) , a ut in G with resp et to S is the edges from a v ertex in S to a v ertex in V ( G ) \ S . The Max ut -problem asks for the size of a largest ut in G . More generally , a k -ut in G is the edges going from S i to S j , i 6 = j , where S 1 , . . . , S k is a partitioning of V ( G ) , and the Max k -ut -problem asks for the size of a largest k -ut. The problem is readily seen to b e iden tial to nding a largest k -olourable subgraph of G . F urthermore, Max k -ut is kno wn to b e APX - omplete for ev ery k ≥ 2 and onsequen tly do es not admit a p olynomial-time appr oximation sheme ( Pt as ). In the absene of a Pt as , it is in teresting to determine the b est p ossible appro ximation ratio c within whi h a problem an b e appro ximated or, alterna- tiv ely the smallest c for whi h it an b e pro v ed that no p olynomial-time appro x- imation algorithm exists (t ypially under some omplexit y-theoreti assumption su h as P 6 = NP ). An appro ximation ratio of . 87856 7 for Max ut w as obtained in 1995 b y Go emans and Williamson [15℄ using semidenite programming. F rieze and Jerrum [14℄ determined lo w er b ounds on the appro ximation ratios for Max k -ut using similar te hniques. Sharp ened results for small v alues of k ha v e later b een obtained b y de Klerk et al. [9℄. Under the assumption that the Unique Games Conje tur e holds, Khot et al. [25℄ sho w ed the appro ximation ratio for k = 2 to b e essen tially optimal and also pro vided upp er b ounds on the appro xi- mation ratio for k > 2 . Håstad [20℄ has sho wn that semidenite programming is a univ ersal to ol for solving the general Max 2-Csp problem o v er an y domain, in the sense that it establishes non-trivial appro ximation results for all of those problems. In this pap er, w e study appro ximabilit y prop erties of a generalised v ersion of Max k -ut alled Max H -Col for undireted graphs H . Jonsson et al. [21℄ ha v e sho wn that, when H is lo op-free, Max H -Col do es not admit a Pt as . Note that if H on tains a lo op, then Max H -Col is a trivial problem. W e presen t appro ximabilit y results for Max H -Col where H is tak en from dieren t families of graphs. Man y of these results turns out to b e lose to optimal under the Unique Games Conjeture. Our approa h is based on analysing appro ximabilit y algorithms applied to problems whi h they are not originally in tended to solv e. This v ague idea will b e laried b elo w. Denote b y G the set of all simple, undireted and nite graphs. A gr aph homomorphism h from G to H is a v ertex map whi h arries the edges in G to edges in H . The existene of su h a map will b e denoted b y G → H . If b oth G → H and H → G , the graphs G and H are said to b e homomorphi al ly e quivalent . This equiv alene will b e denoted b y G ≡ H . F or a graph G ∈ G , let W ( G ) b e the set of weight funtions w : E ( G ) → Q + assigning w eigh ts to edges of G . F or a w ∈ W ( G ) , w e let k w k = P e ∈ E ( G ) w ( e ) denote the total w eigh t of G . No w, W eighte d Maximum H -Colour able Sub gr aph ( Max H -Col ) is the maximisation problem with Instane: An edge-w eigh ted graph ( G, w ) , where G ∈ G and w ∈ W ( G ) . Solution: A subgraph G ′ of G su h that G ′ → H . Measure: The w eigh t of G ′ with resp et to w . Giv en an edge-w eigh ted graph ( G, w ) , denote b y mc H ( G, w ) the measure of the optimal solution to the problem Max H -Col . Denote b y mc k ( G, w ) the (w eigh ted) size of the largest k -ut in ( G, w ) . This notation is justied b y the fat that mc k ( G, w ) = mc K k ( G, w ) . In this sense, Max H -Col generalises Max k -ut . The deision v ersion of Max H -Col , the H -  olouring problem has b een extensiv ely studied (See [17℄ and its man y referenes.) and Hell and Ne²et°il [16℄ ha v e sho wn that the problem is in P if H on tains a lo op or is bipartite, and NP -omplete otherwise. Langb erg et al. [27℄ ha v e studied the appro ximabilit y of Max H -Col when H is part of the input. W e also note that Max H -Col is a sp eialisation of the Max Csp problem. The homomorphism relation → denes a quasi-order, but not a partial order on the set G . The failing axiom is that of an tisymmetry , sine G ≡ H do es not neessarily imply G = H . T o remedy this, let G ≡ denote the set of equiv alene lasses of G under homomorphi equiv alene. The relation → is dened on G ≡ in the ob vious w a y and on this set it is a partial order. In fat, → pro vides a lattie struture on G ≡ and this lattie will b e denoted b y C S . F or a more in-depth treatmen t of graph homomorphisms and the lattie C S , see [17℄. Here, w e endo w G ≡ with a metri d dened in the follo wing w a y: for M , N ∈ G , let d ( M , N ) = 1 − inf G ∈G w ∈W ( G ) mc M ( G, w ) mc N ( G, w ) · inf G ∈G w ∈W ( G ) mc N ( G, w ) mc M ( G, w ) . (1) In Lemma 5 w e will sho w that d satises the follo wing prop ert y:  Let M , N ∈ G and assume that mc M an b e appro ximated within α . Then, mc N an b e appro ximated within (1 − d ( M , N )) · α . Hene, w e an use d for extending previously kno wn appro ximabilit y b ounds on Max H -Col to new and larger lasses of graphs. F or instane, w e an apply Go e- mans and Williamson's algorithm (whi h is in tended for solving Max K 2 -Col ) to Max C 11 -Col (i.e. the yle on 11 v erties) and analyse ho w w ell the problem is appro ximated (w e will see later on that Go emans and Williamson's algorithm appro ximates Max C 11 -Col within 0.79869). In ertain ases, the metri d is related to a w ell-studied graph parameter kno wn as bip artite density b ( H ) [1, 3, 6, 18, 28℄: if H ′ is bipartite subgraph of H with maxim um n um b er of edges, then b ( H ) = e ( H ′ ) e ( H ) . In the end of Setion 2 w e will see that b ( H ) = 1 − d ( K 2 , H ) for all edge-transitiv e graphs H . W e note that while d is in v arian t under homomorphi equiv alene, this is not in general true for bipartite densit y . The pap er is divided in to t w o main parts. Setion 2 is used for pro ving the basi prop erties of d , sho wing that it is w ell-dened on G ≡ , and that it is a metri. After that, w e sho w that d is omputable b y linear programming and that the omputation of d ( M , N ) an b e simplied whenev er M or N is edge-transitiv e. W e onlude this part b y pro viding some examples. The seond part of the pap er uses d for studying the appro ximabilit y of Max H -Col . F or sev eral lasses of graphs, w e in v estigate optimialit y issues b y exploiting inappro ximabilit y b ounds that are onsequenes of the Unique Games Conjeture. Comparisons are also made to the b ounds a hiev ed b y the general Max 2-Csp -algorithm b y Håstad [20℄. Our in v estigation o v ers a sp etrum of graphs, ranging from graphs with few edges and/or on taining large smallest yles to graphs on taining Θ ( n 2 ) edges. Dense graphs are onsidered from t w o p ersp etiv es; rstly as graphs ha ving a n um b er of edges lose to maximal and seondly as graphs from the G ( n, p ) mo del of random graphs, pioneered b y Erd®s and Rén yi [13℄. The te hniques used in this pap er seem to generalise naturally to larger sets of problems. This and other questions are disussed in Setion 4 whi h onludes our pap er. 2 Appro ximation via the Metri d In this setion w e start out b y pro ving basi prop erties of the metri d , that ( G ≡ , d ) is a metri spae, and that pro ximit y of graphs M , N in this spae lets us in terrelate the appro ximabilit y of Max M -Col and Max N -Col . Setions 2.2 and 2.3 are dev oted to sho wing ho w to ompute d . 2.1 The Spae ( G ≡ , d ) W e b egin b y in tro duing a funtion s : G × G → R whi h enables us to express d in a natural w a y and simplify forthoming pro ofs. Let M , N ∈ G and dene s ( M , N ) = inf G ∈G w ∈W ( G ) mc M ( G, w ) mc N ( G, w ) . (2) The denition of d from (1) an then b e written as follo ws: d ( M , N ) = 1 − s ( N , M ) · s ( M , N ) . (3) A onsequene of (2) is that the relation mc M ( G, w ) ≥ s ( M , N ) · mc N ( G, w ) holds for all G ∈ G and w ∈ W ( G ) . Using this observ ation, w e sho w that s ( M , N ) and thereb y d ( M , N ) b eha v es w ell under graph homomorphisms and homomorphi equiv alene. Lemma 1. L et M , N ∈ G and M → N . Then, for every G ∈ G and every weight funtion w ∈ W ( G ) , mc M ( G, w ) ≤ mc N ( G, w ) . Pr o of. If G ′ → M for some subgraph G ′ of G , then G ′ → N as w ell. The lemma immediately follo ws. ⊓ ⊔ Corollary 2. If M and N ar e homomorphi al ly e quivalent, then mc M ( G, w ) = mc N ( G, w ) . Corollary 3. L et M 1 ≡ M 2 and N 1 ≡ N 2 b e two p airs of homomorphi al ly e quivalent gr aphs. Then, for i, j, k , l ∈ { 1 , 2 } , s ( N i , M j ) = s ( N k , M l ) . Pr o of. Corollary 2 sho ws that for all G ∈ G and w ∈ W ( G ) , w e ha v e mc M j ( G, w ) mc N i ( G, w ) = mc M l ( G, w ) mc N k ( G, w ) . No w, tak e the inm um o v er graphs G and w eigh t funtions w on b oth sides. ⊓ ⊔ Corollary 3 sho ws that s and d are w ell-dened as funtions on the set G ≡ . W e an no w sho w that d is indeed a metri on this spae. Lemma 4. The p air ( G ≡ , d ) forms a metri sp a e. Pr o of. P ositivit y and symmetry follo ws immediately from the denition and the fat that s ( M , N ) ≤ 1 for all M and N . Sine s ( M , N ) = 1 if and only if N → M , it also holds that d ( M , N ) = 0 if and only if M and N are homomorphially equiv alen t. That is, d ( M , N ) = 0 if and only if M and N represen t the same mem b er of G ≡ . F urthermore, for graphs M , N and K ∈ G : s ( M , N ) · s ( N , K ) = inf G ∈G w ∈W ( G ) mc M ( G, w ) mc N ( G, w ) · inf G ∈G w ∈W ( G ) mc N ( G, w ) mc K ( G, w ) ≤ inf G ∈G w ∈W ( G ) mc M ( G, w ) mc N ( G, w ) · mc N ( G, w ) mc K ( G, w ) = s ( M , K ) . Therefore, with a = s ( M , N ) · s ( N , M ) , b = s ( N , K ) · s ( K , N ) and c = s ( M , K ) · s ( K, M ) ≥ a · b , d ( M , N ) + d ( N , K ) − d ( M , K ) = 1 − a + 1 − b − (1 − c ) ≥ ≥ 1 − a − b + a · b = (1 − a ) · (1 − b ) ≥ 0 , whi h sho ws that d satises the triangle inequalit y . ⊓ ⊔ W e sa y that a maximisation problem Π an b e appro ximated within c < 1 if there exists a randomised p olynomial-time algorithm A su h that c · Op t ( x ) ≤ E ( A ( x )) ≤ Op t ( x ) for all instanes x of Π . Our next result sho ws that pro ximit y of graphs G and H in d allo ws us to determine b ounds on the appro ximabilit y of Max H -Col from kno wn b ounds on the appro ximabilit y of Max G -Col . Lemma 5. L et M , N , K b e gr aphs. If mc M  an b e appr oximate d within α , then mc N  an b e appr oximate d within α · (1 − d ( M , N )) . If it is NP -har d to appr ox- imate mc K within β , then mc N is not appr oximable within β / (1 − d ( N , K )) unless P = NP . Pr o of. Let A ( G, w ) b e the measure of the solution returned b y an algorithm whi h appro ximates mc M within α . W e kno w that for all G ∈ G and w ∈ W ( G ) w e ha v e the inequalities mc N ( G, w ) ≥ s ( N , M ) · mc M ( G, w ) and mc M ( G, w ) ≥ s ( M , N ) · mc N ( G, w ) . Consequen tly , mc N ( G, w ) ≥ mc M ( G, w ) · s ( N , M ) ≥ A ( G, w ) · s ( N , M ) ≥ mc M ( G, w ) · α · s ( N , M ) ≥ mc N ( G, w ) · α · s ( N , M ) · s ( M , N ) = mc N ( G, w ) · α · (1 − d ( M , N )) . F or the seond part, assume to the on trary that there exists a p olynomial-time algorithm B that appro ximates mc N within β / (1 − d ( N , K )) . A ording to the rst part mc K an then b e appro ximated within (1 − d ( N , K )) · β / (1 − d ( N , K )) = β . This is a on tradition unless P = NP . ⊓ ⊔ 2.2 Exploiting Symmetries W e ha v e seen that the metri d ( M , N ) an b e dened in terms of s ( M , N ) . In fat, when M → N w e ha v e 1 − d ( M , N ) = s ( M , N ) . It is therefore of in terest to nd an expression for s whi h an b e alulated easily . After Lemma 6 (whi h sho ws ho w mc M ( G, w ) dep ends on w ) w e in tro due a dieren t w a y of desribing the solutions to Max M -Col whi h mak es the pro ofs of the follo wing results more natural. In Lemma 7, w e sho w that a partiular t yp e of w eigh t funtion pro vides a lo w er b ound on mc M ( G, w ) /mc N ( G, w ) . Finally , in Lemma 8, w e pro vide a simpler expression for s ( M , N ) whi h dep ends diretly on the automorphism group and thereb y the symmetries of N . This expression b eomes partiularly simple when N is edge-transitiv e. An immediate onsequene of this is that s ( K 2 , H ) = b ( H ) for edge-transitiv e graphs H . The optim um mc H ( G, w ) is sub-linear with resp et to the w eigh t funtion, as is sho wn b y the follo wing lemma. Lemma 6. L et G, H ∈ G , α ∈ Q + and let w, w 1 , . . . , w r ∈ W ( G ) b e weight funtions on G . Then,  mc H ( G, α · w ) = α · mc H ( G, w ) ,  mc H ( G, P r i =1 w i ) ≤ P r i =1 mc H ( G, w i ) . Pr o of. The rst part is trivial. F or the seond part, let G ′ b e an optimal solution to the instane ( G, P r i =1 w i ) of Max H -Col . Then, the measure of this solution equals the sum of the measures of G ′ as a (p ossibly sub optimal) solution to ea h of the instanes ( G, w i ) . ⊓ ⊔ An alternativ e desription of the solutions to Max H -Col is as follo ws: let G and H ∈ G , and for an y v ertex map f : V ( G ) → V ( H ) , let f # : E ( G ) → E ( H ) b e the (partial) edge map indued b y f . In this notation h : V ( G ) → V ( H ) is a graph homomorphism preisely when ( h # ) − 1 ( E ( H )) = E ( G ) or, alternativ ely when h # is a total funtion. The set of solutions to an instane ( G, w ) of Max H -Col an then b e tak en to b e the set of v ertex maps f : V ( G ) → V ( H ) with the measure w ( f ) = X e ∈ ( f # ) − 1 ( E ( H )) w ( e ) . In the remaining part of this setion, w e will use this desription of a solution. Let f : V ( G ) → V ( H ) b e an optimal solution to the instane ( G, w ) of Max H -Col . Dene the w eigh t w f ∈ W ( H ) as follo ws: for ea h e ∈ E ( H ) , let w f ( e ) = X e ′ ∈ ( f # ) − 1 ( e ) w ( e ′ ) mc H ( G, w ) . W e no w pro v e the follo wing result: Lemma 7. L et M , N ∈ G b e two gr aphs. Then, for every G ∈ G , every w ∈ W ( G ) , and any optimal solution f to ( G, w ) of Max N -Col , it holds that mc M ( G, w ) mc N ( G, w ) ≥ mc M ( N , w f ) . Pr o of. Arbitrarily  ho ose an optimal solution g : V ( N ) → V ( M ) to the instane ( N , w f ) of Max M -Col . Then, g ◦ f is a solution to ( G, w ) as an instane of Max M -Col . The w eigh t of this solution is mc M ( N , w f ) · mc N ( G, w ) , whi h implies that mc M ( G, w ) ≥ mc M ( N , w f ) · mc N ( G, w ) , and the result follo ws after division b y mc N ( G, w ) . ⊓ ⊔ Let M and N ∈ G b e graphs and let A = Aut ( N ) b e the automorphism group of N . W e will let π ∈ A at on { u, v } ∈ E ( N ) b y π · { u, v } = { π ( u ) , π ( v ) } . The graph N is edge-transitiv e if and only if A ats transitiv ely on the edges of N . Let ˆ W ( N ) b e the set of w eigh t funtions w ∈ W ( N ) whi h satisfy k w k = 1 and for whi h w ( e ) = w ( π · e ) for all e ∈ E ( N ) and π ∈ Aut ( N ) . Lemma 8. L et M , N ∈ G . Then, s ( M , N ) = inf w ∈ ˆ W ( N ) mc M ( N , w ) . In p artiular, when N is e dge-tr ansitive, s ( M , N ) = mc M ( N , 1 / e ( N )) . Pr o of. The easy diretion go es through as follo ws: s ( M , N ) ≤ inf w ∈ ˆ W ( N ) mc M ( N , w ) mc N ( N , w ) = inf w ∈ ˆ W ( N ) mc M ( N , w ) . F or the rst part of the lemma, it will b e suien t to pro v e that the follo wing inequalit y holds for for some w ′ ∈ ˆ W . mc M ( G, w ) mc N ( G, w ) ≥ mc M ( N , w ′ ) (4) T aking the inm um o v er graphs G and w eigh t funtions w ∈ W ( G ) in the left- hand side of this inequalit y will then sho w that s ( M , N ) ≥ mc M ( N , w ′ ) ≥ inf w ∈ ˆ W ( N ) mc M ( N , w ) . Let A = Aut ( N ) b e the automorphism group of N . Let π ∈ A b e an arbitrary automorphism of N . If f is an optimal solution to ( G, w ) as an instane of Max N -Col , then so is f π = π ◦ f . Let w π = w π ◦ f . By Lemma 7, inequalit y (4) is satised b y w π . Summing π in this inequalit y o v er A giv es | A | · mc M ( G, w ) mc N ( G, w ) ≥ X π ∈ A mc M ( N , w π ) ≥ mc M ( N , X π ∈ A w π ) , where the last inequalit y follo ws from Lemma 6. The w eigh t funtion P π ∈ A w π an b e determined as follo ws. X π ∈ A w π ( e ) = X π ∈ A P e ′ ∈ ( f # ) − 1 ( π · e ) w ( e ′ ) mc N ( G, w ) = | A | | Ae | · P e ′ ∈ ( f # ) − 1 ( Ae ) w ( e ′ ) mc N ( G, w ) , where Ae denotes the orbit of e under A . Th us, w ′ P π ∈ A w π / | A | ∈ ˆ W ( N ) and w ′ satises (4) so the rst part follo ws. F or the seond part, note that when the automorphism group A ats transitiv ely on E ( N ) , there is only one orbit Ae = E ( N ) . Then, the w eigh t funtion w ′ is giv en b y w ′ ( e ) = 1 e ( N ) · P e ′ ∈ ( f # ) − 1 ( E ( N )) w ( e ′ ) mc N ( G, w ) = 1 e ( N ) · mc N ( G, w ) mc N ( G, w ) . ⊓ ⊔ 2.3 T o ols for Computing Distanes F rom Lemma 8 it follo ws that in order to determine s ( M , N ) , it is suien t to minimise mc M ( N , w ) o v er ˆ W ( N ) . W e will no w use this observ ation to desrib e a linear program for omputing s ( M , N ) . F or i ∈ { 1 , . . . , r } , let A i b e the orbits of Aut ( N ) ating on E ( N ) . The measure of a solution f when w ∈ ˆ W ( N ) is equal to P r i =1 w i · f i , where w i is the w eigh t of an edge in A i and f i is the n um b er of edges in A i whi h are mapp ed to an edge in M b y f . Note that giv en a w , the measure of a solution f dep ends only on the v etor ( f 1 , . . . , f r ) ∈ N r . Therefore, tak e the solution spae to b e the set of su h v etors: F = { ( f 1 , . . . , f r ) | f is a solution to ( N , w ) of Max M -Col } Let the v ariables of the linear program b e w 1 , . . . , w r and s , where w i represen ts the w eigh t of ea h elemen t in the orbit A i and s is an upp er b ound on the solutions. min s P i f i · w i ≤ s for ea h ( f 1 , . . . , f r ) ∈ F P i | A i | · w i = 1 w i , s ≥ 0 Giv en a solution w i , s to this program, w ( e ) = w i when e ∈ A i is a w eigh t funtion whi h minimises mc M ( G, w ) . The v alue of this solution is s = s ( M , N ) . Example 9. The whe el gr aph on k v erties, W k , is a graph that on tains a yle of length k − 1 plus a v ertex v not in the yle su h that v is onneted to ev ery other v ertex. W e all the edges of the k − 1 -yle outer e dges and the remaining k − 1 edges sp okes . It is easy to see that W k on tains a maxim um lique of size 4 if k = 4 (in fat, W 4 = K 4 ) and a maxim um lique of size 3 in all other ases. F urthermore, W k is 3-olourable if and only if k is o dd, and 4-olourable otherwise. This implies that for o dd k , the wheel graphs are homomorphially equiv alen t to K 3 . W e will determine s ( K 3 , W n ) for ev en n ≥ 6 using the previously desrib ed onstrution of a linear program. Note that the group ation of Aut ( W n ) on E ( W n ) has t w o orbits, one whi h onsists of all outer edges and one whi h onsists of all the sp ok es. If w e remo v e one outer edge or one sp ok e from W k , then the resulting graph an b e mapp ed homomorphially on to K 3 . Therefore, it sues to  ho ose F = { f , g } with f = ( k − 1 , k − 2) and g = ( k − 2 , k − 1) sine all other solutions will ha v e a smaller measure than at least one of these. The program for W k lo oks lik e this: min s ( k − 1) · w 1 + ( k − 2) · w 2 ≤ s ( k − 2) · w 1 + ( k − 1) · w 2 ≤ s ( k − 1) · w 1 + ( k − 1) · w 2 = 1 w i , s ≥ 0 The solution is w 1 = w 2 = 1 / (2 k − 2) with s ( K 3 , W k ) = s = (2 k − 3) / (2 k − 2) . Example 10. An example where the w eigh ts in the optimal solution to the linear program are dieren t for dieren t orbits is giv en b y s ( K 2 , K 8 / 3 ) . The r ational  omplete gr aph K 8 / 3 has v ertex set { 0 , 1 , . . . , 7 } , whi h is though t of as plaed on a irle with 0 follo wing 7 . There is an edge b et w een an y t w o v erties whi h are at a distane at least 3 from ea h other. Ea h v ertex has distane 4 to exatly one other v ertex, whi h means there are 4 su h edges. These edges form one orbit A 1 and the remaining 8 edges form the other orbit A 2 . There are t w o maximal solutions, f = (0 , 8) and g = (4 , 6) whi h giv es the program: min s 0 · w 1 + 8 · w 2 ≤ s 4 · w 1 + 6 · w 2 ≤ s 4 · w 1 + 8 · w 2 = 1 w i , s ≥ 0 The solution to this program is w 1 = 1 / 20 and w 2 = 1 / 10 with the optim um b eing 4 / 5 . In some ases, it ma y b e hard to determine a desired distane b et w een H and M or N . If w e kno w that H is homomorphially sandwi hed b et w een M and N so that M → H → N , then w e an pro vide an upp er b ound on the distane of H to M or N b y using the distane b et w een M and N . F ormally , w e ha v e: Lemma 11. L et M → H → N . Then, s ( M , H ) ≥ s ( M , N ) and s ( H, N ) ≥ s ( M , N ) . Pr o of. Sine H → N , it follo ws from Lemma 1 that mc H ( G, w ) ≤ mc N ( G, w ) . Th us, s ( M , H ) = inf G ∈G w ∈W ( G ) mc M ( G, w ) mc H ( G, w ) ≥ inf G ∈G w ∈W ( G ) mc M ( G, w ) mc N ( G, w ) = s ( M , N ) . The seond part follo ws similarly . ⊓ ⊔ W e will see sev eral appliations of this lemma in Setions 3.1 and 3.2. 3 Appro ximabilit y of Max H -Col Let A b e an appro ximation algorithm for Max H -Col . Our metho d basially allo ws us to measure ho w w ell A p erforms on other problems Max H ′ -Col . In this setion, w e will apply the metho d to v arious algorithms and v arious graphs. W e do t w o things for ea h kind of graph under onsideration: ompare the p erformane of our metho d with that of some existing, leading, appro xima- tion algorithm and in v estigate ho w lose to optimalit y w e an get. Our main algorithmi to ols will b e the follo wing: Theorem 12 (Go emans and Williamson [15℄). mc 2  an b e appr oximate d within α GW = min 0 <θ <π θ/ π (1 − cos θ ) / 2 ≈ . 878567 . Theorem 13 (F rieze and Jerrum [14℄). mc k  an b e appr oximate d within α k ∼ 1 − 1 k + 2 ln k k 2 . Here, the relation ∼ indiates t w o expressions whose ratio tends to 1 as k → ∞ . W e note that de Klerk et al. [9℄ ha v e presen ted the sharp est kno wn b ounds on α k for small v alues of k ; for instane, α 3 ≥ 0 . 836008 . Let v ( G ) , e ( G ) denote the n um b er of v erties and edges in G , resp etiv ely . Håstad has sho wn the follo wing: Theorem 14 (Håstad [20℄). L et H b e a gr aph. Ther e is an absolute  onstant c > 0 suh that mc H  an b e appr oximate d within 1 − t ( H ) d 2 · (1 − c d 2 log d ) wher e d = v ( H ) and t ( H ) = d 2 − 2 · e ( H ) . W e will ompare the p erformane of this algorithm on Max H -Col with the p erformane of the algorithms in Theorems 12 and 13 analysed using Lemma 5 and estimates of the distane d . This omparison is not en tirely fair sine Hås- tad's algorithm w as probably not designed with the goal of pro viding optimal resultsthe goal w as to b eat random assignmen ts. Ho w ev er, it is the urren tly b est algorithm that an appro ximate Max H -Col for arbitrary H ∈ G . F or this purp ose, w e in tro due t w o funtions, F J k and Hå , su h that, if H is a graph, F J k ( H ) denotes the b est b ound on the appro ximation guaran tee when F rieze and Jerrum's algorithm for Max k -ut is applied to the problem mc H , while Hå ( H ) is the guaran tee when Håstad's algorithm is used to appro ximate mc H . T o b e able to in v estigate the ev en tual near-optimalit y of our appro xima- tion metho d w e will rely on the Unique Games Conjeture (UGC). Khot [24℄ suggested this onjeture as a p ossible diretion for pro ving inappro ximabilit y prop erties of some imp ortan t onstrain t satisfation problems o v er t w o v ariables. W e need the follo wing problem only for stating the onjeture: Denition 15. The Unique L ab el Cover pr oblem L ( V , W, E , [ M ] , { π v, w } ( v, w ) ∈ E ) is the fol lowing pr oblem: Given is a bip artite gr aph with left side verti es V , right side verti es W , and a set of e dges E . The go al is to assign one `lab el' to every vertex of the gr aph, wher e [ M ] is the set of al lowe d lab els. The lab el ling is sup- p ose d to satisfy  ertain  onstr aints given by bije tive maps σ v, w : [ M ] → [ M ] . Ther e is one suh map for every e dge ( v , w ) ∈ E . A lab el ling `satises' an e dge ( v , w ) if σ v, w (lab el( w )) = label( v ) . The optimum of the unique lab el  over pr ob- lem is dene d to b e the maximum fr ation of e dges satise d by any lab el ling. No w, UGC is the follo wing: Conje tur e 16 (Unique Games Conje tur e). F or an y η , γ > 0 , there exists a onstan t M = M ( η , γ ) su h that it is NP -hard to distinguish whether the Unique Lab el Co v er problem with lab el set of size M has optim um at least 1 − η or at most γ . F rom hereon w e assume that UGC is true, whi h giv es us the follo wing inap- pro ximabilit y results: Theorem 17 (Khot et al. [25℄).  F or every ε > 0 , it is NP-har d to appr oximate mc 2 within α GW + ε .  It is NP-har d to appr oximate mc k within (1 − 1 /k +(2 ln k ) /k 2 + O ((ln ln k ) /k 2 )) . 3.1 Sparse Graphs In this setion, w e in v estigate the p erformane of our metho d on graphs whi h ha v e relativ ely few edges, and w e see that the girth of the graphs pla ys a en tral role. The girth of a graph is the length of a shortest yle on tained in the graph. Similarly , the o dd girth of a graph giv es the length of a shortest o dd yle in the graph. Before w e pro eed w e need some fats ab out yle graphs. Note that the o dd yles form a  hain in the lattie C S b et w een K 2 and C 3 = K 3 in the follo wing w a y: K 2 → · · · → C 2 i +1 → C 2 i − 1 → · · · → C 3 = K 3 . The follo wing lemma giv es the v alues of s ( M , N ) for pairs of graphs in this  hain. The v alue dep ends only on the target graph of the homomorphism. Lemma 18. L et k < m b e p ositive, o dd inte gers. Then, s ( K 2 , C k ) = s ( C m , C k ) = k − 1 k . Pr o of. Note that C 2 k +1 6→ K 2 and C 2 k +1 6→ C 2 m +1 . Ho w ev er, after remo v- ing one edge from C 2 k +1 , the remaining subgraph is isomorphi to the path P 2 k +1 whi h in turn is em b eddable in b oth K 2 and C 2 m +1 . Sine C 2 k +1 is edge- transitiv e, the result follo ws from Lemma 8. ⊓ ⊔ With Lemma 18 at hand, w e an pro v e the follo wing: Prop osition 19. L et k ≥ 3 b e o dd. Then, F J 2 ( C k ) ≥ k − 1 k · α GW and Hå ( C k ) = 2 k + c k 2 log k − 2 c k 3 log k . F urthermor e, mc C k  annot b e appr oximate d within k k − 1 · α GW + ε for any ε > 0 . Pr o of. F rom Lemma 18 w e see that s ( K 2 , C k ) = k − 1 k whi h implies (using Lemma 5) that F J 2 ( C k ) ≥ k − 1 k · α GW . F urthermore, mc 2 annot b e appro x- imated within α GW + ε ′ for an y ε ′ > 0 . F rom the seond part of Lemma 5, w e get that mc C k annot b e appro ximated within k k − 1 · ( α GW + ε ′ ) for an y ε ′ . With ε ′ = ε · k − 1 k the result follo ws. Finally , w e see that Hå ( C k ) = 1 − k 2 − 2 k k 2 ·  1 − c k 2 log k  = ck + 2 k 2 log k − 2 c k 3 log k = = 2 k + c k 2 log k − 2 c k 3 log k . Håstad's algorithm do es not p erform partiularly w ell on sparse graphs; this is reeted b y its p erformane on yle graphs C k where the appro ximation guaran tee tends to zero when k → ∞ . W e will see that this trend is apparen t for all graph t yp es studied in this setion. No w w e an on tin ue with a result on a lass of graphs with large girth: Prop osition 20. L et m > k ≥ 4 . If H is a gr aph with o dd girth g ≥ 2 k + 1 and minimum de gr e e ≥ 2 m − 1 2( k +1) , then F J 2 ( H ) ≥ 2 k 2 k +1 · α GW and mc H  annot b e appr oximate d within 2 k +1 2 k · α GW + ε for any ε > 0 . Asymptoti al ly, Hå ( H ) is b ounde d by c n 2 log n + 2( n g/ ( g − 1) ) 3 n 4 n 1 / ( g − 1) − 2 n g/ ( g − 1) n 1 / ( g − 1) c n 4 log n , wher e n = v ( H ) . Pr o of. Lai & Liu [26℄ ha v e pro v ed that if H is a graph with o dd girth ≥ 2 k + 1 and minim um degree ≥ 2 m − 1 2( k +1) , then there exists a homomorphism from H to C 2 k +1 . Th us, K 2 → H → C 2 k +1 whi h implies that 1 − d ( K 2 , H ) ≥ 1 − d ( K 2 , C 2 k +1 ) = 2 k 2 k +1 . By Lemma 5, F J 2 ( H ) ≥ 2 k 2 k +1 · α GW , but mc H annot b e appro ximated within 2 k +1 2 k · α GW + ε for an y ε > 0 . Dutton and Brigham [10℄ sho w that one upp er b ound on e ( H ) has asymptoti order n 1+2 / ( g − 1) . This lets us sa y that Hå ( H ) ∼ 1 − n 2 − 2 · n 1+2 / ( g − 1) n 2 ·  1 − c n 2 log n  = = cn 2 + 2 n (3 g − 1) / ( g − 1) log n − 2 n ( g +1) / ( g − 1) c n 4 log n = = c n 2 log n + 2( n g/ ( g − 1) ) 3 n 4 n 1 / ( g − 1) − 2 n g/ ( g − 1) n 1 / ( g − 1) c n 4 log n . ⊓ ⊔ If w e restrit ourselv es to planar graphs, then it is p ossible to sho w the follo wing: Prop osition 21. L et H b e a planar gr aph with girth at le ast g = 20 k − 2 3 . If v ( H ) = n , then F J 2 ( H ) ≥ 2 k 2 k +1 · α GW and Hå ( H ) ≤ 6 n − 12 n 2 + c n 2 log n − 6 c n 3 log n + 12 c n 4 log n . mc H  annot b e appr oximate d within 2 k +1 2 k · α GW + ε for any ε > 0 . Pr o of. Boro din et al. [7℄ ha v e pro v ed that H is (2 + 1 k ) -olourable whi h is equiv alen t to sa ying that there exists a homomorphism from H to C 2 k +1 . The pro of pro eeds as for Prop osition 20. The planar graph H annot ha v e more than 3 n − 6 edges so Hå ( H ) is b ounded from ab o v e b y 1 − n 2 − 2(3 n − 6) n 2 ·  1 − c n 2 log n  = = cn 2 − 6 n c + 1 2 c + 6 n 3 log n − 12 n 2 log n n 4 log n = = 6 n − 12 n 2 + c n 2 log n − 6 c n 3 log n + 12 c n 4 log n . (In fat, H on tains no more than max { g ( n − 2) / ( g − 2) , n − 1 } edges, but using this only mak es for a more on v oluted expression to study .) ⊓ ⊔ Prop osition 21 an b e strengthened and extended in dieren t w a ys: one is to onsider a result b y Dv o°ák et al. [11℄. They ha v e pro v ed that ev ery planar graph H of o dd-girth at least 9 is homomorphi to the P etersen graph P . The P etersen graph is edge-transitiv e and it is kno wn (f. [3℄) that the bipartite densit y of P is 4 / 5 or, in other w ords, s ( K 2 , P ) = 4 / 5 . Consequen tly , mc H an b e appro ximated within 4 5 · α GW but not within 4 5 · α GW + ε for an y ε > 0 . This is b etter than Prop osition 21 for planar graphs with girth stritly less than 13. Another w a y of extending Prop osition 21 is to onsider graphs em b eddable on higher-gen us surfaes. F or instane, the lemma is true for graphs em b eddable on the pro jetiv e plane, and it is also true for graphs of girth stritly greater than 20 k − 2 3 whenev er the graphs are em b eddable on the torus or Klein b ottle. These b ounds are diret onsequenes of results in Boro din et al. W e onlude the setion b y lo oking at a lass of graphs that ha v e small girth. Let 0 < β < 1 , b e the appro ximation threshold for mc 3 , i.e. mc 3 is appro ximable within β but not within β + ε for an y ε > 0 . Curren tly , w e kno w that α 3 ≤ 0 . 836008 ≤ β ≤ 102 103 [9, 22℄. The wheel graphs W k from Example 9 are homomorphially equiv alen t to K 3 for o dd k and w e onlude (b y Lemma 5) that mc W k has the same appro ximabilit y prop erties as mc 3 in this ase. F or ev en k ≥ 6 , W k is not homomorphially equiv alen t to K 3 , though. Prop osition 22. F or k ≥ 6 and even, F J 3 ( W k ) ≥ α 3 · 2 k − 3 2 k − 2 but mc W k is not appr oximable within β · 2 k − 2 2 k − 3 . Hå ( W k ) = 4 k − 4 k 2 + c k 2 log k − 4 c k 3 log k + 4 c k 4 log k . Pr o of. W e kno w from Example 9 that K 3 → W k and s ( K 3 , W k ) = 2 k − 3 2 k − 2 . The rst part of the result follo ws b y an appliation of Lemma 5. Hå ( W k ) = 1 − t ( W k ) d 2 ·  1 − c d 2 log d  = /d = k, e ( W k ) = 2( k − 1 ) / = = 1 − k 2 − 4( k − 1) k 2 ·  1 − c k 2 log k  = = k 2 c + 4 k 3 log k − 4 kc − 4 k 2 log k + 4 c k 4 log k = = 4 k − 4 k 2 + c k 2 log k − 4 c k 3 log k + 4 c k 4 log k ⊓ ⊔ W e see that F J 3 ( W k ) → α 3 when k → ∞ , while Hå ( W k ) tends to 0. 3.2 Dense and Random Graphs W e will no w study dense graphs, i.e. graphs H on taining Θ ( v ( H ) 2 ) edges. F or a graph H on n v erties, w e ob viously ha v e H → K n . If w e assume that ω ( H ) ≥ r , then w e also ha v e K r → H . Th us, if w e ould determine s ( K r , K n ) , then w e ould use Lemma 11 to alulate a b ound on F J n ( H ) . Let ω ( G ) denote the size of the largest lique in G and χ ( G ) denote the  hro- mati n um b er of G . The T urán graph T ( n, r ) is a graph formed b y partitioning a set of n v erties in to r subsets, with sizes as equal as p ossible, and onneting t w o v erties whenev er they b elong to dieren t subsets. T urán graphs ha v e the follo wing prop erties [31℄:  e ( T ( n, r )) = ⌊  1 − 1 r  · n 2 2 ⌋ ;  ω ( T ( n, r )) = χ ( T ( n, r )) = r ;  if G is a graph su h that e ( G ) > e ( T ( v ( G ) , r )) , then ω ( G ) > r . Lemma 23. L et r and n b e p ositive inte gers. Then, s ( K r , K n ) = e ( T ( n, r )) /e ( K n ) Pr o of. Sine K n is edge-transitiv e, it sues to sho w that mc r ( K n , 1 /e ( K n )) = e ( T ( n, r )) /e ( K n ) . Assume mc r ( K n , 1 /e ( K n )) · e ( K n ) > e ( T ( n, r )) . This implies that there exists an r -partite graph G on k v erties with stritly more than e ( T ( n, r )) edges  this is imp ossible sine ω ( G ) > r and, onsequen tly , χ ( G ) > r . Th us, mc K r ( K n , 1 /e ( K n )) · e ( K n ) = e ( T ( n, r )) b eause T ( n, r ) is an r -partite subgraph of K n . ⊓ ⊔ No w, w e are ready to pro v e the follo wing: Prop osition 24. L et v ( H ) = n and pik r ∈ N , σ ∈ R suh that  1 − 1 r  · n 2 2  ≤ σ · n 2 = e ( H ) ≤ n ( n − 1) 2 . Then, F J n ( H ) ≥ α n · 2 j  1 − 1 r  · n 2 2 k n · ( n − 1) ∼ 1 − 1 r − 1 n + 2 ln n n ( n − 1) Hå ( H ) = 2 σ + c n 2 log n − 2 σ · c n 2 log n . Pr o of. W e ha v e K r → H due to T urán and H → K n holds trivially sine v ( H ) = n . By Lemma 23 s ( K r , K n ) = 2 j  1 − 1 r  · n 2 2 k n · ( n − 1 ) . The rst part of the result follo ws from Lemma 5 sine d ( H, K n ) ≤ d ( K r , K n ) = 1 − s ( K r , K n ) and some straigh tforw ard alulations. Hå ( H ) = 1 − n 2 − σ · n 2 n 2 ·  1 − c n 2 log n  = = c + 2 σ · n 2 log n − 2 σ · c n 2 log n = c n 2 log n + 2 σ − 2 σ · c n 2 log n . ⊓ ⊔ Note that when r and n gro w, F J n ( H ) tends to 1 . This means that, asymptoti- ally , w e annot do m u h b etter. If w e ompare the expression for F J n ( H ) with the inappro ximabilit y b ound for mc n (Theorem 17), w e see that all w e ould hop e for is a faster on v ergene to w ards 1 . As σ satises  1 − 1 r  · 1 2 ≤ σ ≤  1 − 1 n  · 1 2 , w e onlude that Hå ( H ) also tends to 1 as r and n gro w. T o get a b etter grip on ho w Hå ( H ) b eha v es w e lo ok at t w o extreme ases. F or a maximal σ =  1 − 1 r  · 1 2 , Hå ( H ) b eomes 1 − 1 n + c n 3 log n . On the other hand, this guaran tee, for a minimal σ =  1 − 1 r  · 1 2 is 1 − 1 r + c rn 2 log n . A t the same time, it is easy to see that F rieze and Jerrum's algorithm mak es these p oin ts appro ximable within α n (sine, in this ase, H ≡ K n ) and α r (sine T urán's theorem tells us that H → K r holds in this ase), resp etiv ely . Our onlusion is that F rieze and Jerrum's and Håstad's algorithms p erform almost equally w ell on these graphs asymptotially . Another w a y to study dense graphs is via random graphs. Let G ( n, p ) denote the random graph on n v erties in whi h ev ery edge is  hosen randomly and indep enden tly with probabilit y p = p ( n ) . W e sa y that G ( n, p ) has a prop ert y A asymptoti al ly almost sur ely (a.a.s.) if the probabilit y it satises A tends to 1 as n tends to innit y . Here, w e let p = c for some 0 < c < 1 . F or G ∈ G ( n, p ) it is w ell kno wn that a.a.s. ω ( G ) assumes one of at most t w o v alues around 2 ln n ln(1 /p ) [5, 30℄. It is also kno wn that, almost surely χ ( G ) ∼ n 2 ln( np ) ln  1 1 − p  , as np → ∞ [4, 29℄. Let us sa y that χ ( G ) is onen trated in width s if there exists u = u ( n, p ) su h that a.a.s. u ≤ χ ( G ) ≤ u + s . Alon and Kriv elevi h [2℄ ha v e sho wn that for ev ery onstan t δ > 0 , if p = n − 1 / 2 − δ then χ ( G ) is onen trated in width s = 1 . That is, almost surely , the  hromati n um b er tak es one of t w o v alues. Prop osition 25. L et H ∈ G ( n, p ) . When np → ∞ , F J m ( H ) ∼ 1 − 2 m + 2 ln m m 2 + 1 m 2 − 2 ln m m 3 , wher e m = ω ( H ) . Hå ( H ) = p − p n + (1 − p ) · c n 2 log n + pc n 3 log n . Pr o of. Let k = χ ( H ) . F J m ( H ) ≥ α m · s ( K m , K k ) ∼  1 − 1 m + 2 ln m m 2  · 2 j  1 − 1 m  · k 2 2 k k ( k − 1 ) ∼ ∼ ( m 2 − m + 2 ln m )( m − 1) k m 3 ( k − 1) = = k k − 1 − 2 k m ( k − 1) + k m 2 ( k − 1) + 2 k ln m m 2 ( k − 1) − 2 k ln m m 3 ( k − 1) ( ∗∗ ) If n is large, then k ≫ m and ( ∗∗ ) ∼ 1 − 2 m + 2 ln m m 2 + 1 m 2 − 2 ln m m 3 . The exp eted n um b er of edges for a graph H ∈ G ( n, p ) is  n 2  p , so Hå ( H ) = 1 − t ( G ) d 2 · (1 − c d 2 log d ) = /d = n, e ( G ) =  n 2  p/ = = 1 − n 2 − ( n 2 − n ) p n 2 · (1 − c n 2 log n ) = 1 − n − pn + p n · (1 − c n 2 log n ) = = 1 − (1 − p + p n ) · (1 − c n 2 log n ) = = pn 3 log n + nc − pnc − pn 2 log n + pc n 3 log n = = p − p n + (1 − p ) · c n 2 log n + pc n 3 log n ⊓ ⊔ W e see that, in the limiting ase, Hå ( H ) tends to p , while F J m ( H ) tends to 1 . Again, this means that, for large enough graphs, w e annot do m u h b etter. With a b etter analysis, one ould p ossibly rea h an expression for F J m ( H ) that has a faster on v ergene rate. Of ourse, it is in teresting to lo ok at what happ ens for graphs H ∈ G ( n, p ) where np do es not tend to ∞ when n → ∞ . The follo wing theorem lets us do this. Theorem 26 (Erd®s and Rén yi [13℄). L et c b e a p ositive  onstant and p = c n . If c < 1 , then a.a.s. no  omp onent in G ( n, p )  ontains mor e than one yle, and no  omp onent has mor e than ln n c − 1 − ln c verti es. No w w e see that if np → ε when n → ∞ and 0 < ε < 1 , then G ( n, p ) almost surely onsists of omp onen ts with at most one yle. Th us, ea h omp onen t resem bles a yle where, p ossibly , trees are atta hed to ertain yle v erties, and ea h omp onen t is homomorphially equiv alen t to the yle it on tains. Sine w e kno w from Setion 3.1 that F rieze and Jerrum's algorithm p erforms b etter than Håstads algorithm on yle graphs, it follo ws that the same relationship holds in this part of the G ( n, p ) sp etrum. 4 Conlusions and Op en Problems W e ha v e seen that applying F rieze and Jerrum's algorithm to Max H -Col giv es omparable to or b etter results than when applying Håstad's Max 2-Csp al- gorithm for the lasses of graphs w e ha v e onsidered. One p ossible explanation for this is that the analysis of the Max 2-Csp algorithm only aims to pro v e it b etter than a random solution on exp etation, whi h ma y lea v e ro om for strengthenings of the appro ximation guaran tee. A t the same time, w e are proba- bly o v erestimating the distane b et w een the graphs. It is lik ely that b oth results an b e impro v ed. Kap oris et al. [23℄ ha v e sho wn that mc 2 is appro ximable within . 952 for an y giv en a v erage degree d and asymptotially almost all random graphs G in G ( n, m =  d 2 n  ) , where G ( n, m ) is the probabilit y spae of random graphs on n v erties and m edges seleted uniformly at random. In a similar v ein, Co ja- Oghlan et al. [8℄ giv e an algorithm that appro ximates mc k within 1 − O (1 / √ np ) in exp eted p olynomial time, for graphs from G ( n, p ) . It w ould b e in teresting to kno w if these results ould b e arried further, to other graphs G , so that b etter appro ximabilit y b ounds on Max H -Col , for H su h that G → H , ould b e a hiev ed. Erd®s [12℄ has pro v ed that for an y p ositiv e in tegers k and l there exists a graph of  hromati n um b er k and girth at least l . It is ob vious that su h graphs annot b e sandwi hed b et w een K 2 and a yle as w as the ase of the graphs of high girth in Setion 3.1. A dieren t idea is th us required to deal with these graphs. In general, to apply our metho d more preisely , w e need a b etter understanding of the struture of C S and ho w this in terats with our metri d . The idea of dening a metri on a spae of problems whi h relates their appro ximabilit y an b e extended to more general ases. 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