1. Approximation Algorithms for Graph Homomorphism Problems Chaitanya Swamy University of Waterloo Joint work with Michael Langberg and Yuval Rabani Open University Technion

2. Maximum Graph Homomorphism Given: graphs G = (V G,E G ) and H = (V H,E H ) Value of mapping  find a mapping   : V G  V H that maximizes no. of edges of G mapped to edges of H Goal: Maximize |{ (u,v)  E G : (  (u),  (v) )  E H }| G H

3. Maximum Graph Homomorphism Given: graphs G = (V G,E G ) and H = (V H,E H ) Value of mapping  find a mapping   : V G  V H that maximizes no. of edges of G mapped to edges of H Goal: Maximize |{ (u,v)  E G : (  (u),  (v) )  E H }| G H 

4. Maximum Graph Homomorphism Given: graphs G = (V G,E G ) and H = (V H,E H ) Value of mapping  find a mapping   : V G  V H that maximizes no. of edges of G mapped to edges of H Goal: Maximize |{ (u,v)  E G : (  (u),  (v) )  E H }| G H 

5. H = : Max-Cut problem G

6.  Problem is NP-hard, APX-hard even for fixed H Optimization version of H-coloring: decide if there is a mapping  of value |E G | (such a  homomorphism) e.g., when H is a k-clique, H-coloring  k-coloring problem, maximum graph homomorphism (MGH)  Max-k-Cut H-coloring is NP-complete if H is not bipartite and does not contain a self-loop (Hell & Nesetril) G

7. Related Work MGH problem appears to be new. H-coloring: well studied problem; Hell & Nesetril proved that H-coloring is either in P or is NP-complete –restrictive/list H-coloring: various restrictions placed on , e.g., restrictions on {  (u)} for u  V G, or  - 1 (i) for i  V H –counting versions of these problems: Dyer & Greenhill proved a dichotomy theorem for counting # of H-colorings –sampling a random H-coloring Minimum cost homomorphism: find  that minimizes (cost of assigning labels to nodes) + (weights of images of E G ); studied by Cohen et al., Gutin et al., Aggarwal et al. –if edge weights in H form a metric, this is the metric labeling problem (Kleinberg & Tardos)

8. Related Work (contd.) maximum common subgraph: given graphs G, H, find their largest common subgraph  essentially MGH where  is required to be one-one MGH can be reduced to this problem: – blow up each i  V H to an independent set of size |V G | – replace each edge (i,j)  E H by complete bipartite graph G H

9. Related Work (contd.) maximum common subgraph: given graphs G, H, find their largest common subgraph  essentially MGH where  is required to be one-one MGH can be reduced to this problem: – blow up each i  V H to an independent set of size |V G | – replace each edge (i,j)  E H by complete bipartite graph G H Kann: (B+ 1 )-approx. when degrees in G, H are ≤ B.

10. A Trivial 0.5-approximation 1 ) Fix an edge (i,j) of H 2) Map each u  V G to i or j randomly with probability ½. G H G

11. A Trivial 0.5-approximation 1 ) Fix an edge (i,j) of H 2) Map each u  V G to i or j randomly with probability ½. Each edge of G is mapped to (i,j) with probability ½,  expected value of mapping = |E G |/2  get 0.5-approximation algorithm (can derandomize) H G OPT MGH (G,H) ≥ MaxCut(G) ≥ |E G |/2

12. More generally, for a subset N  V H define its density r(N) = ( 2|E(N)| ) / |N| 2 Mapping each u  V G randomly to a label in N maps r(N). |E G | edges of G in expectation  gives an r(N)-approximation algorithm e.g., if H has a triangle, get a 2/3-approximation if H has a k-clique, get a ( 1 – 1 /k)-approximation In general, factor of 0.5 might be the best possible!

13. Informal Statement of Result There is no (0.5+  )-approximation algorithm for MGH, unless certain average-case instances of subgraph isomomorphism can be solved in polynomial time. G n,p  distribution on n-vertex graphs where each edge is chosen independently with probability p Our average-case instances are related to G n,p Main question: how hard is subgraph isomomorphism on a pair of random graphs G  G n,p and H  G n,q where q >> p > ln(n)/n?

14. The Roadmap Main Lemma: If H is triangle-free with k nodes, and G  G n,p where p=c. ln(k)/n with n, c suitably large, then with high probability (over all G’s), OPT(G,H) ≤ ( 1 +  )|E G |/2 So, if G is a subgraph of H, OPT(G,H) = |E G | if G is not a subgraph of H, OPT(G,H) ≤ ( 1 +  )|E G |/2 whp. A (0.5+  )-approximation algorithm can be used to distinguish between these two cases Inapproximability result based on the assumption that this is hard when G, H are drawn from a suitable distribution on triangle-free graphs Formulate this precisely as a refutation problem factor 2 gap

15. The Refutation Problem Let  n,p = distribution on n-node  -free graphs obtained by taking G  G n,p, removing edges randomly till no  s remain Refutation problem: Find a poly-time algorithm that given G  n,p and H  n,q, where q >> p = c. ln(n)/n, (a) returns “yes” if G  H, (b) returns “no” with probability ≥ ½ [With very high probability G will not be a subgraph of H.] A (0.5+  )-approx. algorithm A yields a refutation algorithm: if G  H, then A (G,H) ≥ (0.5+  )|E G | otherwise, let G be obtained by removing edges from G’  G n,p OPT(G,H) ≤ OPT(G’,H) ≤ ( 1 +  )|E G’ |/2  ( 1 +  )c. n ln(n)/4 |E G’ |  c. n ln(n)/2 and (# of  ’s in G’) ≤ c 3. ln 3 (n)  n 1 /2 whp.  |E G | ≥ ( 1 –  )c. n ln(n)/2, A (G,H) ≤ OPT(G,H) ≤ ( 1 +4  )|E G |/2

16. Refutation Problem (contd.) Feige initiated the use of average-case complexity to prove hardness results, where average-case hardness translates to hardness of a refutation problem Can make refutation problem harder and more robust: require algorithm to say “yes” if G has a subgraph of size |E G |( 1 -  ) isomorphic to H How hard is the refutation problem? Open. But, local analysis does not work – return “yes” iff all “small” subgraphs of G are subgraphs of H. Also can make G have  (ln(n)/lnln(n)) girth. We set q >> p, to be “far” from graph isomorphism which is poly-time solvable for random graphs

17. Main Lemma and Proof Lemma: Let  ≤ 0.5. If H is triangle-free with k nodes, G  G n,p where p=c. ln(k)/n with n ≥ n 0 (  ), c ≥ c 0 (  ), then whp. (a) OPT(G,H) ≤ ( 1 +  )c. n ln(k)/4, (b) |E G | ≥ ( 1 –  )c. n ln(k)/2, so (c) OPT(G,H) ≤ ( 1 +4  )|E G |/2 Proof: (a) Fix a mapping . For a random G  G n,p, Value of  = V(  ) = ∑ (i,j)  E H ∑ u,v  V G :  (u)=i,  (v)=j X uv E[V(  )] = p. ∑ (i,j)  E H |  - 1 (i)| |  - 1 (j)|

18. Turan’s Theorem An n-node graph that is K r+ 1 -free has at most ( 1 - 1 /r). n 2 /2 edges. Corollary: Let H be a n-node graph that is K r+ 1 -free. Let w:V H  Z + be a wt. function such that ∑ i w i = n. Then, ∑ (i,j)  E H w i.w j ≤ ( 1 - 1 /r). n 2 /2 Proof: 1 1 2 2 H H’ Blow i  V H to independent set of size w i to get H’ H’ is also K r+ 1 -free – use Turan on H’

19. Main Lemma and Proof Lemma: Let  ≤ 0.5. If H is triangle-free with k nodes, G  G n,p where p=c. ln(k)/n with n ≥ n 0 (  ), c ≥ c 0 (  ), then whp. (a) OPT(G,H) ≤ ( 1 +  )c. n ln(k)/4, (b) |E G | ≥ ( 1 –  )c. n ln(k)/2, so (c) OPT(G,H) ≤ ( 1 +4  )|E G |/2 Proof: (a) Fix a mapping . For a random G  G n,p, Value of  = V(  ) = ∑ (i,j)  E H ∑ u,v  V G :  (u)=i,  (v)=j X uv E[V(  )] = p. ∑ (i,j)  E H |  - 1 (i)| |  - 1 (j)|

20. Main Lemma and Proof Lemma: Let  ≤ 0.5. If H is triangle-free with k nodes, G  G n,p where p=c. ln(k)/n with n ≥ n 0 (  ), c ≥ c 0 (  ), then whp. (a) OPT(G,H) ≤ ( 1 +  )c. n ln(k)/4, (b) |E G | ≥ ( 1 –  )c. n ln(k)/2, so (c) OPT(G,H) ≤ ( 1 +4  )|E G |/2 Proof: (a) Fix a mapping . For a random G  G n,p, Value of  = V(  ) = ∑ (i,j)  E H ∑ u,v  V G :  (u)=i,  (v)=j X uv E[V(  )] = p. ∑ (i,j)  E H |  - 1 (i)| |  - 1 (j)| ≤ p. n 2 /4(by Turan) V(  ) is sum of independent random variables, so Pr[V(  ) > ( 1 +  )E[V(  )]] ≤ e –O(n ln(k)) k n total mappings, so by union bound, whp. V(  ) ≤ ( 1 +  )c. n ln(k)/4 for all   OPT(G,H) ≤ ( 1 +  )c. n ln(k)/4 whp.

21. (b) E[|E G |] = p. n(n– 1 )/2  c. n ln(k)/2 By Chernoff bounds, |E G | ≥ ( 1 –  )c. n ln(k)/2 whp. (c) Therefore, OPT(G,H) ≤ ( 1 +4  )|E G |/2 Refutation problem: Find a poly-time algorithm that given G  n,p and H  n,q, where q >> p = c. ln(n)/n, (a) returns “yes” if G  H, (b) returns “no” with probability ≥ ½ A (0.5+  )-approx. algorithm yields an algorithm for the refutation problem

22. Other Results Can get an 0.5+  ( 1 /|V H | ln(|V H |))-approximation using SDP – gives improvements for any fixed H Prelabeled MGH: a partial labeling  ’:U  V H is also given and output  has to be an extension of  ’. Encodes the Multiway-Uncut problem: given G and terminal-set T  V G, partition V G into |T| parts with terminal in each part, to maximize (# uncut edges) Here H is |T|-self loops,  ’:T  V H is a bijection Get a.8535-approx. using LP rounding.

23. Open Questions Hardness of refutation problem: is subgraph isomorphism solvable in polynomial time when G  G n,p and H  G n,q ? Dense instances: G has  (n 2 ) edges, H is arbitrary; can one get a PTAS? Can get a quasi-PTAS and a PTAS for Max-k-Cut and in general when H is vertex-transitive Directed setting: improve upon trivial 0.25-approx. Encodes Max-Acyclic-Subgraph (nothing better than 0.5 known). Prelabeled MGH: improve upon 1 /3-approximation.

24. Thank You.