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

Approximation Algorithms for Graph Homomorphism

ProblemsChaitanya Swamy

University of Waterloo

Joint work with Michael Langberg and Yuval

RabaniOpen University Technion

Maximum Graph Homomorphism

Given: graphs G = (VG,EG) and H = (VH,EH)

Value of mapping

find a mapping : VGVH that

maximizes no. of edges of G mapped to edges of H

Goal: Maximize |{(u,v)EG : ((u),(v))EH }|

G H



Value of mapping




G H



Value of mapping




G H

•H = : Max-Cut problem

G

•H = : Max-Cut problem

Problem is NP-hard, APX-hard even for fixed H

•Optimization version of H-coloring: decide if there is a mapping of value |EG| (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

Related WorkMGH 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 uVG, or -1(i) for iVH

– 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 EG); 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)

Related Work (contd.)• maximum common subgraph: given graphs G, H,

find their largest common subgraph essentially MGH where is required to be one-oneMGH can be reduced to this problem:– blow up each iVH to an independent set of size |VG|

– replace each edge (i,j)EH by complete bipartite graph

G H

Related Work (contd.)• maximum common subgraph: given graphs G, H,

find their largest common subgraph essentially MGH where is required to be one-oneMGH can be reduced to this problem:– blow up each iVH to an independent set of size |VG|

– replace each edge (i,j)EH by complete bipartite graph

G H

Kann: (B+1)-approx. when degrees in G, H are ≤ B.

A Trivial 0.5-approximation

1) Fix an edge (i,j) of H2) Map each uVG to i or j randomly with probability ½.

HG

A Trivial 0.5-approximation

1) Fix an edge (i,j) of H2) Map each uVG to i or j randomly with probability ½.

Each edge of G is mapped to (i,j) with probability ½, expected value of mapping = |EG|/2

get 0.5-approximation algorithm (can derandomize)

HGOPTMGH(G,H)

≥ MaxCut(G)

≥ |EG|/2

More generally, for a subset N VH define its density r(N) = (2|E(N)|) / |N|2

Mapping each uVG randomly to a label in

N maps r(N).|EG| 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!

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.

Gn,p distribution on n-vertex graphs where each

edge is chosen independently with probability p

Our average-case instances are related to Gn,p

Main question: how hard is subgraph isomomorphism on a pair of random graphs GGn,p and HGn,q where q >> p > ln(n)/n?

The RoadmapMain Lemma: If H is triangle-free with k nodes, and

GGn,p where p=c.ln(k)/n with n, c suitably large, then with high probability (over all G’s), OPT(G,H) ≤ (1+)|EG|/2

So, if G is a subgraph of H, OPT(G,H) = |EG|

if G is not a subgraph of H, OPT(G,H) ≤ (1+)|EG|/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

The Refutation ProblemLet n,p = distribution on n-node -free graphs obtained bytaking GGn,p, removing edges randomly till no s remainRefutation problem: Find a poly-time algorithm that

given Gn,p and Hn,q, where q >> p = c.ln(n)/n,

(a) returns “yes” if GH, (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 GH, then A(G,H) ≥ (0.5+)|EG|

•otherwise, let G be obtained by removing edges from G’Gn,p

OPT(G,H) ≤ OPT(G’,H) ≤ (1+)|EG’|/2 (1+)c.n

ln(n)/4

|EG’| c.n ln(n)/2 and (# of ’s in G’) ≤

c3.ln3(n)n1/2 whp.

|EG| ≥ (1–)c.n ln(n)/2,

A(G,H) ≤ OPT(G,H) ≤ (1+4)|EG|/2

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 |EG|(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

Main Lemma and ProofLemma: Let ≤ 0.5. If

H is triangle-free with k nodes,

GGn,p where p=c.ln(k)/n with n ≥ n0(), c ≥ c0(), then whp.

(a) OPT(G,H) ≤ (1+)c.n ln(k)/4, (b) |EG| ≥ (1–

)c.n ln(k)/2, so

(c) OPT(G,H) ≤ (1+4)|EG|/2

Proof: (a) Fix a mapping . For a random GGn,p,

Value of = V() = ∑(i,j)EH ∑u,vVG :(u)=i, (v)=j Xuv

E[V()] = p.∑(i,j)EH |-1(i)| |-1(j)|

Turan’s Theorem

An n-node graph that is Kr+1-free has at most

(1-1/r).n2/2 edges.

Corollary: Let H be a n-node graph that is Kr+1-free. Let w:VHZ+ be a wt. function such that ∑i wi = n. Then, ∑(i,j)EH wi.wj ≤ (1-

1/r).n2/2

Proof:

1

1 22H H’

Blow iVH to independent set of size wi to get H’ H’ is also Kr+1-free – use Turan on H’




(a) OPT(G,H) ≤ (1+)c.n ln(k)/4, (b) |EG| ≥ (1–

)c.n ln(k)/2, so

(c) OPT(G,H) ≤ (1+4)|EG|/2



E[V()] = p.∑(i,j)EH |-1(i)| |-1(j)|




(a) OPT(G,H) ≤ (1+)c.n ln(k)/4, (b) |EG| ≥ (1–

)c.n ln(k)/2, so

(c) OPT(G,H) ≤ (1+4)|EG|/2



E[V()] = p.∑(i,j)EH |-1(i)| |-1(j)| ≤ p.n2/4 (by Turan)V() is sum of independent random variables, so

Pr[V() > (1+)E[V()]] ≤ e–O(n ln(k))

kn 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.

(b) E[|EG|] = p.n(n–1)/2 c.n ln(k)/2

By Chernoff bounds, |EG| ≥ (1–)c.n ln(k)/2 whp.

(c) Therefore, OPT(G,H) ≤ (1+4)|EG|/2

Refutation problem: Find a poly-time algorithm that given Gn,p and Hn,q,

where q >> p = c.ln(n)/n, (a) returns “yes” if GH, (b) returns “no” with probability ≥ ½

A (0.5+)-approx. algorithm yields an algorithm for the refutation problem

Other Results

• Can get an 0.5+(1/|VH| ln(|VH|))-approximation using SDP – gives improvements for any fixed H

• Prelabeled MGH: a partial labeling ’:UVH is also given and output has to be an extension of ’.Encodes the Multiway-Uncut problem: given G and terminal-set TVG, partition VG into |T| parts with terminal in each part, to maximize (# uncut edges)Here H is |T|-self loops, ’:TVH is a bijection

Get a .8535-approx. using LP rounding.

Open Questions

• Hardness of refutation problem: is subgraph isomorphism solvable in polynomial time when GGn,p and HGn,q?

• Dense instances: G has (n2) 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.

Thank You.

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