Approximation Algorithms for Graph Homomorphism Problems Chaitanya Swamy University of Waterloo...

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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 }|

Value of mapping

•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)

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

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

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.

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 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.

Approximation Algorithms for Graph Homomorphism Problems Chaitanya Swamy University of Waterloo...

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