Chapter 2. Greedy StrategyII. Submodular function

Ding-Zhu Du

What is a submodular function?

Consider a function f on all subsets of a set E.f is submodular if

( ) ( ) ( ) ( )f A f B f A B f A B

Set-Cover

Given a collection C of subsets of a set E, find a minimum subcollection C’ of C such that every element of E appears in a subset in C’ .

Example of Submodular Function

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Meaning of Submodular

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Theorem

Greedy Algorithm produces an approximation within ln n +1 from optimal.

The same result holds for weighted set-cover.

Weighted Set Cover

Given a collection C of subsets of a set E and a weight function w on C, find a minimum total-weight subcollection C’ of C such that every element of E appears in a subset in C’ .

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Subset Interconnection Design

• Given m subsets X1, …, Xm of set X, find a graph G with vertex set X and minimum number of edges such that for every i=1, …, m, the subgraph G[Xi] induced by Xi is connected.

fi

For any edge set E, define fi(E) to be the number of connected components of the subgraph of (X,E), induced by Xi.

• Function -fi is submodular.

Rank

• All acyclic subgraphs form a matroid. • The rank of a subgraph is the cardinality of a

maximum independent subset of edges in the subgraph.

• Let Ei = {(u,v) in E | u, v in Xi}.• Rank ri(E)=ri(Ei)=|Xi|-fi(E).• Rank ri is sumodular.

Potential Function r1+ ּּּּּּּּּ+rm

Theorem Subset Interconnection Design has a (1+ln m)-approximation.

r1(Φ)+ ּּּּּּּּּ+rm(Φ)=0 r1(e)+ ּּּּּּּּּ+rm(e)<m for any edge

Connected Vertex-Cover

• Given a connected graph, find a minimum vertex-cover which induces a connected subgraph.

• For any vertex subset A, p(A) is the number of edges not covered by A.

• For any vertex subset A, q(A) is the number of connected component of the subgraph induced by A.

• -p is submodular.• -q is not submodular.

|E|-p(A)

• p(A)=|E|-p(A) is # of edges covered by A.• p(A)+p(B)-p(A U B) = # of edges covered by both A and B > p(A ∩ B)

-p-q

• -p-q is submodular.

Theorem

• Connected Vertex-Cover has a (1+ln Δ)-approximation.

• -p(Φ)=-|E|, -q(Φ)=0.• |E|-p(x)-q(x) < Δ-1• Δ is the maximum degree.

Theorem

• Connected Vertex-Cover has a 3-approximation.

Weighted Connected Vertex-Cover

Given a vertex-weighted connected graph,find a connected vertex-cover with minimumtotal weight.

Theorem Weighted Connected Vertex-Coverhas a (1+ln Δ)-approximation.

This is the best-possible!!!

Thanks, End