Chapter 2 Greedy Strategy II Submodular function DingZhu

Chapter 2. Greedy Strategy II. 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

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

Greedy Algorithm

Analysis

Analysis

What’s we need?

Actually, this inequality holds if and only if f is submodular and (monotone increasing)

Meaning of Submodular • The earlier, the better! • Monotone decreasing gain!

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 totalweight subcollection C’ of C such that every element of E appears in a subset in C’.

Greedy Algorithm

A General Problem

Greedy Algorithm

A General Theorem Remark:

Proof

1 2 3

zek Ze 2 ze 1

Subset Interconnection Design • Given m subsets X 1, …, 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 r 1++rm Theorem Subset Interconnection Design has a (1+ln m)-approximation. r 1(Φ)++rm(Φ)=0 r 1(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 minimum total weight. Theorem Weighted Connected Vertex-Cover has a (1+ln Δ)-approximation. This is the best-possible!!!

Thanks, End