Lecture 23 Greedy Strategy. What is a submodular function?. Consider a function f on all subsets of a set E . f is submodular if. Set-Cover.
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Consider a function f on all subsets of a set E.
f is submodular if
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’ .
Actually, this inequality holds if and only if f is submodular and
Greedy Algorithm produces an approximation within ln n +1 from optimal.
The same result holds for 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’ .
For any edge set E, define fi(E) to be the number of connected components of the subgraph of (X,E), induced by Xi.
Theorem Subset Interconnection Design has a (1+ln m)-approximation.
r1(e)+ּּּ+rm(e)<m for any edge
= # of edges covered by both A and B
> p(A ∩ B)
Given a vertex-weighted connected graph,
find a connected vertex-cover with minimum
Theorem Weighted Connected Vertex-Cover
has a (1+ln Δ)-approximation.
This is the best-possible!!!