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II. Submodular function

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’ .

What is a submodular function?

Consider a function f on all subsets of a set E.

f is submodular if

for all subsets A, B of E.

{ Use f(C) as the potential function}

Greedy Algorithm produces an approximation within ln +1 from optimal,

where = max S C |S| .

This inequality holds if and only if f is submodular and

(monotone increasing)

- The earlier, the better!
- Monotone decreasing gain!

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

The same result holds for min 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’ .

Remark:

The last inequality follows from the

submodularity of f .

- Given a vertex-weighted graph G = (V,E), find a vertex cover of the min weight.
- Potential Function: f(A) = the number of edges covered by vertices in A.
- f is submodular:

f(A) + f(B) – f(A U B) is the number of edges covered by both A and B, and hence >f(A ∩B).

- Given a vertex-weighted hypergraph H = (V,C), find a hitting set A of the min total weight.

(In a hypergraphH = (V,C), C is a collection of subsets

of V. Also, a hitting set is a subset of V that

intersects each set in C.)

- f(A) [= the number of subsets in C that intersects A] is submodular.

Both Min-WHS and Min-WVC have polynomial time (ln + 1)-approximation, where is the max degree of the input (hyper)graph.

- For nonweighted cases, Min-VC has a polynomial-time 2-approx, and it could be generalized to Min-WVC!! (see Chap 8)

- 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.

Weighted version

Minimize the total edge weight in G.

What is the potential function ?

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

Function– fi is submodular.

Consider two edge sets A and B.

Observe that if an edge in A connects two connected

components of G [B] into one component of G [AUB],

Then this edge also connects two components of

G [AB into one component of G [A].

So, f (B) – f (A U B)

= # of components of G [B] reduced by edges in A

< # of components of G [AB] reduced by edges in A

= f (AB) – f (A).

- 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 riis submodular.

In fact, the rank of any matroid is submodular.

Proof. Consider two independent subsets A, B.

Let I 1 be a maximal indep set in AB, I 2 a

maximal indep set in A that contains I 1, and

I 3 a maximal indep set in AUB that contains I 2.

Since (E,C) is a matroid,

|I 1|=rank(AB), |I 2|=rank(A), |I 3|=rank(AUB).

Let J = I 3 – I 2. From the maximality of I 2, we

know that J is a subset of B. Therefore, I 1UJ

is an indep set in B, |I 1|+|J| < rank(B);

or,

rank(AB) +rank(AUB) – rank(A) < rank(B).

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

r1(Φ)+ּּּ+rm(Φ)=0

r1(e)+ּּּ+rm(e)<m for any edge

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

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

For any vertex subset A, h(A) is the number of connected components of the subgraph induced by A.

– g is submodular (same as in Min-WVC)

– ug(C) = |N(u) – C|= number of neighbors

of u that are not in C

- – u h(C) = [number of connected components in G|C subgraph induced by C that are adjacent to u] – 1
- In the following, A a subset of B but

– u h(A) < – u h(B)

A B

We prove this by a new characterization of submodularity:

f is submodular (and monotone increasing)

iff for all A, x, y,

where =

p(C) = – g(C) – h(C): v up(C) <0

Case 1. u = v.

- v up(C) <0 iff up(C) >0
- If u is not adjacent to any x in C, then

– ug(C) = deg(u). – uh(C) = –1

and so up(C) >0 because deg(u) > 1.

- If u is adjacent to some x in C, then

– uh(C) >0

Case 2. u is not adjacent to v

Case 3. u is adjacent to v

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

–g(Φ) = –|E|, – h(Φ) = 0.

|E|– g(x) – h(x) <Δ – 1

Δ is the maximum degree.

- Connected Vertex-Cover has a 3-approximation.

Need at most |M|-1 vertices to connect a

matching M.

Weighted Connected Vertex-Cover

Given a vertex-weighted connected graph,

find a connected vertex-cover with minimum

total weight.

TheoremWeighted Connected Vertex-Cover

has a (1+ln Δ)-approximation.

This is the best-possible!!!

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