1. APPROXIMATION ALGORITHMS VERTEX COVER – MAX CUT PROBLEMS
SELIM KALAYCI
FIU-SCS
04/13/2005

2. Motivation
Some optimization problems are “NP-hard” (by hardness of related decision problem), there is no poly-time algorithm unless P = NP.
• largest clique
• smallest vertex cover
• largest maximum cut
• . . .
But: sometimes sub-optimal solutions are kind of OK
• pretty large clique
• pretty small vertex cover
• pretty large maximum cut
if algorithms run in poly time (preferably small exponents).

3. APPROXIMATION ALGORITHMS
Definition:Approximation algorithm
An approximation algorithm for a problem is a polynomial-time algorithm that, when given input I, outputs an element of FS(I).
Feasible solution set
A feasible solution is an object of the right type but not necessarily an optimal one.
FS(I) is the set of feasible solutions for I.

4. APPROXIMATION RATIO 1 ≤ Max (O/O*, O*/O) ≤ ρ(n)
Approximation algorithm has approximation ratio of ρ(n), if for any input of size n, the outcome O of its solution is within factor ρ(n) of outcome of optimal solution O*, i.e.
1 ≤ Max (O/O*, O*/O) ≤ ρ(n)
Minimization
problem
Maximization
problem

5. APPROXIMATION RATIO
An algorithm with guaranteed approximation ratio of ρ(n) is called a ρ(n)-approximation algorithm.
A 1-approximation algorithm is optimal, and the larger the ratio, the worse the solution.
• For many NP-complete problems, constant-factor approximations(i.e. computed clique is always at least half the size of maximum-size clique),
• Sometimes best known approx ratio grows with n,

6. VERTEX COVER PROBLEM
Problem: Given graph G = (V,E), find smallest s.t. if (u, v) E, then u V′ or v V′ or both.
Decision problem is NP-complete (3SAT ≤p VC, Sipser 7.5)
Optimization problem is at least as hard.

7. VERTEX COVER ALGORITHM
Here is a trivial 2-approximation algorithm
Input is some graph G = (V,E).
APPROX-VERTEX-COVER
1: C ← Ø ;
2: E′ ← E
3: while E′ ≠ Ø; do
4: let (u, v) be an arbitrary edge of E′
5: C ← C {(u, v)}
6: remove from E′ all edges incident on either u or v
7: end while

8. VERTEX COVER EXAMPLE Input Graph Optimal result, Size 3 a b c e d f g

9. VECTOR COVER ALGORITHM EXAMPLEb c e d f g a b c e d f g
Step 1: choose edge (c,e)
Step 2: choose edge (d,g) a b c e d f g a b c e d f g
Step 3: choose edge (a,b)
Result: Size 6

10. PROOF Theorem. APPROX-VERTEX-COVER is a poly-time
2-approximation algorithm.
Proof. The running time is trivially bounded by O(V * E)
(at most |E| iterations, each of complexity at most O(V )).
Correctness: C clearly is a vertex cover.

11. PROOF cont.
Size of the cover: let A denote set of edges that are picked
({(c, e), (d, g), (a, b)} in example).
1. In order to cover edges in A, any vertex cover, in particular an optimal cover C*, must include at least one endpoint of each edge in A.
By construction of the algorithm, no two edges in A share an endpoint (once edge is picked, all edges incident on either endpoint are removed).
Therefore, no two edges in A are covered by the same vertex in C*, and |C*| ≥ |A|.
2. When an edge is picked, neither endpoint is already in C, thus
|C| = 2|A|.
Combining (1) and (2) yields
|C| = 2|A| ≤ 2|C*|
q.e.d

12. MAX CUT PROBLEM Given an undirected graph G (V,E):
Separate vertices of G into two disjoint sets S and T, such that number of cut edges is maximum. (Maximization Problem)
Cut edge: An edge between a node in S and a node in T.

13. MAX-CUT ALGORITHM Another 2-approximation algorithm : Given G (V,E)
1- S ← Ø and T ← V
2- If moving a single node from S to T, or from T to S increases the number of cut edges, make the move. Repeat 2.
3- Output the number of cut edges.

14. QUESTIONS