Advanced Approximation Algorithms II

1. Advanced Approximation AlgorithmsII How to find a heavy weightcut in a graph

2. The MAX CUT Problem • Given an undirectedgraph G=(V,E) with edgeweightsw:E->R, divide the vertices in two parts so that the sum of the weights of all edgesgoing from one part to the other, is maximized.

3. The MAX CUT Problem 5 8 8 3 11 9 12 1 5 11 7 2 13 4 9

4. MAX CUT vs. MIN CUT • MIN CUT is polynomial time solvable (cf. Max flow algorithms, and Karger’salgorithm) • MAX CUT is NP-hard, evenwhen the edgeweights are the same, and no vertex has morethanthreeneighbors. • Note that MAX CUT is trivial on bipartitegraphs.

5. Overview Wewillseetwopolynomial time techniques to obtain an approximative solution to the MAX CUT problem with a provableworstcase approximation ratio. • Approximation based on localsearchrefinement. • Approximation based on relaxation to semi-definiteprogramming.

6. LocalSearch • Singleflipneighborhood: As long as thereexists a vertex that willincrease the cutweightif it is put in the other part, move it.

7. LocalSearch 5 8 8 3 11 9 12 1 5 11 7 2 13 4 9

8. Analysis • The optimum can not be morethan the sum of all weights. • In the approximation found, eachvertexadds at leasthalf of itsneighborhood’sweight to the solution. We get approximation ratio apx/opt>= 0.5

9. Run Time Analysis? • There is no guarantee the number of flips, and hence the algorithmrun time, is polynomiallybounded! • Trick: Onlyflipifnetgain is >= 2ew(A,B)/n • We get 0.5-e approximation. • Eachflipimprove the weight at least a factor (1+e/n). After n/e flipswehave at leastdoubled the weight. • Weneed at most log2(OPT) doublings…

10. MathematicalPrograming • Formulate your problem as an integer program (IP). • Provide a relaxation of the IP to a linear program (LP) or semidefinite program (SDP), i.e. somethingweknowhow to solveefficiently. • Round the fractional solution to the LP or SDP to an integer solution approximating the original IP

11. How to Obtain an Approximation Guarantee OPT(SDP) Bound on Approximation factor OPT(IP) Actual approximation factor OPT(SDP)Rounded

12. IntegerProgramming • Introduceone {-1,1} variable xi for eachvertex i in V. • Solve for an assignment to the variables that maximizes

13. 2D Relaxation • Introduceone 2-dimensional real vector of unitlength variable yi for eachvertex i in V. • Solve for an assignment to the variables that maximizes

14. 2D Relaxation Can’t be Worse

15. RandomizedRounding of 2D i xi=-1 j Fij yi xj=1 yj yk k xk=1

16. Approximation Ratio

17. RoundingRatio 0.87856

18. Semidefinite Programming