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Advanced Approximation Algorithms for the MAX CUT Problem in Graphs

The MAX CUT problem involves partitioning the vertices of an undirected graph such that the sum of the weights of the edges between the partitions is maximized. This NP-hard problem contrasts with the polynomially solvable MIN CUT problem. We explore two polynomial-time techniques to obtain approximative solutions for MAX CUT: local search refinement and semi-definite programming. The former employs a single flip neighborhood strategy to enhance cut weight, while the latter involves formulating the problem as a linear program, providing a strong approximation guarantee with a ratio of 0.87856.

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Advanced Approximation Algorithms for the MAX CUT Problem in Graphs

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  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

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