Semi-Definite Algorithm for Max-CUT Ran Berenfeld May 10,2005

Agenda : The Max-Cut problem. Goemans-Williamson algorithm. Semi-Definite programming. Other applications.

The Max-Cut Problem : Let be a complete, undirected graph, With edge weights. Find a cut that maximizes

Observations : General definition set weight=1 if edges are un-weighted. set weight=0 for non complete graph. NP-Hard [Karp 72’] approximation is easy. This presentation – [Goemans-Williamson 94’] shows -approximation where [Karloff ’99, Feige-Schechtman ’99] – Goemans Williamson have an integralitty gap of

GW strategy for Max-Cut Graph QP VP SDP 1.Write problem as a Quadratic Problem. (with integer solutions) 2.Relax to vector programming. 3.Vector programming is equal to semi-definite programming (SDP). 4.Solve SDP. Approx

Graph QP Assign a variable to each vertex. Let for vertices in

QP VP Replace each with. Old objective value is achieved setting where Approx

QP VP Approx Motivation : heavy weighted vertices will be “far” away from each other. 1000

VP SDP we’ll show later that VP is equal to SDP.

SDP we’ll also show later how SDP is polynomial time solvable to any accuracy degree. But first lets analyze the approximation ratio.

Suppose are the vectors solution to our VP. To obtain a cut from the solution : Randomly pick a vector on the unit sphere, and let SDP

Let and be vectors in the VP solution. By the choice of it follows that Pr[the edge is in the cut]= Pr[ ] And so the expected weight of the cut produced by the algorithm is : Approximation Analysis :

If the angle between and is, there is an area of size where can satisfy

Current conclusion : The optimal solution to VP is no less then the optimal cut. So it follows : Now we set And obtain : !

QP SDP Integralitty gap : 01 VP feasible solution and fractional OPT OPT-F 01 QP solutions and the optimal solution OPT 01 Find integral solution of cost OPT-F

SDP A real, symmetric matrix is positive semi-definite if (TFAE) : 1. for all x. 2.all eigenvalues of are non negative. 3.there exist a matrix so that. Notations: means is positive semi Definite. is the convex of all symmetric Matrices.

SDP Define (Frobenius product) :. Where and all ‘s are symmetric. Then SDP in general form is :

VP SDP 1.Replace with. 2.Demand that the matrix be Symmetric and positive semi-definite. It follows that both problems (VP and SDP) are equal.

SDP It’s easy to show that SDP can be solved in polynomial time using the Ellipsoid method. Other methods exists that are much more practical…

SDP The Ellipsoid method A convex set in is described using a set of restrictions We need to find a point in the set. We need to be able, for each point To provide a separating hyperplane (in polynomial time)

SDP The Ellipsoid method The method starts with a large ellipsoid containing. At each step, if the current point is not in,we use the separating hyperplane to find a (significantlly) smaller ellipsoid.

SDP The SDP Problem : We treat the matrix as a vector in. The set of symmetric,positive Semi-definite matrices is convex. It follows the set of feasible solution is convex.

SDP The SDP Problem : Finding a separating hyperplane : If is not symmetric, is a S.H If is not positive semi-definite, it has a Negative eigenvalue. Let be the Eigenvector. Then Is a separating H.P. Any constraint violated is a S.H

SDP The SDP Problem : Finally, the SDP for Max-Cut has a well defined Dual problem. Which is another SDP program with the same objective Value. Intersecting the Primal and Dual program Creates a convex set, which is not empty If the program is feasible, and contains only optimal points.

Some examples :

SDP Use SDP to -approximate MAX-2SAT The input is a 2-CNF formula, over variables. Need to find an assignment so that the weight of the satisfied clauses is maximal. A weight to each clause,

SDP Use SDP to -approximate MAX-2SAT Assign a {-1,1} variables, Also add a special {-1,1} variable, which will determine the mapping between {-1,1} to {True/False}

SDP Use SDP to -approximate MAX-2SAT Given any boolean formula C, we want v(C) to be 1 if the formula is true,0 otherwise. For example if then

SDP Use SDP to -approximate MAX-2SAT Another example :

SDP Use SDP to -approximate MAX-2SAT This way we can change the 2-CNF to a QP in the form :

SDP Use SDP to -approximate MAX-2SAT Relax the program to

SDP Use SDP to -approximate MAX-2SAT The expected weight E[V] : And the same analysis will work here to show that this algorithm is an -approximate.

Semi-Definite Algorithm for Max-CUT Ran Berenfeld May 10,2005