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Semi-Definite Algorithm for Max-CUT Ran Berenfeld May 10,2005
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Agenda : The Max-Cut problem. Goemans-Williamson algorithm. Semi-Definite programming. Other applications.
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The Max-Cut Problem : Let be a complete, undirected graph, With edge weights. Find a cut that maximizes
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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
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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
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Graph QP Assign a variable to each vertex. Let for vertices in
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QP VP Replace each with. Old objective value is achieved setting where Approx
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QP VP Approx Motivation : heavy weighted vertices will be “far” away from each other. 1000
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VP SDP we’ll show later that VP is equal to SDP.
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SDP we’ll also show later how SDP is polynomial time solvable to any accuracy degree. But first lets analyze the approximation ratio.
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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
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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 :
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If the angle between and is, there is an area of size where can satisfy
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Current conclusion : The optimal solution to VP is no less then the optimal cut. So it follows : Now we set And obtain : !
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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
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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.
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SDP Define (Frobenius product) :. Where and all ‘s are symmetric. Then SDP in general form is :
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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.
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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…
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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)
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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.
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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.
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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
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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.
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Some examples :
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11 1000
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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,
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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}
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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
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SDP Use SDP to -approximate MAX-2SAT Another example :
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SDP Use SDP to -approximate MAX-2SAT This way we can change the 2-CNF to a QP in the form :
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SDP Use SDP to -approximate MAX-2SAT Relax the program to
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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.
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Semi-Definite Algorithm for Max-CUT Ran Berenfeld May 10,2005
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