Semidefinite Programming Lecture 22: Apr 2

Semidefinite Programming What is semidefinite programming? A relaxation of quadratic programming. A special case of convex programing. A generalization of linear programming. Can be optimized in polynomial time. What is it good for? Shannon capacity Perfect graphs Approximation algorithms Image segmentation and clustering Constraint satisfaction problems Number theory, quantum computation, etc…

Maximum Cut (Maximum Cut) Given an undirected graph, with an edge weight w(e) on each edge e, find a partition (S,V-S) of V so as to maximum the total weight of edges in this cut, i.e. edges that have one endpoint in S and one endpoint in V-S.

Similar to vertex cover. Maximum Cut When is computing maximum cut easy? When we are given a bipartite graph. The maximum cut problem can also be interpreted as the problem of finding a maximum bipartite subgraph. There is a simple greedy algorithm with approximation ratio ½. Similar to vertex cover.

Quadratic Program for MaxCut The two sides of the partition. if they are on opposite sides. if they are on the same side.

Quadratic Program for MaxCut This quadratic program is called strict quadratic program, because every term is of degree 0 or degree 2. This is unlikely to be solved in polynomial time, otherwise P=NP.

Vector Program for MaxCut

Vector Program for MaxCut This is a relaxation of the strict quadratic program (why?) Vector program: linear inequalities over inner products. Vector program = semidefinite program. Can be “solved” in polynomial time (ellipsoid, interior point).

Geometric Interpretation Think of as an n-dimensional vector. Contribute more to the objective if the angle is bigger.

Demonstration Rubber band method. László Lovász

Algorithm (Max-Cut Algorithm) Solve the vector program. Let be an optimal solution. Pick r to be a uniformly distributed vector on the unit sphere . Let

Analysis Claim:

Analysis Suppose and has an edge. Contribution to semidefinite program: Contribution to the solution: Approximation Ratio:

Analysis Proof: Linearity of expection. Let W be the random variable denoting the weight of edges in the cut. Claim:

Try to find a tight example. Algorithm (Max-Cut Algorithm) Solve the vector program. Let be an optimal solution. Pick r to be a uniformly distributed vector on the unit sphere . Let Repeat a few times to get a good approximation with high probability. This algorithm performs extremely well in practice. Try to find a tight example.

Remarks Hard to imagine a combinatorial algorithm with the same performance. Assuming the “unique games conjecture”, this algorithm is the best possible! That is, it is NP-hard to find a better approximation algorithm!

Constraint Satisfaction Problems (Max-2-SAT) Given a formula in which each clause contains two literals, find a truth assignment that satisfies the maximum number of clauses. e.g. An easy algorithm with approximation ratio ½. An LP-based algorithm with approximation ratio ¾. An SDP-based algorithm with approximation ratio 0.87856.

Vector Program for MAX-2-SAT Given a formula in which each clause contains two literals, find a truth assignment that satisfies the maximum number of clauses. Additional variable (trick): A variable is set to be true if: A variable is set to be false if:

Vector Program for MAX-2-SAT Denote v(C) to be the value of a clause C, which is defined as follows. Consider a clause containing 2 literals, e.g. . Its value is:

Vector Program for MAX-2-SAT Objective: where a(ij) and b(ij) is the sum of coefficients.

Algorithm (MAX-2-SAT Algorithm) Solve the vector program. Let be an optimal solution. Pick r to be a uniformly distributed vector on the unit sphere . Let be the “true” variables.

Analysis Term-by-term analysis. First consider the second term. Contribution to semidefinite program: Contribution to the solution: Approximation Ratio:

Analysis Term-by-term analysis. Consider the first term. Contribution to semidefinite program: Contribution to the solution: Approximation Ratio:

Algorithm (MAX-2-SAT Algorithm) Solve the vector program. Let be an optimal solution. Pick r to be a uniformly distributed vector on the unit sphere . Let be the “true” variables. This is a 0.87856-approximation algorithm for MAX-2-SAT. Can be improved to 0.931!

More SDP in approximation algorithms Sparsest cut: O(√log n) Constraint satisfaction problems Correlation clustering Graph colouring A very powerful tool in the design of approximation algorithms. Useful to know geometry and algebra. Next two lectures: SDP and perfect graphs.

Summary: Topics Classical problem: TSP, Steiner trees Covering problem: vertex cover, set cover Packing problem: knapsack, bin packing Graph partitioning problem: multiway cut, multi-cut Job scheduling: makespan, general assignment Network design: Steiner network, degree constrained spanning trees Constraint satisfaction: maximum cut, max 2-SAT

Summary: Techniques Combinatorial arguments: TSP, Steiner trees, multiway cut Greedy algorithm and Randomized rounding: set cover Dynamic programming: knapsack, bin packing Region Growing: multi-cut Iterative relaxation: scheduling, network design problems Primal-dual method: vertex cover, multi-cut in tree Semidefinite programming: maximum cut, constraint satisfaction

Concluding Remarks Learn to design better heuristics. e.g. iterative rounding, SDP performs extremely well in practice. Use LP or SDP as a good way to estimate the optimal value. e.g. help to test the performance of a heuristic. Relax the search space to a convex set.

Remaining Schedule Apr 10-11: NO CLASSES Apr 17-18: Last week of lecture (perfect graphs, SDP) Apr 23-24: Project presentation (15 minutes) May 1: Project report deadline May 8: Homework 3 deadline