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Slide 2 Arora: SDP + Approx Survey Semidefinite Programming and Approximation Algorithms for NP-hard Problems: A Survey Sanjeev Arora Princeton University http://intractability.princeton.edu Slide 3 Arora: SDP + Approx Survey NP-completeness Pragmatic Researcher Why the fuss? I am perfectly content with approximately optimal solutions. (e.g., cost within 10% of optimum) Bad News: NP-hard for many problems. (PCPs) Good news: Possible for a few problems. (Approximation Algorithms) Thousands of problems are NP-complete (TSP, Scheduling, Circuit layout, Machine Learning,..) Slide 4 Arora: SDP + Approx Survey Approximation Algorithms MAX-3SAT: Given 3-CNF formula, find assignment maximizing the number of satisfied clauses. An -approximation algorithm is one that for every formula, produces in polynomial time an assignment that satisfies at least OPT/ clauses. ( > 1). Good News: [KarloffZwick97] 8/7-approximation algorithm. Bad News: [Hastad97] If P NP then for every > 0, an (8/7 - )-approximation algorithm does not exist. Many similar results... Slide 5 Arora: SDP + Approx Survey Example: 2-approximation for Min Vertex Cover G= (V, E) Vertex Cover = Set of vertices that touches every edge LP Relaxation Claim: Value at least OPT/2 Proof: Rounding most Proof: On Complete Graph K n, OPT = n-1 but setting all x i = 1/2 gives feasible LP soln 0 x i 1 Slide 6 Arora: SDP + Approx Survey General Philosophy Interested in: NP-hard Minimization Problem Value = OPT Write tractable relaxation value= Round to get a solution of cost = Approximation ratio = Integrality gap Slide 7 Arora: SDP + Approx Survey SDP = Generalization of linear programming; vector programming Graph Vector Representation. (Inner products satisfy some linear constraints) Developed in 1970s as one of many flavors of nonlinear optimization. Can be solved in poly time (GLS81). Has many applications in operations research, control theory, approximation algorithms for NP-hard problems. Slide 8 Arora: SDP + Approx Survey Main Idea in SDP: Simulate nonlinear programming by convex program Nonlinear program for Vertex Cover Homogenized SDP relaxation: New variable intended to stand for Vector Programs. Slide 9 Arora: SDP + Approx Survey Take home message SDP gives best approximation known for host of NP-hard problems (and algorithms can be made highly efficient): Vertex Cover Sparsest Cut and most graph partitioning problems Graph coloring Max-cut, and every Constraint Satisfaction Problem. Analysis of these algorithms used interesting geometric ideas, which have had other applications. Compelling evidence from complexity theory that no poly-time algorithm can do better than many of these SDP-based algorithms.(Novel interplay between SDP, reductions, high-dimension geometry.) Slide 10 Arora: SDP + Approx Survey Outline: SDPs & Approximation SDP and its use in approximation: Generations 1 & 2 Understanding SDPs high dimensional geometry Faster algorithms (multiplicative update rule) Limitations of SDPs, Unique Games Conjecture Future directions Open problems Slide 11 Arora: SDP + Approx Survey How do you understand these vector programs? Ans. Interesting geometric analysis Slide 12 Arora: SDP + Approx Survey Understanding SDPs Understanding phenomena in high-dimensional geometry computes c-approximation for c < 2 iff following is true Vertex Cover SDP Every graph in this family has an independent set of size Thm [Frankl-Rodl87] False. Vertices: n unit vectors Edges: almost-antipodal pairs RnRn [GK96] Slide 13 Arora: SDP + Approx Survey SDP rounding: The two generations Generation 1: *Uses random hyperplane as in [GW]; * Edge-by-edge analysis Max-2SAT and Max-CUT [GW94] ;Graph coloring [KMS95]; MAX-3SAT [KZ97]; Algorithms for Unique Games;.. Generation 2: Global rounding and analysis Graph partitioning problems [ARV04], Graph deletion and directed partitioning problems [ACMM05], New analysis of graph coloring [ACC06] Disproof of UGC for expanding constraints [AKKSTV08] Recently, generation 1.5: Squish n solve rounding. k-CSPs[RS09] Slide 14 Arora: SDP + Approx Survey 1 st Generation Rounding: Ratio 1.13.. for MAX-CUT [Goemans- Williamson93] G = (V,E) Find that maximizes capacity. Quadratic Programming Formulation Semidefinite Relaxation [DP 91, GW 93] Slide 15 Arora: SDP + Approx Survey Randomized Rounding (1 st Gen) [GW 93] v6v6 v2v2 v3v3 v5v5 RnRn v1v1 Form a cut by partitioning v 1,v 2,...,v n around a random hyperplane. SDP OPT vjvj vivi ij Old math rides to the rescue... Slide 16 Arora: SDP + Approx Survey Surely this bizarre algorithm is not the right way to solve max cut?? Slide 17 Arora: SDP + Approx Survey Fact 1: No rounding algorithm can produce a better solution out of this SDP [Feige-Schechtman] Fact 2: If P NP then impossible to get 1.06-approximation in poly time [Hastad97] Fact 3: If unique games conjecture is true, no better than 1.13-approximation possible in poly time.[KKMO05] (i.e., algorithm on prev. slide is optimal) Edges between all pairs of vectors making an angle 138 degrees. Slide 18 Arora: SDP + Approx Survey 2 nd Generation: for c-balanced separator G= (V, E); constant c >0 Goal: Find cut s.t. each side contains at least c fraction of nodes and minimized 1 SDP: Triangle inequality Angle subtended by the line joining two of them on the third is non-obtuse; condition. Slide 19 Arora: SDP + Approx Survey Rounding algorithm for -approximation [ARV04] 1. Pick random hyperplane 2. Remove points in slab of width 3. Remove any pair (i, j) that lie on opp. sides of slab but 4. Call remaining sets S, T. Do BFS from S to T according to distance 5. Output level of BFS tree with least # of edges. S T ST Heart of analysis: Shows |S|, |T| = (n); Large well-separated sets Slide 20 Arora: SDP + Approx Survey Geometric fact underlying the analysis ( restatement of [ARV04] Structure Theorem by [AL06]) (expander : | (S)| (|S|) ) Vertices: unit vectors satisfying triangle inequality Edges: If then no graph in this family is an expander. Proof: Difficult chaining argument. (Aside: Has been used to prove that l 1 embeds into l 2 with distortion [CGR05,ALN06] ) = Slide 21 Arora: SDP + Approx Survey Next few slides: Results showing Approximation is hard assuming Unique-Games-Conjecture (UGC) Recall: Integrality gap of an SDP = min c st Arora: SDP + Approx Survey SDPs and MW Updates: Primal-dual algorithm Known: MW Update rule --> Approx. solutions to LPs [PST91, Y95, GK97,..etc.] [AK07] Matrix MW update rule that uses formal analogy between psd matrices and nonnegative real #s. (Spl. Case: LPs= SDPs with 0s on offdiagonals) [Other ingredients: flow computations, eigenvalues, dimension reduction tricks, etc.] experts constraints payoffs slack in constraint [Golden-Thompson] Slide 41 Arora: SDP + Approx Survey Embeddings and Cuts Thm[LLR94, AR94]: Integrality gap for SDP for Nonuniform Sparsest Cut = Min distortion of any embedding of into Rounding algorithm of [ARV04] gives insight into structure of ; basis of new embeddings Hardness results for sparsest cut yielded insights at the heart of the embedding impossibility results. Slide 42 Arora: SDP + Approx Survey Limitations of SDPs For many problems, we know neither an NP-hardness result (via PCPs) nor a good SDP-based approach. 2nd Generation results: Large families of LPs or SDPs dont work [ABL02], [ABLT06]: Proving integrality gaps without knowing the LP. Much subsequent work, especially on families obtained from lift and project ideas) 1st generation results: Specific SDPs dont work Can we show that known SDPs dont work?? Slide 43 Arora: SDP + Approx Survey Example: 2-approximation for Min Vertex Cover G= (V, E) Vertex Cover = Set of vertices that touches every edge LP Relaxation Claim: Value at least OPT/2 Proof: Rounding most Proof: On Complete Graph K n, OPT = n-1 but setting all x i = 1/2 gives feasible LP soln Slide 44 Next, briefly Arora: SDP + Approx Survey Connection between analysis of SDPs and Geometric Embedding of Metric Spaces Slide 45 [CGR05,ALN05]: Yes, C possible for X = Arora: SDP + Approx Survey Geometric embeddings of metric spaces x (X, d): metric space y d(x, y) f f(x) f(y) C = distortion Thm (Bourgain85) For every X, there is f s.t. C= O(log n). Open qs since then: is it possible to achieve smaller C for concrete X, say X = ? Via [LLR94,AR94] implies approx for general sparsest cut Slide 46 Arora: SDP + Approx Survey Main issue: Local versus Global Example: [Erdos] There are graphs on n vertices that cannot be colored with 100 colors yet every subgraph on 0.01 n vertices is 3-colorable. LP relaxations or SDP relaxations concern local conditions. How well do such local conditions capture global property in question? Results for MAX-k-SAT [], [AAT05], Vertex Cover[ABLT06], [STT07a+b] MAX-CUT, Vertex Cover etc.[CMM08] Lifted SDPs. Connections to Proof Complexity.