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Semidefinite Programming and Approximation Algorithms for NP-hard Problems: A Survey Sanjeev Arora Princeton University http://intractability.princeton.edu 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 (GLS’81). Has many applications in operations research, control theory, approximation algorithms for NP-hard problems. Arora: SDP + Approx Survey Take home message… SDP gives best approximation known for host of NP-hard problems (and algorithms can be made shockingly 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….) Arora: SDP + Approx Survey Main Idea in SDP: “Simulate” nonlinear programming by convex program Nonlinear program for Vertex Cover SDP relaxation: New variable intended to stand for “Vector Programs.” Arora: SDP + Approx Survey Homogenized 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 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 Arora: SDP + Approx Survey = Integrality gap How do you understand these vector programs? Ans. Interesting geometric analysis Arora: SDP + Approx Survey Understanding SDPs <--> Understanding phenomena in high-dimensional geometry Vertex Cover SDP computes c-approximation for c < 2 iff following is true [GK96] Vertices: n unit vectors Edges: almost-antipodal pairs Rn Every graph in this family has an independent set of size Thm [Frankl-Rodl’87] False. Arora: SDP + Approx Survey SDP rounding: The two generations First generation: *Uses random hyperplane as in [GW]; * Edge-by-edge analysis Max-2SAT and Max-CUT [GW’94] ;Graph coloring [KMS’95]; MAX-3SAT [KZ’97]; Algorithms for Unique Games;.. Second generation: Global rounding and analysis Graph partitioning problems [ARV’04], Graph deletion and directed partitioning problems [ACMM’05], New analysis of graph coloring [ACC’06] Disproof of UGC for expanding constraints [AKKSTV’08] Recently, generation 1.5: “Squish ‘n solve” rounding. k-CSPs [RS’09] Arora: SDP + Approx Survey 1st Generation Rounding: Ratio 1.13.. for MAX-CUT [GoemansWilliamson’93] G = (V,E) Find that maximizes capacity Quadratic Programming Formulation Semidefinite Relaxation [DP ’91, GW ’93] Arora: SDP + Approx Survey . Randomized Rounding (1st Gen) Rn v2 v1 v6 v3 v5 [GW ’93] Form a cut by partitioning v1,v2,...,vn around a random hyperplane. SDPOPT vi ij vj Old math rides to the rescue... Arora: SDP + Approx Survey Fact 1: No rounding algorithm can produce a better solution out of this SDP [Feige-Schechtman] “Edges between all pairs of vectors making an angle 138 degrees.” Fact 2: If P NP then impossible to get 1.06-approximation in poly time [Hastad’97] Fact 3: If “unique games conjecture” is true, no better than 1.13-approximation in poly time.[KKMO’05] (i.e., algorithm on prev. slide is optimal) Arora: SDP + Approx Survey 2nd Generation: for c-balanced separator G= (V, E); constant c >0 1 -1 Goal: Find cut s.t. each side contains at least c fraction of nodes and minimized SDP: “Triangle inequality” Arora: SDP + Approx Survey Angle subtended by the line joining two of them on the third is non-obtuse; “ “ condition. Rounding algorithm for -approximation [ARV’04] 1. Pick random hyperplane S T 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 S T 5. Output level of BFS tree with least # of edges. Heart of analysis: Shows |S|, |T| = W(n); “Large well-separated sets” Arora: SDP + Approx Survey Geometric fact underlying the analysis (restatement of [ARV04] “Structure Theorem” by [AL06]) Vertices: unit vectors satisfying “triangle inequality” If then no graph in this family is an “expander.” (“expander” : |(S)| > W(|S|) ) Edges: Proof: Difficult “chaining” argument. (Aside: Has been used to prove that l1 embeds into l2 with distortion [CGR’05,ALN’06]) 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 <= c Let “tough instance” = problem instance with integrality gap c These play crucial role in above reductions!! Arora: SDP + Approx Survey Unique Games Given: Number p, and m equations in n vars of the form: Promise: Either there is a solution that satisfies fraction of constraints or no solution satisfies even fraction. UGC [Khot’02]: Deciding which case holds is NP-hard. [Raghavendra’08; building upon [KKMO’04][MOO’05]] UGC For every MAX-CSP, the simplest SDP relaxation is the best possible poly-time approximation. Arora: SDP + Approx Survey Anatomy of a UGC-based hardness result (eg Khot-Regev, KKMO, Raghavendra08) Variables Equations Interpret as a graph Prove using harmonic analysis that near-optimum solns correspond to good Equations solution to the unique game Variables Replace edges/vertices with gadgets involving “integrality gap instance” Arora: SDP + Approx Survey Generation 1.5 rounding: Squish-n-solve (Raghavendra-Steurer’09; provably optimal approximation for all k-CSPs if UGC is true) Project to random t-dim subspace; t =O(1) Merge nodes whose vectors are close together; get instance of size exp(t) and solve it optimally. If g= approx. ratio for this algorithm then it is also the integrality gap for SDP, and also the best possible approx. ratio (assuming UGC). Arora: SDP + Approx Survey Issue of Running Time Solving SDPs with m constraints takes time. m =n3 in some of these SDPs! Next few slides: Often, can reduce running time: O(n2) or O(n3) [AHK’05], [AK’07] even O(n) for CSPs! [St’10]; O(n1.5 +m) for sparsest cut [S’09] Main idea: “Primal-dual schema.” Solve to approximate optimality; using insights from the rounding algorithms. “Multiplicative Weight-Update Rule for psd matrices” Arora: SDP + Approx Survey Primal-dual approach for SDP relaxations (contd.) [A., Kale’07] At step t: Primal player: PSD matrix Xt; candidate primal Dual player: “Candidate slack matrix” Mt Let me run the rounding algorithm on Pt, get a primal integer candidate and point out how pitiful it is. Primal player: Xt+1 = exp(- t Mt) (Analysis uses formal analogy between real #s and symmetric matrics: [Other ingredients: flow computations, eigenvalues, dimension reduction tricks, etc.; simplified by [KRV07], [KRVV08]] Arora: SDP + Approx Survey Future directions…. Arora: SDP + Approx Survey Direction 1: How to soup up the relaxation • Other forms of convex relaxations instead of SDP? (e.g. geometric programming, convex programming) For a start, re-derive existing approximation ratios (eg 0.878.. For MAX-CUT) using the above. Arora: SDP + Approx Survey Direction 2: Understand “Lifted” SDP relaxations Recall: SDP tries to “simulate” nonlinear programming; Variable for Why not take it to the next level? Variables for products of up to k variables. This is the main idea of Lovasz-Schrijver’91, Sherali-Adams, Lasserre etc. Yields better approx for hypergraph I.S. [CS’08] Don’t seem to help for some problems: MAX-k-SAT [BGHMP’06], [AAT’05], Vertex Cover[ABLT’06],[STT’07a+b] MAX-CUT, Vertex Cover etc. [CMM’08] MAX-3SAT (Sch’09) (v. subtle and beautiful ideas!) Arora: SDP + Approx Survey Direction 3: graph expansion [ARV]; SDP w/ triangle ineq. Approx Ratio O(1) approx is UGC hard; as is improving Cheeger for constt [Cheeger’72]; eigenvalue Expansion Lots of space for improvement even if UGC holds Also, “small set expansion” problem is v. close to UG and progress on it would possibly give progress on UG. [RS’10] Arora: SDP + Approx Survey Direction 4: Subexponential algorithms • Inspiration: Unique Games with completeness 1- can be approximately solved in exp(n) time [ABS’10] (For problems like MAX3SAT, no such algorithms exist if 3SAT has no subexp. algorithms) Intriguing possibility: Many of the UGC-hard problems have subexponential algorithms. Another interesting idea: derive [ABS’10] algorithm for UG using SDPs or Lasserre relaxations. Arora: SDP + Approx Survey Direction 5: SDP in Avg. Case Complexity Problems like 3SAT seem difficult not only in the worst case but also “on average.” (Needs careful definition!) Theory of Avg Case complexity exists, but doesn’t usually apply to problems of practical interest (e.g., random 3SAT). Recent development: Interreducibility among some “average case” problems of interest. [Feige’01]; e.g Easy “optimal” 7/8-hardness of MAX-3SAT. SDP is used in the reduction! (Used to weed out “uninteresting” cases) Arora: SDP + Approx Survey Direction 6: Applications to quantum computing • PSD matrix of trace 1 = “density matrix”, a way to describe a mixed quantum state Recent result QIP=PSPACE (JUW’10) uses the [AK’07] Primal-dual framework. (“Fast NC computation of near-optimum quantum state”) Expertise in designing specialized matrices for SDP integrality gaps may prove useful in QC… Arora: SDP + Approx Survey Open problems • Can Lasserre relaxations compute nontrivial approximations to Vertex Cover, MAX-CUT, etc? (ruled out already for MAX-3SAT [Sch’08]) • Generation 3 rounding? • Iterative rounding for SDPs? • Resolve UGC (eg disprove by giving truly subexp. algorithms) • SDP as a proof technique---apply to open problems of circuit complexity, communication complexity etc. Looking forward to many developments THANK YOU! Arora: SDP + Approx Survey Classical MW update rule (Example: predicting the market) 1$ won for correct prediction 1$ lost for incorrect prediction • N “experts” on TV • Can we perform as good as the best expert in hindsight? Thm[Going back to Hannan, 1950s] Yes. Arora: SDP + Approx Survey Weighted Majority Algorithm (LW’94) Losses M1t M2 t • For each expert, weight wi. Initially wi 1 • Follow expert i w/ prob. proportional to wit •Update weights according to M3t Claim: Expected per-round loss of our algorithm … Arora: SDP + Approx Survey Lagrangian method to approximately solve LPs (PST’91, many others) Losses M1t M2 t M3t … Arora: SDP + Approx Survey Experts = Dual constraints ; maintain weighting • For eachLP expert, weight w i. Initially wi 1 Loss vector = expert Slack vector of candidate dualtosoln • Follow i w/ prob. proportional w it •Update weights according to Claim: a few rounds; the average of all loss Claim:After Expected per round loss of ourthealgorithm vectors is an approximately feasible dual soln. Lagrangian method to approximately solve LPs (PST’91, many others since) x = weighting of n experts; updated via multiplicative update Loss vector Expected loss Only 1 constraint ! Claim: Expected per round loss of our algorithm Average per-round loss of expert i = i’th coordinate of Arora: SDP + Approx Survey ! Lagrangian method to approximately solve SDPs (A.,Kale ‘07) psd x =x=weighting PSD matrix; of n updated experts; according updated via to multiplicative update Loss vector Expected loss Claim: Expected per round loss of our algorithm Arora: SDP + Approx Survey ! SDPs and MW Updates: Primal-dual algorithm Known: MW Update rule --> Approx. solutions to LPs [PST’91, Y’95, GK’97,..etc.] “experts” <-> constraints “payoffs” <-> “slack in constraint” [AK’07] Matrix MW update rule that uses formal analogy between psd matrices and nonnegative real #s. [Golden-Thompson] (Spl. Case: LPs= SDPs with 0’s on offdiagonals) [Other ingredients: flow computations, eigenvalues, dimension reduction tricks, etc.] 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. 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. Can we show that known SDPs don’t work?? 1st generation results: Specific SDPs don’t work 2nd Generation results: Large families of LPs or SDPs don’t work [ABL’02], [ABLT’06]: “Proving integrality gaps without knowing the LP.” Much subsequent work, especially on families obtained from “lift and project” ideas) 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” most Claim: Value at least OPT/2 Proof: On Complete Graph Kn, Proof: “Rounding” OPT = n-1 but setting all xi = 1/2 gives feasible LP soln Arora: SDP + Approx Survey Background: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: [KarloffZwick’97] 8/7-approximation algorithm. Bad News: [Hastad’97] If P NP then for every > 0, an (8/7 -)-approximation algorithm does not exist. Many similar results... Arora: SDP + Approx Survey Next, briefly Connection between analysis of SDPs and Geometric Embedding of Metric Spaces Arora: SDP + Approx Survey Geometric embeddings of metric spaces (X, d): metric space y d(x, y) f(y) f x f(x) C = distortion Thm (Bourgain’85) 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 = ? [CGR’05,ALN05]: Yes, C Via [LLR94,AR94] implies Arora: SDP + Approx Survey possible for X = approx for general sparsest cut 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 [], [AAT’05], Vertex Cover[ABLT’06], [STT’07a+b] MAX-CUT, Vertex Cover etc. [CMM’08] “Lifted SDPs.” Connections to Proof Complexity. Arora: SDP + Approx Survey