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The Max-Cut problem: Election recounts? Majority vs. Electoral College? 7812

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The 2-Lin(mod 3) problem: Simultaneously satisfy as many as you can.

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2-Variable Constraint Satisfaction Problems (“2-CSPs”) Variables:x 1, x 2, x 3, …, x n Label Set: (= allowed values for the variables) Input:Constraints 1, 2, …, N on pairs of variables. Output:Assignment satisfying as many constrs. as possible. “Optimization”, “Approximation Algorithms”

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Graph version x1x1 x2x2 x3x3 x5x5 x4x4 x6x6 x7x7 x8x8 x9x9 x 10 = {, }

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2-CSP examples Max-Cut: = {0,1}, ’s of the form “ x i x j ” 2-Lin(mod 3): = 3, ’s of the form “x i = x j + c” 2-SAT: = {0,1}, ’s are Vertex-Cover: Input: A graph. Goal: Select as few vertices as possible s.t. all edges are “covered”. Coloring 3-colorable graphs: Input: a 3-colorable graph. Goal: Legally color it using as few colors as possible. (running example),,,

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1. Egg on our face re complexity of algorithms for 2–CSPs. Story of the talk 2. Efficient “Property Testing” algs. ) Hardness for CSPs 3. Remarkably efficient (2–query!) Property Testing algs. exist.

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Complexity theory dictum “Essentially every natural algorithmic problem has been shown to be in P (polynomial time) or NP-hard.” (Exceptions: Factoring, Graph-Isomorphism.)

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This is a lie. Given a graph, find a cut achieving at least 90% of the max cut. Given a 3-colorable graph, color it using at most 100 colors. Find a vertex cover at most 1.99 times the minimum. Find a 2-SAT assignment satisfying 95% of the maximum. Given (1− )-satisfiable 2-Lin(mod p) system, satisfy (1/p) /2 fraction. Find a cut within factor log log log n of the sparsest cut. (1 − 1/2k)-approximate Max-k-Cut … 1.49-approximate metric TSP 1.54-approximate minimum Steiner tree (.1 log n)-approximate asymm. metric TSP 1.3-approximate minimum multiway cut 1.51-approximate minimum uncapacitated metric facility location (log n)-approximate bandwidth of graphs (log n) 1/3 -approximating 0-Extension.92-approximating MAX-E3-Set-Splitting 1.5-approximate rectangle tiling O(1)-approximate minimum linear arrangement O(1)-approximate minimum feedback arc.51-approximate satisfiable betweenness instances

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We gotta do something about this! 1. Prove problems are in P. Seems we need a radically new algorithmic idea. Max-Cut: Most recent working algorithmic idea was from ’89–’92… Goemans-Williamson ’94 proved it always finds a cut achieving ¸ 87. 8567 % of the optimum. 2. Prove new NP-hardness results. Even after much effort, only some success. Not much for 2-CSPs. (“PCP Theorem” [AS’92, ALMSS’92] + “Parallel Repetition Theorem” [Raz’95])

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“Unique Games Conjecture” [Khot’02] 2-CSPs? [Khot’02] [KR’03] [KKMO’04] [MOO’05] [DMR’06] ( [MOO’05] ) [KV’05] Max-k-CSP Max-3-CSP [ST’06] [O’0?] [KO’06] 2-Lin(mod 2) Vertex-Cover Max-Cut 2-SAT 2-Lin(mod p) Coloring 3-Colorable Graphs Sparsest Cut Max-Cut-Gain ( 87. 8567 % ) A general theory is developing.

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2. Efficient “Property Testing” algs. ) Hardness for CSPs Story of the talk 3. Remarkably efficient (2–query!) Property Testing algs. exist. 1. Egg on our face re complexity of algorithms for 2–CSPs.

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Property Testing = “Constant Time Algorithms” = “The art of uninformed decisions” Input: A “huge” object: e.g., truth table f : m ! . Output: YES or NO, depending on whether it has property P. Caveat: You want to answer in constant time. What you can do: Read f(x) for a few random x, say f(x 1 ), …, f(x k ). Apply a “test”, ( f(x 1 ), …, f(x k ) ) ) YES / NO.

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Testing “Dictatorships” For CSP hardness reductions, relevant P is being a “Dictatorship”: f(x) = x i f a Dictatorship ) test outputs YES with prob. ¸ p YES f “very non-Dictatorial” ) test outputs YES with prob. · p NO

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Testing “Dictatorships” “ k-query, -based, (p YES, p NO ) Dictatorship test ” (for an unknown f : m ! ) 1. x 1, x 2, …, x k chosen at random (somehow) from m 2. ( f(x 1 ), f(x 2 ), …, f(x k ) ) is output, either YES or NO Requirement: f is a Dictatorship ) Pr[output YES] ¸ p YES f “very non-Dictatorial” ) Pr[output YES] · p NO

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CSP hardness Rule of Thumb NP-hardness (or “Unique Games Conjecture”) reduction for: “Satisfying a fraction of the optimum, given a k-CSP instance with constraints.” p NO p YES Remark: This idea is old: from [BGS’95]. Novelty: 2-query Dictatorship tests exist! Why? “ k-query, -based, (p YES, p NO ) Dictatorship test ”

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Rule of Thumb example: Max-Cut Max-Cut: 2-CSP over {0,1} with constraints of form “ x i x j ”. 2-query, “ ”-based, (90%, 80%) Dictator test for f : {0,1} m ! {0,1} ) “Assuming UGC, it is NP-hard to find cuts that achieve 88.888 % of the optimal cut.” 1. Pick x, y 2 {0,1} m in some clever random way. 2. Query f(x), f(y) and output YES iff f(x) f(y). ) Dictatorships pass w.p. ¸ 90%, “Totally not Dictatorships” pass w.p. · 80%.

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Story of the talk 1. Egg on our face re complexity of algorithms for 2–CSPs. 2. Efficient “Property Testing” algs. ) Hardness for CSPs 3. Remarkably efficient (2–query!) Property Testing algs. exist.

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f 2-query, “ ”-based Dictatorship test? m voters winner f : {0,1} m ! {0,1} Voting:0 & 1 are two parties.m voters.f is voting rule.

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2-query, “ ”-based Dictatorship test? [KKMO’04] suggestion: Election #1: Each voter flips a coin. Election #2: Each voter, with probability 90%, reverses their vote. Test: Winner #1 Winner #2. Prob[ Dictatorship passes ]: Prob[ Majority passes ]: Prob[ Electoral College passes ]: ¼ 79.5% 90% ¼ 70.1%

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Majority Is The Highest [KKMO’04] conjectured, [MOO’05] proved: “Majority is the non-Dictator passing the test with highest probability.” Hence: “ ”-based, (, Hence: UGC-hardness of finding cut within ● [GW’94] is optimal Max Cut alg., assuming UGC ● Resolves conjectures [Kalai’03,’04] in theory of voting, also problems [ADFS ’04] in combinatorics. ● Result can be used to prove (sometimes improve) essentially all known UGC reductions. Consequences: 79.5% 90% 79.5 % ) Dictatorship test. of Max Cut. ¼ 88.4 %

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“Unique Games Conjecture” [Khot’02] 2-CSPs? [Khot’02] [KR’03] [KKMO’04] [MOO’05] [DMR’06] ( [MOO’05] ) [KV’05] Max-k-CSP Max-3-CSP [ST’06] [O’0?] [KO’06] 2-Lin(mod 2) Vertex-Cover Max-Cut 2-SAT 2-Lin(mod p) Coloring 3-Colorable Graphs Sparsest Cut Max-Cut-Gain ( 87. 8567 % ) A general theory is developing. [MOO’05]

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The proof that Majority is the highest 1.Generalize Central Limit Theorem. “Sums of random 0’s and 1’s ! Gaussians.” “Polynomials of random 0’s and 1’s ! polynomials of Gaussians.” 2. m Gaussians is like uniform distribution on m-dim. sphere. Problem becomes a cut problem on the sphere. Specifically “Min-Bisection”. 3. For small noise params (angles), essentially similar to finding the blob of half-volume w/ smallest perimeter. (Connections to Double Bubble problem.)

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Open problems I’m thinking about 1. Prove Unique Games Conjecture. ( [FKO] : trying to give reduction from Max-Cut hardness.) 2. Analyze various other constant-query Dictatorship tests. 3. Change from the “Dictatorship test f : {0,1} m ! {0,1}” paradigm. ( [KO’06] has some partial work on this.)

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