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

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

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”

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

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),,,

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.

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

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 

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 ¸ % 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])

“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 ( % ) A general theory is developing.

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.

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.

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

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

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 ” 

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 % 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%.

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.

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.

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%

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 %

“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 ( % ) A general theory is developing. [MOO’05]

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.)

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.)