Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP Yuichi Yoshida Kyoto Univ. & Preferred Infrastructure 2011 年 5 月 26 日木曜日
Polynomial-Time Approximation for Max CSP 2011 年 5 月 26 日木曜日
Max CSP (Constraint Satisfaction Problem) • Given variables and constraints on them. Satisfy constraints as many as possible by assigning values to variables. • Ex.: Max Cut (, Max k -SAT, Max E3LIN2) Input I : ( v 1 ⊕ v 2 ), ( v 1 ⊕ v 3 ), ( v 1 ⊕ v 4 ), ( v 2 ⊕ v 3 ), ( v 2 ⊕ v 4 ) v 1 v 2 β = (0,0,1,1) opt( I ) = val( I , β ) = 4 v 3 v 4 • In this talk, we mainly deal with Max Cut. But, we can use the same machinery to “any” CSP. constant domain size constant arity 2011 年 5 月 26 日木曜日
Known Results for Poly-Time Approximation • Since Max CSP is NP-Hard in general, approximation have been considered. • Approximation by SDP (semidefinite programmings) • Hardness by PCP/Unique Games Conjecture CSP SDP UG-Hard Max k -SAT ? 0.787 or more [ABZ06] Max Cut 0.878... [GW95] 0.878...+ ε [KKMO04] Max Dicut 0.874 or more [LLZ06] ? Max k -CSP poly( k )/2 k [CMM09] poly( k )/2 k [ST06] 2011 年 5 月 26 日木曜日
Known Results for Poly-time Approximation [Rag08] (informal) For every CSP Λ , under Unique Games Conjecture, “BasicSDP” + a certain rounding is the best possible poly-time approximation algorithm. CSP SDP UG-Hard Max k- SAT coincides (up to coincides (up to ε ) Max Cut coincides (up to coincides (up to ε ) ≈ 0.878... Max Dicut coincides (up to ε ) coincides (up to Max k -CSP coincides (up to ε ) coincides (up to 2011 年 5 月 26 日木曜日
Constant-Time Approximation for Max CSP 2011 年 5 月 26 日木曜日
Const-Time Approximation for Max CSP • We want faster approximation algorithms! ( O(1) time) • A value x is an ( α , ε )-approximation to x * if: α x * - ε n ≦ x ≦ x * • An assignment β is ( α , ε )-approximate assignment for an input I if val( I , β ) is ( α , ε )-approximation to opt( I ) 2011 年 5 月 26 日木曜日
Bounded-Degree Model • It takes Ω ( n ) to read the whole input. Thus, we read it through an oracle. • An input I = ( V , P ) is given as an oracle O I : V × [ t ] → P ( t = degree bound). O I ( v , i ) = the i -th constraint incident to v . • Query complexity: # of accesses to the oracle. 2011 年 5 月 26 日木曜日
Can we show something similar to [Rag08] on ( α , ε ) -approximation? 2011 年 5 月 26 日木曜日
Our Results • (informal) For every CSP Λ , unconditionally, ”BasicLP”+ a certain rounding is the best possible constant-time approximation algorithm. CSP O(1) queries (via LP) need Ω ( √ n ) queries Max k- SAT coincides (up to coincides (up to ε ) ≈ 0.75 Max Cut coincides (up to coincides (up to ε ) ≈ 0.5 Max Dicut coincides (up to ε ) ≈ 0.5 coincides (up to Max k -CSP coincides (up to coincides (up to ε ) ≈ 2/2 k 2011 年 5 月 26 日木曜日
Our Results in Detail • lp( I ) : the optimal value of BasicLP for an input I . opt( I ) • Integrality Gap: α Λ = inf lp( I ) I ∈ Λ [Theorem] For every CSP Λ : ∀ ε >0 , there exists a constant-time ( α Λ - ε , ε )- approximation algorithm. ∀ ε >0 , ∃ δ >0 , any algorithm that outputs ( α Λ + ε , δ ) - approximation to opt( I ) with prob ≥ 2/3 requires Ω ( √ n ) queries. Query complexity: exp(exp(poly( qst / ε ))) 2011 年 5 月 26 日木曜日
Our Results in Detail • lp( I ) : the optimal value of BasicLP for an input I . opt( I ) • Integrality Gap: α Λ = inf lp( I ) I ∈ Λ [Theorem] For every CSP Λ : ∀ ε >0 , there exists a constant-time ( α Λ - ε , ε )- approximation algorithm. ∀ ε >0 , ∃ δ >0 , any algorithm that outputs ( α Λ + ε , δ ) - With prob ≥ 2/3, gives an oracle access to some approximation to opt( I ) with prob ≥ 2/3 requires ( α Λ - ε , ε ) -approx assignment β . Once we succeed, we Ω ( √ n ) queries. can compute β v in constant time for each v . Query complexity: exp(exp(poly( qst / ε ))) 2011 年 5 月 26 日木曜日
Our Results in Detail • lp( I ) : the optimal value of BasicLP for an input I . opt( I ) • Integrality Gap: α Λ = inf lp( I ) I ∈ Λ [Theorem] For every CSP Λ : ∀ ε >0 , there exists a constant-time ( α Λ - ε , ε )- approximation algorithm. ∀ ε >0 , ∃ δ >0 , any algorithm that outputs ( α Λ + ε , δ ) - approximation to opt( I ) with prob ≥ 2/3 requires Ω ( √ n ) queries. Query complexity: exp(exp(poly( qst / ε ))) 2011 年 5 月 26 日木曜日
Upper Bound in More Detail opt( I ) • Integrality gap curve: S Λ ( c ) = inf I ∈ Λ , m lp( I ) ≥ cm • For every CSP Λ , ε > 0 , a constant-time algorithm exists satisfying the following: For an input I with lp( I ) = cm ( c ∈ (0,1] ) , with prob ≥ 2/3 , it gives an oracle access to β such that S Λ ( c - ε ) m - ε n ≦ val( I , β ) ≦ opt( I ) . 1.0 opt( I ) ≧ cm opt( I ) ≦ S Λ ( c - ε ) m - ε n are distinguishable in const time. 0 0 1.0 2011 年 5 月 26 日木曜日
Lower Bounds in More Detail opt( I ) • Integrality gap curve: S Λ ( c ) = inf I ∈ Λ , m lp( I ) ≥ cm • For every CSP Λ , c ∈ [0, 1], ε >0 , any algorithm satisfying the following requires Ω ( √ n ) queries: For an input I with opt( I )= cm , with prob ≥ 2/3 , it outputs a value x such that ( S Λ ( c )+ ε ) m ≦ x ≦ opt( I ) . 1.0 opt( I ) ≧ cm opt( I ) ≦ ( S Λ ( c )+ ε ) m are indistinguishable in const time. 0 0 1.0 2011 年 5 月 26 日木曜日
Comparison to [Rag08] This work [Rag08] For every CSP, BasicLP is For every CSP, BasicSDP is the best algorithm. the best algorithm. Unconditional Assuming UGC Lower bounds hold for Lower bounds do not hold satisfiable instances for satisfiable instances 2011 年 5 月 26 日木曜日
Property Testing • An input I is ε -far from satisfiability: we need to remove ε tn constraints to make I satisfiable. • CSP Λ is testable : we can decide with prob ≥ 2/3 whether an input of CSP Λ is satisfiable or ε -far. • If lp( I ) = m implies opt( I ) = m , then CSP Λ is testable in constant time. (If integrality gap curve is continuous at c = 1 .) • If not, testing CSP Λ requires Ω ( √ n) queries. 2011 年 5 月 26 日木曜日
Proof sketch: Lower bounds ( α Λ + ε , δ )-approximation needs Ω ( √ n ) queries. 2011 年 5 月 26 日木曜日
BasicLP for Max Cut e • Consider the following IP. u v • x v , i : indicating v has a value i ∈ {0,1} • µ e , β : indicating e has an assignment β ∈ {0,1} 2 max Σ e w e (µ e ,01 + µ e ,10 ) s.t. x v ,0 + x v ,1 = 1 ∀ v µ e ,00 + µ e ,01 = x v ,0 ∀ e = ( v , u ) µ e ,10 + µ e ,11 = x v ,1 ∀ e = ( v , u ) x v , i ∈ {0,1} ∀ v , i µ e , β ∈ {0,1} ∀ e , β 2011 年 5 月 26 日木曜日
BasicLP for Max Cut e • Relax the IP to LP. u v • x v : probability distribution of value of v . • µ e : probability distribution of assignment to e . max Σ e w e (µ e ,01 + µ e ,10 ) s.t. x v ,0 + x v ,1 = 1 ∀ v µ e ,00 + µ e ,01 = x v ,0 ∀ e = ( v , u ) µ e ,10 + µ e ,11 = x v ,1 ∀ e = ( v , u ) ∀ v , i x v , i ≧ 0 ∀ e , β µ e , β ≧ 0 2011 年 5 月 26 日木曜日
Proof Strategy • Choose I s.t. lp( I ) = cm , opt( I ) ≈ α Λ cm . • Create two distributions of inputs using I : • D I opt : generates J s.t. opt( J ) ≦ ( α Λ c + ε ) m . • D I lp : generates J s.t. opt( J ) ≧ cm . I opt( I ) = 2 / 3, lp( I ) = 1 e µ e ,00 = µ e ,11 =0 u v µ e ,01 = µ e ,10 =1/2 2011 年 5 月 26 日木曜日
Yao’s minimax principle • Let A be a deterministic algorithm supposed to output: • “Yes” if J is generated by D I opt • ”No” if J is generated by D I lp • Let G I opt , G I lp be the distribution of subgraphs seen by A running on D I opt , D I lp , respectively. • It suffices to show that G I opt and G I lp are “close” when # of queries is o( √ n ) 2011 年 5 月 26 日木曜日
Construction of D I opt 1. Make clusters by duplicating each vertex of I . 2. Make an expander for each edge of I . • The optimal assignment for J and I are similar. • With prob 1-o(1) , opt( J ) ≦ ( α Λ c + ε ) m J I e u v 2011 年 5 月 26 日木曜日
Construction of D I lp 1. Make clusters by duplicating each vertex of I. 2. Make an expander for each edge of I using µ . • The optimal assignment for J can be made from µ. • opt( J ) ≧ cm J I µ e ,01 µ e ,01 e u v µ e ,10 µ e ,10 µ e ,00 = µ e ,11 =0 µ e ,01 = µ e ,10 =1/2 2011 年 5 月 26 日木曜日
• As long as we do not find a cycle, G I opt and G I lp behaves identically. G I lp G I opt 2011 年 5 月 26 日木曜日
• As long as we do not find a cycle, G I opt and G I lp behaves identically. G I lp G I opt 2011 年 5 月 26 日木曜日
• As long as we do not find a cycle, G I opt and G I lp behaves identically. G I lp G I opt 2011 年 5 月 26 日木曜日
• As long as we do not find a cycle, G I opt and G I lp behaves identically. G I lp G I opt 2011 年 5 月 26 日木曜日
• As long as we do not find a cycle, G I opt and G I lp behaves identically. G I lp G I opt 2011 年 5 月 26 日木曜日
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