A Formal Theory for the Complexity Class Associated with the Stable - PowerPoint PPT Presentation

A Formal Theory for the Complexity Class Associated with the Stable Marriage Problem Dai Tri Man Lˆ e Joint work with Stephen Cook and Yuli Ye Department of Computer Science University of Toronto Canada LCC 2011 1 / 17

Two Aspects of Proof Complexity Propositional Proof Complexity (Pitassi’s invited talk) 1 ◮ the lengths of proofs of tautologies in various proof systems Bounded Arithmetic 2 ◮ the power of weak formal systems to prove theorems of interest in computer science Both are closely related to mainstream complexity theory (2) and (1) are related by “propositional translations” ◮ a proof in theory T � uniform short proofs in propositional proof system P T ◮ bounded arithmetic = uniform version of propositional proof complexity “bounded”: induction axioms are restricted to bounded formulas 2 / 17

Two Aspects of Proof Complexity Propositional Proof Complexity (Pitassi’s invited talk) 1 ◮ the lengths of proofs of tautologies in various proof systems Bounded Arithmetic 2 ◮ the power of weak formal systems to prove theorems of interest in computer science Both are closely related to mainstream complexity theory (2) and (1) are related by “propositional translations” ◮ a proof in theory T � uniform short proofs in propositional proof system P T ◮ bounded arithmetic = uniform version of propositional proof complexity “bounded”: induction axioms are restricted to bounded formulas 2 / 17

Bounded Reverse Mathematics [Cook-Nguyen ’10] Motivation Classify theorems according to the computational complexity of concepts needed to prove them. Program in Chapter 9 Introduce a general method for associating 1 a canonical minimal theory VC for certain complexity classes C AC 0 ⊆ C ⊆ P Given a theorem τ , try to find the smallest 2 complexity class C such that VC ⊢ τ 3 / 17

Bounded Reverse Mathematics [Cook-Nguyen ’10] “As a matter of fact, the subject of the book can almost be thought as developing the proof theory that is missing from the descriptive complexity approach to understanding complexity classes through logic.” [Atserias ’11] 3 / 17

Outline of the talk The complexity class CC 1 ◮ Interesting natural complete problems: stable marriage, lex-first maximal matching, comparator circuit value problem. . . Use the Cook-Nguyen method to define a theory for CC 2 Discuss many open problems related to CC 3 4 / 17

Outline of the talk The complexity class CC 1 ◮ Interesting natural complete problems: stable marriage, lex-first maximal matching, comparator circuit value problem. . . Use the Cook-Nguyen method to define a theory for CC 2 Discuss many open problems related to CC 3 4 / 17

Comparator Circuits Originally invented for sorting, e.g., ◮ Batcher’s O (log 2 n )-depth sorting Comparator gate networks (’68) p p ∧ q ◮ Ajtai-Koml´ x • os-Szemer´ edi (AKS) O (log n )-depth sorting networks (’83) q y p ∨ q � Can also be considered as boolean circuits. Example w 0 • • 1 0 0 0 w 1 • 1 0 1 � w 2 1 1 w 3 • 0 1 0 � w 4 0 1 1 � w 5 0 0 0 � 5 / 17

Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian’s PhD thesis ’90], [Mayr-Subramanian ’92] 6 / 17

Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian’s PhD thesis ’90], [Mayr-Subramanian ’92] decision problems AC 0 many-one-reducible to Ccv � � CC = 2 Complete problems: stable marriage, lex-first maximal matching. . . ◮ CC ∗ = decision problems AC 0 oracle-reducible to Ccv � � 3 ◮ Needed when developing a Cook-Nguyen style theory for CC ◮ The function class FCC ∗ is closed under compostion NC 1 ⊆ NL ⊆ CC ⊆ CC Subr ⊆ CC ∗ ⊆ P 6 / 17

Comparator Circuit Value ( Ccv ) Problem (decision) 1 w 0 • • Given a comparator circuit with specified 1 w 1 � • 1 w 2 Boolean inputs, determine the output ? 0 w 3 � • value of a designated wire. 0 w 4 � 0 w 5 � Complexity classes CC Subr = � � decision problems log-space many-one-reducible to Ccv 1 ◮ [Subramanian’s PhD thesis ’90], [Mayr-Subramanian ’92] decision problems AC 0 many-one-reducible to Ccv � � CC = 2 Complete problems: stable marriage, lex-first maximal matching. . . ◮ CC ∗ = decision problems AC 0 oracle-reducible to Ccv � � 3 ◮ Needed when developing a Cook-Nguyen style theory for CC ◮ The function class FCC ∗ is closed under compostion NC 1 ⊆ NL ⊆ CC ⊆ CC Subr ⊆ CC ∗ ⊆ P 6 / 17

Stable Marriage Problem (search version) (Gale-Shapley ’62) Given n men and n women together with their preference lists Find a stable marriage between men and women, i.e., a perfect matching 1 satisfies the stability condition: no two people of the opposite sex like 2 each other more than their current partners Preference lists a x y Men: b y x x a b Women: y a b 7 / 17

Stable Marriage Problem (search version) (Gale-Shapley ’62) Given n men and n women together with their preference lists Find a stable marriage between men and women, i.e., a perfect matching 1 satisfies the stability condition: no two people of the opposite sex like 2 each other more than their current partners Preference lists a x a x y Men: b y x y x a b b Women: y a b stable marriage 7 / 17

Stable Marriage Problem (search version) (Gale-Shapley ’62) Given n men and n women together with their preference lists Find a stable marriage between men and women, i.e., a perfect matching 1 satisfies the stability condition: no two people of the opposite sex like 2 each other more than their current partners Preference lists a x a x a x y Men: b y x y y x a b b b Women: y a b stable marriage unstable marriage 7 / 17

Stable Marriage Problem (search version) (Gale-Shapley ’62) Given n men and n women together with their preference lists Find a stable marriage between men and women, i.e., a perfect matching 1 satisfies the stability condition: no two people of the opposite sex like 2 each other more than their current partners Preference lists a x a x a x y Men: b y x y y x a b b b Women: y a b stable marriage unstable marriage Stable Marriage Problem (decision version) Is a given pair of ( m , w ) in the man-optimal (woman-optimal) stable marriage? 7 / 17

Lex-first maximal matching problem Lex-first maximal matching Let G be a bipartite graph. Successively match the bottom nodes x , y , z , . . . to the least available top node a c b y x z w 8 / 17

Lex-first maximal matching problem Lex-first maximal matching Let G be a bipartite graph. Successively match the bottom nodes x , y , z , . . . to the least available top node a c b y x z w 8 / 17

Lex-first maximal matching problem Lex-first maximal matching Let G be a bipartite graph. Successively match the bottom nodes x , y , z , . . . to the least available top node a c b y x z w 8 / 17

Lex-first maximal matching problem Lex-first maximal matching Let G be a bipartite graph. Successively match the bottom nodes x , y , z , . . . to the least available top node a c b y x z w 8 / 17

Lex-first maximal matching problem Lex-first maximal matching Let G be a bipartite graph. Successively match the bottom nodes x , y , z , . . . to the least available top node a c b y x z w Lex-first maximal matching problem (decision) Is a given edge { u , v } in the lex-first maximal matching of G ? 8 / 17

Reducing lex-first maximal matching to Ccv a c b d y x z x • • • • 1 0 y • • • • 1 0 z • • • • 1 0 a 0 1 � � 0 b 1 � � c 0 1 � � 0 d 0 � 9 / 17

Reducing Ccv to lex-first maximal matching p 0 p 1 � q 0 q 1 • p 0 q 0 p 1 q 1 y x 10 / 17

Reducing Ccv to lex-first maximal matching p 0 p 1 1 1 � q 0 q 1 • 1 1 p 0 p 0 q 0 q 0 p 1 q 1 y x 10 / 17

Reducing Ccv to lex-first maximal matching p 0 p 1 1 1 � q 0 q 1 • 1 1 p 0 p 0 q 0 q 0 p 1 p 1 q 1 q 1 y x 10 / 17

Reducing Ccv to lex-first maximal matching p 0 p 1 0 1 � q 0 q 1 • 1 0 p 0 q 0 q 0 p 1 q 1 y x 10 / 17

Reducing Ccv to lex-first maximal matching p 0 p 1 0 1 � q 0 q 1 • 1 0 p 0 p 0 q 0 q 0 p 1 p 1 q 1 y x 10 / 17

Outline of the talk The complexity class CC 1 ◮ Interesting natural complete problems: stable marriage, lex-first maximal matching, comparator circuit value problem. . . Use the Cook-Nguyen method to define a theory for CC 2 Discuss many open problems related to CC 3 11 / 17