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Background Query A Stable Marriage Requires Communication Complexity Communication Complexity Proofs Yannai A. Gonczarowski Open Problems The Hebrew University of Jerusalem and Microsoft Research January 5, 2015 Joint work with: Noam

  1. Background Partial Answers Query • The Gale-Shapley algorithm makes Complexity Θ ( n 2 ) queries of size Θ (log n ) each. Communication Complexity • The size of the input is Θ ( n 2 log n ). Proofs Open • Improving upon GS requires Problems random access to the input. Theorem (Ng and Hirschberg, 1990) In a model with two unit-cost queries: “what is woman w ’s ranking of man m ?” and “which man does woman w rank at place k ?” (and the dual queries), finding a stable marriage requires Θ ( n 2 ) queries. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 4 / 16

  2. Background Partial Answers Query • The Gale-Shapley algorithm makes Complexity Θ ( n 2 ) queries of size Θ (log n ) each. Communication Complexity • The size of the input is Θ ( n 2 log n ). Proofs Open • Improving upon GS requires Problems random access to the input. Theorem (Ng and Hirschberg, 1990) In a model with two unit-cost queries: “what is woman w ’s ranking of man m ?” and “which man does woman w rank at place k ?” (and the dual queries), finding a stable marriage requires Θ ( n 2 ) queries. (= Θ ( n 2 log n ) bits, like Gale-Shapley.) Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 4 / 16

  3. Background Partial Answers Query • The Gale-Shapley algorithm makes Complexity Θ ( n 2 ) queries of size Θ (log n ) each. Communication Complexity • The size of the input is Θ ( n 2 log n ). Proofs Open • Improving upon GS requires Problems random access to the input. Theorem (Ng and Hirschberg, 1990) In a model with two unit-cost queries: “what is woman w ’s ranking of man m ?” and “which man does woman w rank at place k ?” (and the dual queries), finding a stable marriage requires Θ ( n 2 ) queries. (= Θ ( n 2 log n ) bits, like Gale-Shapley.) Theorem (Chou and Lu, 2010) If one is allowed to separately query each of the log n bits of the answer to queries such as “which man does woman w rank at place k ?”, then Θ ( n 2 log n ) queries are still required. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 4 / 16

  4. Background Challenges and Contribution Query Complexity Communication Complexity Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  5. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  6. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  7. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs • Can randomized algorithms do any Open Problems better? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  8. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs • Can randomized algorithms do any Open Problems better? Theorem (Main Result for Query Complexity) Any randomized (or deterministic) algorithm that uses any type of Boolean queries to the women’s and to the men’s preferences to solve any of the following problems requires Ω ( n 2 ) queries in the worst case: Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  9. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs • Can randomized algorithms do any Open Problems better? Theorem (Main Result for Query Complexity) Any randomized (or deterministic) algorithm that uses any type of Boolean queries to the women’s and to the men’s preferences to solve any of the following problems requires Ω ( n 2 ) queries in the worst case: 1 finding a marriage close to being stable, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  10. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs • Can randomized algorithms do any Open Problems better? Theorem (Main Result for Query Complexity) Any randomized (or deterministic) algorithm that uses any type of Boolean queries to the women’s and to the men’s preferences to solve any of the following problems requires Ω ( n 2 ) queries in the worst case: 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  11. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs • Can randomized algorithms do any Open Problems better? Theorem (Main Result for Query Complexity) Any randomized (or deterministic) algorithm that uses any type of Boolean queries to the women’s and to the men’s preferences to solve any of the following problems requires Ω ( n 2 ) queries in the worst case: 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, 3 determining whether a given pair is contained in some/every stable marriage, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  12. Background Challenges and Contribution Query • Can a more powerful model Complexity Communication (allowing more complex queries) Complexity allow for faster algorithms? Proofs • Can randomized algorithms do any Open Problems better? Theorem (Main Result for Query Complexity) Any randomized (or deterministic) algorithm that uses any type of Boolean queries to the women’s and to the men’s preferences to solve any of the following problems requires Ω ( n 2 ) queries in the worst case: 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, 3 determining whether a given pair is contained in some/every stable marriage, 4 finding ε n pairs that appear in some/every stable marriage. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 5 / 16

  13. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  14. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity • Alice and Bob wish to perform some computation. Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  15. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity • Alice and Bob wish to perform some computation. Proofs • The computation depends on X , held by Alice, and on Y , Open Problems held by Bob. X Y Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  16. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity • Alice and Bob wish to perform some computation. Proofs • The computation depends on X , held by Alice, and on Y , Open Problems held by Bob. To perform it, they exchange information. X Y Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  17. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity • Alice and Bob wish to perform some computation. Proofs • The computation depends on X , held by Alice, and on Y , Open Problems held by Bob. To perform it, they exchange information. • The communication cost of a given protocol for such a computation is the number of bits that Alice and Bob exchange under this protocol in the worst case. X Y Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  18. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity • Alice and Bob wish to perform some computation. Proofs • The computation depends on X , held by Alice, and on Y , Open Problems held by Bob. To perform it, they exchange information. • The communication cost of a given protocol for such a computation is the number of bits that Alice and Bob exchange under this protocol in the worst case. • The communication complexity of the computation is the least communication cost of any protocol for it. X Y Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  19. Background Communication Complexity (Yao, 1979) Query We prove our result by considering the communication Complexity complexity of these problems. Communication Complexity • Alice and Bob wish to perform some computation. Proofs • The computation depends on X , held by Alice, and on Y , Open Problems held by Bob. To perform it, they exchange information. • The communication cost of a given protocol for such a computation is the number of bits that Alice and Bob exchange under this protocol in the worst case. • The communication complexity of the computation is the least communication cost of any protocol for it. • Randomized communication complexity is defined analogously using randomized protocols (with success rate X bounded away from 1 2 ). Y Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 6 / 16

  20. Background Communication Complexity of Stability Query Complexity Theorem (Main Result for Communication Complexity) Communication Complexity Let Alice hold the women’s preferences and let Bob hold the Proofs men’s preferences. The randomized (and deterministic) Open communication complexity of each of the following problems Problems is Ω ( n 2 ) : Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 7 / 16

  21. Background Communication Complexity of Stability Query Complexity Theorem (Main Result for Communication Complexity) Communication Complexity Let Alice hold the women’s preferences and let Bob hold the Proofs men’s preferences. The randomized (and deterministic) Open communication complexity of each of the following problems Problems is Ω ( n 2 ) : 1 finding a marriage close to being stable, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 7 / 16

  22. Background Communication Complexity of Stability Query Complexity Theorem (Main Result for Communication Complexity) Communication Complexity Let Alice hold the women’s preferences and let Bob hold the Proofs men’s preferences. The randomized (and deterministic) Open communication complexity of each of the following problems Problems is Ω ( n 2 ) : 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 7 / 16

  23. Background Communication Complexity of Stability Query Complexity Theorem (Main Result for Communication Complexity) Communication Complexity Let Alice hold the women’s preferences and let Bob hold the Proofs men’s preferences. The randomized (and deterministic) Open communication complexity of each of the following problems Problems is Ω ( n 2 ) : 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, 3 determining whether a given pair is contained in some/every stable marriage, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 7 / 16

  24. Background Communication Complexity of Stability Query Complexity Theorem (Main Result for Communication Complexity) Communication Complexity Let Alice hold the women’s preferences and let Bob hold the Proofs men’s preferences. The randomized (and deterministic) Open communication complexity of each of the following problems Problems is Ω ( n 2 ) : 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, 3 determining whether a given pair is contained in some/every stable marriage, 4 finding ε n pairs that appear in some/every stable marriage. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 7 / 16

  25. Background Communication Complexity of Stability Query Complexity Theorem (Main Result for Communication Complexity) Communication Complexity Let Alice hold the women’s preferences and let Bob hold the Proofs men’s preferences. The randomized (and deterministic) Open communication complexity of each of the following problems Problems is Ω ( n 2 ) : 1 finding a marriage close to being stable, 2 determining whether a given marriage is stable or far from, 3 determining whether a given pair is contained in some/every stable marriage, 4 finding ε n pairs that appear in some/every stable marriage. A lower bound on the communication complexity of each problem immediately implies the same lower bound on the number of Boolean queries in any algorithm for the same problem, even if arbitrary preprocessing of all the women’s preferences and of all the men’s preferences is allowed. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 7 / 16

  26. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  27. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  28. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  29. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  30. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  31. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  32. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  33. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  34. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  35. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  36. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  37. Background Previous Results Regarding CC of Finding Stability Query Complexity Theorem (Segal, 2007) Communication Complexity Any deterministic communication protocol among all 2 n Proofs participants for finding a stable marriage requires Ω ( n 2 ) bits of Open Problems communication. Theorem (Chou and Lu, 2010) Any deterministic and noninteractive communication protocol among all 2 n participants for finding an approximately-stable marriage requires Ω ( n 2 log n ) bits of communication. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 8 / 16

  38. On the Gap Between n 2 and n 2 log n Background Query • Our lower bound of Θ ( n 2 ) queries for verification is tight, Complexity Communication even in a weak comparison model. Complexity Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 9 / 16

  39. On the Gap Between n 2 and n 2 log n Background Query • Our lower bound of Θ ( n 2 ) queries for verification is tight, Complexity Communication even in a weak comparison model. Complexity • What about our Θ ( n 2 ) lower bound for finding a stable Proofs marriage? Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 9 / 16

  40. On the Gap Between n 2 and n 2 log n Background Query • Our lower bound of Θ ( n 2 ) queries for verification is tight, Complexity Communication even in a weak comparison model. Complexity • What about our Θ ( n 2 ) lower bound for finding a stable Proofs marriage? Open Problems • We do not know of any o ( n 2 log n ) algorithm for finding a stable marriage, even randomized, even in the strong 2-party communication model. . . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 9 / 16

  41. On the Gap Between n 2 and n 2 log n Background Query • Our lower bound of Θ ( n 2 ) queries for verification is tight, Complexity Communication even in a weak comparison model. Complexity • What about our Θ ( n 2 ) lower bound for finding a stable Proofs marriage? Open Problems • We do not know of any o ( n 2 log n ) algorithm for finding a stable marriage, even randomized, even in the strong 2-party communication model. . . • . . . nor do we have any improved ω ( n 2 ) lower bound, even for deterministic algorithms and even in the weak comparison model. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 9 / 16

  42. On the Gap Between n 2 and n 2 log n Background Query • Our lower bound of Θ ( n 2 ) queries for verification is tight, Complexity Communication even in a weak comparison model. Complexity • What about our Θ ( n 2 ) lower bound for finding a stable Proofs marriage? Open Problems • We do not know of any o ( n 2 log n ) algorithm for finding a stable marriage, even randomized, even in the strong 2-party communication model. . . • . . . nor do we have any improved ω ( n 2 ) lower bound, even for deterministic algorithms and even in the weak comparison model. Open Problem Consider the comparison model for stable marriage that only allows for queries of the form “does man m prefer woman w 1 over woman w 2 ?” and, dually, “does woman w prefer man m 1 over man m 2 ?”. How many such queries are required, in the worst case, to find a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 9 / 16

  43. Background Main Tool: The Disjointness Problem Query x = ( x i ) n y = ( y i ) n Complexity Let n ∈ N and let ¯ i =1 and ¯ i =1 be bit vectors. y ) � ¬ � n Communication The disjointness function is DISJ(¯ x , ¯ i =1 ( x i ∧ y i ), i.e., Complexity 0 if and only if there exists i s.t. x i = y i = 1. Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 10 / 16

  44. Background Main Tool: The Disjointness Problem Query x = ( x i ) n y = ( y i ) n Complexity Let n ∈ N and let ¯ i =1 and ¯ i =1 be bit vectors. y ) � ¬ � n Communication The disjointness function is DISJ(¯ x , ¯ i =1 ( x i ∧ y i ), i.e., Complexity 0 if and only if there exists i s.t. x i = y i = 1. Proofs Open Theorem (CC of DISJ (Kalyanasundaram and Schintger, Problems 1992; see also Razborov, 1992)) The randomized (and deterministic) communication complexity x ∈ { 0 , 1 } n is held by Alice and of calculating DISJ(¯ x , ¯ y ) , where ¯ y ∈ { 0 , 1 } n is held by Bob, is Θ ( n ) . ¯ Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 10 / 16

  45. Background Main Tool: The Disjointness Problem Query x = ( x i ) n y = ( y i ) n Complexity Let n ∈ N and let ¯ i =1 and ¯ i =1 be bit vectors. y ) � ¬ � n Communication The disjointness function is DISJ(¯ x , ¯ i =1 ( x i ∧ y i ), i.e., Complexity 0 if and only if there exists i s.t. x i = y i = 1. Proofs Open Theorem (CC of DISJ (Kalyanasundaram and Schintger, Problems 1992; see also Razborov, 1992)) The randomized (and deterministic) communication complexity x ∈ { 0 , 1 } n is held by Alice and of calculating DISJ(¯ x , ¯ y ) , where ¯ y ∈ { 0 , 1 } n is held by Bob, is Θ ( n ) . Moreover, this lower bound ¯ holds even for unique disjointness , i.e., if it is given that ¯ x � ≤ 1 . � � and ¯ y are either disjoint or uniquely intersecting : � ¯ x ∩ ¯ y Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 10 / 16

  46. Background Main Tool: The Disjointness Problem Query x = ( x i ) n y = ( y i ) n Complexity Let n ∈ N and let ¯ i =1 and ¯ i =1 be bit vectors. y ) � ¬ � n Communication The disjointness function is DISJ(¯ x , ¯ i =1 ( x i ∧ y i ), i.e., Complexity 0 if and only if there exists i s.t. x i = y i = 1. Proofs Open Theorem (CC of DISJ (Kalyanasundaram and Schintger, Problems 1992; see also Razborov, 1992)) The randomized (and deterministic) communication complexity x ∈ { 0 , 1 } n is held by Alice and of calculating DISJ(¯ x , ¯ y ) , where ¯ y ∈ { 0 , 1 } n is held by Bob, is Θ ( n ) . Moreover, this lower bound ¯ holds even for unique disjointness , i.e., if it is given that ¯ x � ≤ 1 . � � and ¯ y are either disjoint or uniquely intersecting : � ¯ x ∩ ¯ y All of our results follow from defining suitable embeddings of DISJ into various problems regarding stable marriages, i.e., mapping ¯ x into preferences for the women and and ¯ y into preferences for the men, such that solving the stability-related problem reveals the value of DISJ(¯ x , ¯ y ). Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 10 / 16

  47. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Communication Complexity Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  48. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  49. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  50. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems x = ( x i • Let ¯ j ) ( i , j ) ∈ P and y = ( y i ¯ j ) ( i , j ) ∈ P . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  51. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems x = ( x i • Let ¯ j ) ( i , j ) ∈ P and y = ( y i ¯ j ) ( i , j ) ∈ P . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  52. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems x = ( x i • Let ¯ j ) ( i , j ) ∈ P and y = ( y i ¯ j ) ( i , j ) ∈ P . Woman i : Men j Men j Man i > j = 1 > s.t. x i s.t. x i j = 0 Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  53. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems x = ( x i • Let ¯ j ) ( i , j ) ∈ P and y = ( y i ¯ j ) ( i , j ) ∈ P . Woman i : Men j Men j Man i > j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i Women i j = 1 > Woman j > s.t. y i s.t. y i j = 0 Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  54. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems x = ( x i • Let ¯ j ) ( i , j ) ∈ P and y = ( y i ¯ j ) ( i , j ) ∈ P . Woman i : Men j Men j Man i > j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i Women i j = 1 > Woman j > s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  55. Background Verifying / Finding an Exactly-Stable Marriage Query • P � [ n ] 2 \ { ( i , i ) | i ∈ [ n ] } Complexity Communication — The pairs of distinct Complexity elements of [ n ] � { 1 , . . . , n } . Proofs Open • Note: | P | = n · ( n − 1). Problems x = ( x i • Let ¯ j ) ( i , j ) ∈ P and y = ( y i ¯ j ) ( i , j ) ∈ P . Woman i : Men j Men j Man i > j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i Women i j = 1 > Woman j > s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  56. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity need not rank all candidates. Proofs Open Problems Woman i : Men j Men j Man i > j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i Women i j = 1 > Woman j > s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  57. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity need not rank all candidates. Proofs Open • For stability, we additionally Problems require that each person is married to someone who is ranked on their preference list. Woman i : Men j Men j Man i > j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i Women i j = 1 > Woman j > s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  58. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity need not rank all candidates. Proofs Open • For stability, we additionally Problems require that each person is married to someone who is ranked on their preference list. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  59. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity Theorem (Gale and Shapley, 1962) need not rank all candidates. Proofs Open There exists a men-optimal stable marriage. • For stability, we additionally Problems require that each person is married to someone who is ranked on their preference list. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  60. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity Theorem (Gale and Shapley, 1962) need not rank all candidates. Proofs Open There exists a men-optimal stable marriage. • For stability, we additionally Problems require that each person is Theorem (McVitie and Wilson, 1971) married to someone who is ranked on their preference list. The men-optimal stable marriage = the women-worst stable marriage. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  61. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity Theorem (Gale and Shapley, 1962) need not rank all candidates. Proofs Open There exists a men-optimal stable marriage. • For stability, we additionally Problems require that each person is Theorem (McVitie and Wilson, 1971) married to someone who is ranked on their preference list. The men-optimal stable marriage = the women-worst stable marriage. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Theorem (Rural Hospitals Theorem (Roth, 1984)) Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i Each participant is either single in all stable marriages s.t. y i s.t. y i j = 0 or married in all stable marriages. Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  62. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity need not rank all candidates. Proofs Open • For stability, we additionally Problems require that each person is married to someone who is ranked on their preference list. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 But what about finding a stable marriage? Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  63. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity need not rank all candidates. Proofs Open • For stability, we additionally Problems require that each person is married to someone who is ranked on their preference list. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 If this marriage is stable, then it is the unique stable marriage. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  64. Background Verifying / Finding an Exactly-Stable Marriage Query Complexity Marriage with partial preferences: Communication • A partial preferences list Complexity need not rank all candidates. Proofs Open • For stability, we additionally Problems require that each person is married to someone who is ranked on their preference list. Woman i : Men j Man i > ✟✟✟✟✟ ❍❍❍❍❍ Men j j = 1 > s.t. x i s.t. x i j = 0 Man j : Women i j = 1 > Woman j > ✟✟✟✟✟ ❍❍❍❍❍ Women i s.t. y i s.t. y i j = 0 Woman i and man j are blocking ⇔ x i j = 1 and y i j = 1 If this marriage is stable, then it is the unique stable marriage. (Embedding the problem w/partial preferences into the problem w/full preferences is not hard.) Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 11 / 16

  65. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  66. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  67. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Details of our Main CC Theorem The CC of each of the following is Ω ( n 2 ) : Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  68. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Details of our Main CC Theorem The CC of each of the following is Ω ( n 2 ) : 1 finding a (1 − ε ) -stable marriage, for fixed 0 ≤ ε < 1 2 . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  69. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Details of our Main CC Theorem The CC of each of the following is Ω ( n 2 ) : 1 finding a (1 − ε ) -stable marriage, for fixed 0 ≤ ε < 1 2 . 2 determining whether a given marriage is stable or ε -unstable, for fixed 0 ≤ ε < 1 , Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  70. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Details of our Main CC Theorem The CC of each of the following is Ω ( n 2 ) : 1 finding a (1 − ε ) -stable marriage, for fixed 0 ≤ ε < 1 2 . 2 determining whether a given marriage is stable or ε -unstable, for fixed 0 ≤ ε < 1 , 3 determining whether a given pair is contained in some/every stable marriage, Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  71. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Details of our Main CC Theorem The CC of each of the following is Ω ( n 2 ) : 1 finding a (1 − ε ) -stable marriage, for fixed 0 ≤ ε < 1 2 . 2 determining whether a given marriage is stable or ε -unstable, for fixed 0 ≤ ε < 1 , 3 determining whether a given pair is contained in some/every stable marriage, 4 finding ε n pairs that appear in some/every stable marriage, for fixed 0 ≤ ε < 1 . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  72. Background Distance from Stability Query • For any pair of perfect marriages, we define the divorce Complexity distance between them to be the number of pairs married Communication Complexity in the first but not in the second (equivalently, vice versa ). Proofs • We say that a marriage is ( 1 − ε ) -stable if its divorce Open Problems distance from some stable marriage is no more than ε n . We say that it is ε -unstable otherwise. Details of our Main CC Theorem The CC of each of the following is Ω ( n 2 ) : 1 finding a (1 − ε ) -stable marriage, for fixed 0 ≤ ε < 1 2 . 2 determining whether a given marriage is stable or ε -unstable, for fixed 0 ≤ ε < 1 , 3 determining whether a given pair is contained in some/every stable marriage, 4 finding ε n pairs that appear in some/every stable marriage, for fixed 0 ≤ ε < 1 . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 12 / 16

  73. Background Differentiating Between Stability and ε -Instability Query Complexity Communication Complexity Proofs Open Problems Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  74. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  75. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  76. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. • Color δ = δ ( ε ) of the black suites in red. Let x = ( x i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } and y = ( y i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } . Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  77. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. • Color δ = δ ( ε ) of the black suites in red. Let x = ( x i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } and y = ( y i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } . • Preferences (l2r within sets): Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  78. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. • Color δ = δ ( ε ) of the black suites in red. Let x = ( x i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } and y = ( y i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } . • Preferences (l2r within sets): • Gray: Red > Black > Gray. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  79. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. • Color δ = δ ( ε ) of the black suites in red. Let x = ( x i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } and y = ( y i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } . • Preferences (l2r within sets): • Gray: Red > Black > Gray. • Black: Gray > Rest. Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  80. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. • Color δ = δ ( ε ) of the black suites in red. Let x = ( x i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } and y = ( y i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } . • Preferences (l2r within sets): • Gray: Red > Black > Gray. • Black: Gray > Rest. • Red-suite woman i : Red-suite men j > Gray > Rest. s.t. x i j = 1 Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

  81. Background Differentiating Between Stability and ε -Instability Query • Key idea: embed unique Complexity Communication disjointness so that small Complexity changes in the preferences Proofs yield very large changes in the Open Problems structure of stable marriages. • Color δ = δ ( ε ) of the black suites in red. Let x = ( x i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } and y = ( y i ¯ j ) i , j ∈{ 1 ,..., δ n 2 } . • Preferences (l2r within sets): • Gray: Red > Black > Gray. • Black: Gray > Rest. • Red-suite woman i : Red-suite men j > Gray > Rest. s.t. x i j = 1 • Red-suite man j : Red-suite women i > Gray > Rest. s.t. y i j = 1 Yannai A. Gonczarowski (HUJI&MSR) A Stable Marriage Requires Communication January 5, 2015 13 / 16

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