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HolographicAlgorithms Beyond Matchgates Heng Guo (joint work with Jin-Yi Cai and Tyson Williams ) University of Wisconsin-Madison Kbenhavn July 11th 2014 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 1 / 22 Counting

  1. HolographicAlgorithms Beyond Matchgates Heng Guo (joint work with Jin-Yi Cai and Tyson Williams ) University of Wisconsin-Madison København July 11th 2014 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 1 / 22

  2. Counting Perfect Matchings Perfect Matchings f 1 f 2 f 1 f 2 f 3 f 4 f 3 f 1 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 2 / 22

  3. Counting Perfect Matchings Perfect Matchings f 1 f 2 f 1 f 2 f 3 f 4 f 3 f 1 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 2 / 22

  4. Counting Perfect Matchings Perfect Matchings f 1 f 2 f 1 f 2 f 3 f 4 f 3 f 1 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 2 / 22

  5. However, for planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61] . The FKT algorithm is based on Pfaffian orientations of planar graphs. FKT Algorithm Counting Perfect Matchings is # P -hard [Valiant 79] in general graphs. Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 3 / 22

  6. The FKT algorithm is based on Pfaffian orientations of planar graphs. FKT Algorithm Counting Perfect Matchings is # P -hard [Valiant 79] in general graphs. However, for planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61] . Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 3 / 22

  7. FKT Algorithm Counting Perfect Matchings is # P -hard [Valiant 79] in general graphs. However, for planar graphs, there is a polynomial time algorithm [Kastelyn 61 & 67, Temperley and Fisher 61] . The FKT algorithm is based on Pfaffian orientations of planar graphs. Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 3 / 22

  8. Put functions Exact-One ( EO ) on nodes and make edges variables. # PM is just the partition function: EO d E v v V E 0 1 PM Counting Perfect Matchings Revisited A systematic way to view # PM . Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 4 / 22

  9. # PM is just the partition function: EO d E v E 0 1 v V PM Counting Perfect Matchings Revisited A systematic way to view # PM . Put functions Exact-One ( EO ) on nodes and make edges variables. EO 3 EO 4 EO 3 EO 3 EO 4 EO 3 EO 4 EO 3 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 4 / 22

  10. # PM is just the partition function: EO d E v E 0 1 v V PM Counting Perfect Matchings Revisited A systematic way to view # PM . Put functions Exact-One ( EO ) on nodes and make edges variables. EO 3 EO 4 EO 3 EO 4 EO 4 EO 3 EO 4 EO 3 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 4 / 22

  11. Counting Perfect Matchings Revisited A systematic way to view # PM . Put functions Exact-One ( EO ) on nodes and make edges variables. # PM is just the partition function: ∑ ∏ # PM = EO d ( σ | E ( v ) ) . σ : E �→ { 0 , 1 } v ∈ V EO 3 EO 4 EO 3 EO 4 EO 4 EO 3 EO 4 EO 3 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 4 / 22

  12. It is parameterized by a function set with f v . Also known as: Read-Twice #CSP , Tensor Networks, Graphical Models… Holant Problems The (Boolean) Holant problem on instance Ω is to evaluate Holant Ω = ∑ ∏ f v ( σ | E ( v ) ) , v ∈ V σ : E �→ { 0 , 1 } a sum over all edge assignments σ : E → { 0 , 1 } . Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 5 / 22

  13. Also known as: Read-Twice #CSP , Tensor Networks, Graphical Models… Holant Problems The (Boolean) Holant problem on instance Ω is to evaluate Holant Ω = ∑ ∏ f v ( σ | E ( v ) ) , v ∈ V σ : E �→ { 0 , 1 } a sum over all edge assignments σ : E → { 0 , 1 } . It is parameterized by a function set F with f v ∈ F . Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 5 / 22

  14. Tensor Networks, Graphical Models… Holant Problems The (Boolean) Holant problem on instance Ω is to evaluate Holant Ω = ∑ ∏ f v ( σ | E ( v ) ) , v ∈ V σ : E �→ { 0 , 1 } a sum over all edge assignments σ : E → { 0 , 1 } . It is parameterized by a function set F with f v ∈ F . Also known as: Read-Twice #CSP , Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 5 / 22

  15. Graphical Models… Holant Problems The (Boolean) Holant problem on instance Ω is to evaluate Holant Ω = ∑ ∏ f v ( σ | E ( v ) ) , v ∈ V σ : E �→ { 0 , 1 } a sum over all edge assignments σ : E → { 0 , 1 } . It is parameterized by a function set F with f v ∈ F . Also known as: Read-Twice #CSP , Tensor Networks, Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 5 / 22

  16. Holant Problems The (Boolean) Holant problem on instance Ω is to evaluate Holant Ω = ∑ ∏ f v ( σ | E ( v ) ) , v ∈ V σ : E �→ { 0 , 1 } a sum over all edge assignments σ : E → { 0 , 1 } . It is parameterized by a function set F with f v ∈ F . Also known as: Read-Twice #CSP , Tensor Networks, Graphical Models… Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 5 / 22

  17. f y 1 x 1 x 2 EO 3 EO 3 y 2 The function f on the right is (2,0,0,1). This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings View some functions together as a new one. Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  18. f The function f on the right is (2,0,0,1). This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  19. The function f on the right is (2,0,0,1). This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 f Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  20. This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 f The function f on the right is (2,0,0,1). Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  21. This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 f The function f on the right is (2,0,0,1). Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  22. This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 f The function f on the right is (2,0,0,1). Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  23. This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 f The function f on the right is (2,0,0,1). Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  24. This is also called tensor contraction. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 x 1 x 2 View some functions together as a new one. EO 3 EO 3 y 2 f The function f on the right is (2,0,0,1). Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

  25. Given functions f 1 , f 2 , and a partition x 1 and x 2 of variables x , the contraction g is: g x f 1 x 1 y f 2 x 2 y y If a set of functions is tractable, then any function expressible by is also tractable. Functions Expressible by Perfect Matchings y 1 View some functions together as a new one. x 1 x 2 EO 3 EO 3 y 2 f The function f on the right is (2,0,0,1). This is also called tensor contraction. Heng Guo (CS, UW-Madison) General Holographic Algorithms ICALP 2014 6 / 22

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