THE COMPLEXITY OF NECKLACE SPLITTING, CONSENSUS-HALVING AND - - PowerPoint PPT Presentation

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THE COMPLEXITY OF NECKLACE SPLITTING, CONSENSUS-HALVING AND DISCRETE HAM SANDWICH From the papers: Consensus-Halving is PPA-Complete (STOC 2018). The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches (STOC 2019). joint works

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SLIDE 1

THE COMPLEXITY OF NECKLACE SPLITTING,
 CONSENSUS-HALVING AND DISCRETE HAM SANDWICH

From the papers:

Consensus-Halving is PPA-Complete (STOC 2018). The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches (STOC 2019). joint works with with P. W. Goldberg.

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SLIDE 2

NECKLACE SPLITTING (WITH TWO THIEVES)

An open necklace with an even number of beads of each of n colours. Cut the necklace into parts using n cuts. Assign a label (A or B) to each part (the name of the thief that gets it). Goal: A partition such that A and B have the same number of beads of each colour.

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SLIDE 3

A A B B

NECKLACE SPLITTING (WITH TWO THIEVES)

An open necklace with an even number of beads of each of n colours. Cut the necklace into parts using n cuts. Assign a label (A or B) to each part (the name of the thief that gets it). Goal: A partition such that A and B have the same number of beads of each colour.

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SLIDE 4

THE HISTORY OF NECKLACE SPLITTING

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SLIDE 5

THE HISTORY OF NECKLACE SPLITTING

  • Alon. Splitting Necklaces (Advances in Mathematics 1987).
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SLIDE 6

THE HISTORY OF NECKLACE SPLITTING

  • Alon. Splitting Necklaces (Advances in Mathematics 1987).

Alon and West. The Borsuk-Ulam Theorem and the Bisection of Necklaces (American Mathematical Society 1986).

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SLIDE 7

THE HISTORY OF NECKLACE SPLITTING

  • Alon. Splitting Necklaces (Advances in Mathematics 1987).

Alon and West. The Borsuk-Ulam Theorem and the Bisection of Necklaces (American Mathematical Society 1986). Bhatt and Leiserson. How to Assemble Tree Machines (STOC 1982).

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SLIDE 8

THE HISTORY OF NECKLACE SPLITTING

  • Alon. Splitting Necklaces (Advances in Mathematics 1987).

Alon and West. The Borsuk-Ulam Theorem and the Bisection of Necklaces (American Mathematical Society 1986). Bhatt and Leiserson. How to Assemble Tree Machines (STOC 1982). Hobby and Rice. A Moment Problem in L1 Approximation (American Mathematical Society 1965).

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SLIDE 9

THE HISTORY OF NECKLACE SPLITTING

  • Alon. Splitting Necklaces (Advances in Mathematics 1987).

Alon and West. The Borsuk-Ulam Theorem and the Bisection of Necklaces (American Mathematical Society 1986). Bhatt and Leiserson. How to Assemble Tree Machines (STOC 1982). Hobby and Rice. A Moment Problem in L1 Approximation (American Mathematical Society 1965).

  • Neyman. Un Théorème d’ Existence (C.R. Academie de Science 1942).
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SLIDE 10

A TOTAL PROBLEM

Total problem: A solution always exists. Proof by the Borsuk-Ulam Theorem (1933):

Let f : Sn → ℝn be a continuous function. Then, there exists x ∈ Sn such that f(x) = f(−x) . f(x)

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SLIDE 11

FINDING A SOLUTION

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SLIDE 12

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution?

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SLIDE 13

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution?

  • Alon. Non-constructive Proofs

in Combinatorics (International Congress of Mathematicians, 1990).

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SLIDE 14

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution? Despite Alon’s cautious optimism, no such algorithms exist!

  • Alon. Non-constructive Proofs

in Combinatorics (International Congress of Mathematicians, 1990).

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SLIDE 15

CONSENSUS-HALVING

  • F. Simmons and F. Su. Consensus-halving via theorems of Borsuk-Ulam and Tucker.

Mathematical Social Sciences, (2003).

A set of n agents with valuation functions

  • ver an interval (a resource).

These functions are explicitly representable (in time poly(n)) and bounded. Example: Piecewise constant functions. Halving: Cut the interval into pieces and label each piece by either (+) or (-). Consensus-halving: For each agent i, it holds that vi(+)= vi(-)

+ +

  • +

+

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SLIDE 16

CONSENSUS-HALVING

A solution that uses n cuts is guaranteed to exist. Simmons and Su (2003). There are instances for which n-1 cuts are not enough. Simmons and Su (2003).

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SLIDE 17

APPROXIMATE CONSENSUS- HALVING

For each agent i, it holds that |vi(+)-vi(-)| ≤ ε

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SLIDE 18

FINDING A SOLUTION

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SLIDE 19

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution?

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SLIDE 20

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution? Simmons and Su’s proof is constructive, but not polynomial-time.

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SLIDE 21

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution? Simmons and Su’s proof is constructive, but not polynomial-time. Actually:

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SLIDE 22

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution? Simmons and Su’s proof is constructive, but not polynomial-time. Actually: Consensus-Halving is a continuous analogue of Necklace-Splitting with two thieves.

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SLIDE 23

FINDING A SOLUTION

Is there an efficient algorithm for finding a solution? Simmons and Su’s proof is constructive, but not polynomial-time. Actually: Consensus-Halving is a continuous analogue of Necklace-Splitting with two thieves. Alon’s proof (1987) of existence for NS goes via CH.

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SLIDE 24

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 25

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 26

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 27

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 28

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 29

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 30

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 31

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 32

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 33

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 34

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 35

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 36

FROM CONSENSUS-HALVING TO NECKLACE SPLITTING

A A B B

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SLIDE 37

FROM NECKLACE SPLITTING TO CONSENSUS-HALVING

Idea: Simulate value blocks by beads
 Denser blocks => more beads.

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SLIDE 38

IN TERMS OF COMPLEXITY…

To prove computational hardness for NS, it suffices to prove computational hardness for ε-CH.

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SLIDE 39

DISCRETE HAM SANDWICH

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SLIDE 40

DISCRETE HAM SANDWICH

d sets of n points in d- dimensional Euclidean space. Find a hyperplane that splits all point sets in half.

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SLIDE 41

DISCRETE HAM SANDWICH

d sets of n points in d- dimensional Euclidean space. Find a hyperplane that splits all point sets in half.

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SLIDE 42

HAM SANDWICHES THROUGHOUT THE YEARS

  • Steinhaus. A Note on the Ham Sandwich Theorem (Mathesis Polska 1938).

Stone and Turkey. Generalized ‘’Sandwich’’ Theorems (Duke Mathematical Journal 1942). Edelsbrunner and Waupotitsch. Computing a Ham-Sandwich Cut in Two Dimensions (Symbolic Computation 1986). Lo, Matoušek and Steiger. Ham-Sandwich Cuts in R^d (STOC 1992). Lo, Matoušek and Steiger. Algorithms for Ham-Sandwich Cuts (Discrete and Computational Geometry 1994).

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SLIDE 43

FINDING A SOLUTION

Total problem: A solution always exists. Again, by the Borsuk-Ulam Theorem.

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SLIDE 44

FROM DISCRETE HAM SANDWICH TO NECKLACE SPLITTING

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SLIDE 45

FROM DISCRETE HAM SANDWICH TO NECKLACE SPLITTING

Consider the moment curve (α, α2, …, αd), for α ∈ [0,1] .

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SLIDE 46

FROM DISCRETE HAM SANDWICH TO NECKLACE SPLITTING

Consider the moment curve (α, α2, …, αd), for α ∈ [0,1] .

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SLIDE 47

FROM DISCRETE HAM SANDWICH TO NECKLACE SPLITTING

Consider the moment curve (α, α2, …, αd), for α ∈ [0,1] . 1 α

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SLIDE 48

FROM DISCRETE HAM SANDWICH TO NECKLACE SPLITTING

Consider the moment curve (α, α2, …, αd), for α ∈ [0,1] . 1 α Insert a red point at (α, α2, …, αd) .

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SLIDE 49

FROM DISCRETE HAM SANDWICH TO NECKLACE SPLITTING

Consider the moment curve (α, α2, …, αd), for α ∈ [0,1] . 1 α Insert a red point at (α, α2, …, αd) . The two thieves take alternating pieces.

First thief Second thief

Any hyperplane intersects the moment curve in at most d points.

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SLIDE 50

IN TERMS OF COMPLEXITY…

To prove computational hardness for NS, it suffices to prove computational hardness for ε-CH.

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SLIDE 51

IN TERMS OF COMPLEXITY…

To prove computational hardness for NS, it suffices to prove computational hardness for ε-CH. To prove computational hardness for DHS, it suffices to prove computational hardness for NS.

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SLIDE 52

IN TERMS OF COMPLEXITY…

To prove computational hardness for NS, it suffices to prove computational hardness for ε-CH. To prove computational hardness for DHS, it suffices to prove computational hardness for NS. It suffices to prove computational hardness for ε-CH.

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SLIDE 53

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists.

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SLIDE 54

COMPLEXITY CLASSES

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SLIDE 55

COMPLEXITY CLASSES

TFNP

Meggido and Papadimitriou (Theoretical Computer Science 1991). “Total” Search Problems, for which a solution is guaranteed to exist and can be verified in polynomial time.

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SLIDE 56

COMPLEXITY CLASSES

TFNP

Meggido and Papadimitriou (Theoretical Computer Science 1991). “Total” Search Problems, for which a solution is guaranteed to exist and can be verified in polynomial time.

PPA

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem LEAF.

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SLIDE 57

COMPLEXITY CLASSES

TFNP

Meggido and Papadimitriou (Theoretical Computer Science 1991). “Total” Search Problems, for which a solution is guaranteed to exist and can be verified in polynomial time.

PPA

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem LEAF.

PPAD

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem END-OF-LINE.

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SLIDE 58

COMPLEXITY CLASSES

TFNP

Meggido and Papadimitriou (Theoretical Computer Science 1991). “Total” Search Problems, for which a solution is guaranteed to exist and can be verified in polynomial time.

PPA

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem LEAF.

PPAD

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem END-OF-LINE.

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SLIDE 59

COMPLEXITY CLASSES

TFNP

Meggido and Papadimitriou (Theoretical Computer Science 1991). “Total” Search Problems, for which a solution is guaranteed to exist and can be verified in polynomial time.

PPA

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem LEAF.

PPAD

Papadimitriou (Journal of Computer and System Sciences, 1994). Problems reducible to the problem END-OF-LINE.

PWPP

PLS PPP

PPADS

CLS FP

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SLIDE 60

SUCCESS OF PPAD

Daskalakis, Goldberg and Papadimitriou. The Complexity of Computing a Nash equilibrium. (SIAM Journal of Computing, 2009). Chen, Deng and Tang Settling the Complexity of Computing 2-Player Nash Equilibria. (Journal of the ACM, 2009).

2011 SIAM Outstanding Paper Prize 2008 Kalai Prize 2008 ACM Doctoral Dissertation Award

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SLIDE 61

PPAD

END-OF-LINE: Input: A (exponentially large, with 2n vertices, implicitly given) directed graph, where each vertex has in-degree and out- degree at most 1 and a vertex with in-degree 0. Output: A vertex with in-degree or out-degree 0.

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SLIDE 62

END-OF-LINE

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SLIDE 63

END-OF-LINE

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SLIDE 64

END-OF-LINE

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SLIDE 65

END-OF-LINE

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SLIDE 66

END-OF-LINE

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SLIDE 67

END-OF-LINE

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SLIDE 68

END-OF-LINE

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SLIDE 69

END-OF-LINE

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SLIDE 70

END-OF-LINE

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SLIDE 71

END-OF-LINE

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SLIDE 72

END-OF-LINE

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SLIDE 73

END-OF-LINE

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SLIDE 74

END-OF-LINE

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SLIDE 75

END-OF-LINE

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SLIDE 76

PPA

LEAF: Input: An undirected (exponentially large, implicitly given) undirected graph where each vertex has degree at most 2 and a vertex of degree 1. Output: Another vertex of degree 1.

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SLIDE 77

PPAD AND PPA

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SLIDE 78

PPAD AND PPA

PPAD

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SLIDE 79

PPAD AND PPA

PPAD Stands for “Polynomial Parity Argument on a Directed graph”.

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SLIDE 80

PPAD AND PPA

PPAD Stands for “Polynomial Parity Argument on a Directed graph”. A problem is in PPAD if it is polynomial-time reducible to END-OF-LINE.

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SLIDE 81

PPAD AND PPA

PPAD Stands for “Polynomial Parity Argument on a Directed graph”. A problem is in PPAD if it is polynomial-time reducible to END-OF-LINE. A problem is PPAD-hard if END-OF-LINE is polynomial-time reducible to it.

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SLIDE 82

PPAD AND PPA

PPAD Stands for “Polynomial Parity Argument on a Directed graph”. A problem is in PPAD if it is polynomial-time reducible to END-OF-LINE. A problem is PPAD-hard if END-OF-LINE is polynomial-time reducible to it. PPA

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SLIDE 83

PPAD AND PPA

PPAD Stands for “Polynomial Parity Argument on a Directed graph”. A problem is in PPAD if it is polynomial-time reducible to END-OF-LINE. A problem is PPAD-hard if END-OF-LINE is polynomial-time reducible to it. PPA Stands for “Polynomial Parity Argument”.

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SLIDE 84

PPAD AND PPA

PPAD Stands for “Polynomial Parity Argument on a Directed graph”. A problem is in PPAD if it is polynomial-time reducible to END-OF-LINE. A problem is PPAD-hard if END-OF-LINE is polynomial-time reducible to it. PPA Stands for “Polynomial Parity Argument”. Containment and hardness defined with respect to polynomial-time reductions to/ from LEAF.

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SLIDE 85

THE COMPLEXITY OF THE THREE PROBLEMS.

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SLIDE 86

THE COMPLEXITY OF THE THREE PROBLEMS.

They are all in PPA.


[Papadimitriou 1994, F.R., Frederiksen, Goldberg and Zhang 2019, F.R. and Goldberg 2019].

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SLIDE 87

THE COMPLEXITY OF THE THREE PROBLEMS.

They are all in PPA.


[Papadimitriou 1994, F.R., Frederiksen, Goldberg and Zhang 2019, F.R. and Goldberg 2019].

Simmons and Su’s proof already almost an “in PPA” result.

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SLIDE 88

THE COMPLEXITY OF THE THREE PROBLEMS.

They are all in PPA.


[Papadimitriou 1994, F.R., Frederiksen, Goldberg and Zhang 2019, F.R. and Goldberg 2019].

Simmons and Su’s proof already almost an “in PPA” result. What about hardness?

slide-89
SLIDE 89

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists.

slide-90
SLIDE 90

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA

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SLIDE 91

A DEEPER LOOK INTO PPA-COMPLETE PROBLEMS

Let’s see what we have to reduce from!

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SLIDE 92

COMPLETE PROBLEMS FOR PPA AND PPAD

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SLIDE 93

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitiou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

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SLIDE 94

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitiou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2020). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein

  • bottle. Deng, Feng, Liu and Qi (2015).

Not many more …

slide-95
SLIDE 95

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitiou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2020). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein

  • bottle. Deng, Feng, Liu and Qi (2015).

Not many more …

Consider a triangulated simplex and a polynomial-time machine (or a circuit) that assigns labels to the vertices of the triangulation…

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SLIDE 96

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitiou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2020). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein

  • bottle. Deng, Feng, Liu and Qi (2015).

Not many more …

slide-97
SLIDE 97

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitiou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2020). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein

  • bottle. Deng, Feng, Liu and Qi (2015).

Not many more …

Consider a triangulated hypergrid and a polynomial-time machine (or a circuit) that assigns labels to the vertices of the triangulation…

slide-98
SLIDE 98

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitiou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2020). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein

  • bottle. Deng, Feng, Liu and Qi (2015).

Not many more …

slide-99
SLIDE 99

“NATURAL” PPA-COMPLETE PROBLEMS?

slide-100
SLIDE 100

“NATURAL” PPA-COMPLETE PROBLEMS?

Papadimitriou (1994)

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SLIDE 101

“NATURAL” PPA-COMPLETE PROBLEMS?

Papadimitriou (1994) Grigni (2001)

slide-102
SLIDE 102

“NATURAL” PPA-COMPLETE PROBLEMS?

Papadimitriou (1994) Grigni (2001) Aisenberg, Bonet and Buss (2020)

slide-103
SLIDE 103

NATURAL PROBLEMS

Problems that do not have a circuit explicit in their definition.
 (Papadimitriou (1994), Grigni (2001), Aisenberg, Bonet and Buss (2020)). Problems that were identified independently from the work on TFNP . (Goldberg (2019), Algorithms UK).

slide-104
SLIDE 104

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA

slide-105
SLIDE 105

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA PPA: a lonely class No natural complete problems

slide-106
SLIDE 106

NATURAL PROBLEMS

Problems that do not have a circuit explicit in their definition.
 (Papadimitriou (1994), Grigni (2001), Aisenberg, Bonet and Buss (2020)). Problems that were identified independently from the work on TFNP . (Goldberg (2019), Algorithms UK).

slide-107
SLIDE 107

NATURAL PROBLEMS

Problems that do not have a circuit explicit in their definition.
 (Papadimitriou (1994), Grigni (2001), Aisenberg, Bonet and Buss (2020)). Problems that were identified independently from the work on TFNP . (Goldberg (2019), Algorithms UK).

Necklace Splitting and Consensus-Halving are natural! Two birds with one stone?

slide-108
SLIDE 108

A NATURAL PPA-COMPLETE PROBLEM

slide-109
SLIDE 109

A NATURAL PPA-COMPLETE PROBLEM

F.R. and Goldberg. Consensus-Halving is PPA-complete. (STOC 2018).

slide-110
SLIDE 110

A NATURAL PPA-COMPLETE PROBLEM

F.R. and Goldberg. Consensus-Halving is PPA-complete. (STOC 2018).

When ε is inversely exponential.

inversely exponential inversely polynomial constant problem becomes easier hardness becomes more difficult to prove

slide-111
SLIDE 111

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA PPA: a lonely class No natural complete problems

slide-112
SLIDE 112

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA PPA: a lonely class No natural complete problems CH is PPA-complete

[F.R. and Goldberg, 2018]

slide-113
SLIDE 113

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA PPA: a lonely class No natural complete problems CH is PPA-complete

[F.R. and Goldberg, 2018]

PPA now has a natural complete problem.

slide-114
SLIDE 114

NECKLACES AND SANDWICHES

F.R. and Goldberg. The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches (STOC 2019).

slide-115
SLIDE 115

NECKLACES AND SANDWICHES

F.R. and Goldberg. The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches (STOC 2019). We prove that ε-CONSENSUS HALVING is PPA-Complete for inverse- polynomial ε.

slide-116
SLIDE 116

NECKLACES AND SANDWICHES

F.R. and Goldberg. The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches (STOC 2019). We prove that ε-CONSENSUS HALVING is PPA-Complete for inverse- polynomial ε. This implies that NECKLACE SPLITTING is PPA-Complete.

slide-117
SLIDE 117

NECKLACES AND SANDWICHES

F.R. and Goldberg. The Complexity of Splitting Necklaces and Bisecting Ham Sandwiches (STOC 2019). We prove that ε-CONSENSUS HALVING is PPA-Complete for inverse- polynomial ε. This implies that NECKLACE SPLITTING is PPA-Complete. This also implies that DISCRETE HAM SANDWICH is PPA-Complete.

slide-118
SLIDE 118

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA PPA: a lonely class No natural complete problems CH is PPA-complete

[F.R. and Goldberg, 2018]

PPA now has a natural complete problem.

slide-119
SLIDE 119

THE STATE OF THE WORLD

Necklace Splitting
 always exists. Discrete Ham Sandwich
 always exists. ε-Consensus-Halving
 always exists. in PPA in PPA in PPA PPA: a lonely class No natural complete problems CH is PPA-complete

[F.R. and Goldberg, 2018]

PPA now has a natural complete problem. NS is PPA-complete DHS is PPA-complete

[F.R. and Goldberg 2019] [F.R. and Goldberg 2019]

PPA now has natural complete problems.

slide-120
SLIDE 120

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitriou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2015). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein bottle. Deng, Feng, Liu and Qi (2015).

slide-121
SLIDE 121

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitriou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2015). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein bottle. Deng, Feng, Liu and Qi (2015). CONSENSUS-HALVING, NECKLACE SPLITTING, DISCRETE HAM SANDWICH. F.R., Goldberg (2018, 2019)

slide-122
SLIDE 122

COMPLETE PROBLEMS FOR PPA AND PPAD

SPERNER, BROUWER, KAKUTANI Papadimitriou (1994). NASH Daskalakis, Goldberg and Papadimitriou (2005, 2009), Chen and Deng (2007, 2009). EXCHANGE ECONOMY Papadimitriou (1994), Chen, Paparas and Yiannakakis (2013). ENVY-FREE CAKE CUTTING Deng, Qi and Saberi (2009, 2012). Many more …

SPERNER for non-orientable spaces Grigni (2001), Friedl, Ivanyos, Santha and Verhoeven (2006). 2D-TUCKER, BORSUK-ULAM Aisenberg, Bonet and Buss (2015). OCTAHEDRAL TUCKER Deng, Feng and Kulkarni (2017). TUCKER, SPERNER on Möbius band and Klein bottle. Deng, Feng, Liu and Qi (2015). CONSENSUS-HALVING, NECKLACE SPLITTING, DISCRETE HAM SANDWICH. F.R., Goldberg (2018, 2019) More?

slide-123
SLIDE 123

NECKLACE SPLITTING WITH MANY THIEVES

slide-124
SLIDE 124

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete.

slide-125
SLIDE 125

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

slide-126
SLIDE 126

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

F.R., Hollender, Sotiraki and Zampetakis. 
 A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting (SODA 2020).

slide-127
SLIDE 127

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

F.R., Hollender, Sotiraki and Zampetakis. 
 A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting (SODA 2020).

Necklace Splitting with p thieves is in PPA-p, for p a prime power.

slide-128
SLIDE 128

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

F.R., Hollender, Sotiraki and Zampetakis. 
 A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting (SODA 2020).

Necklace Splitting with p thieves is in PPA-p, for p a prime power. What about hardness?

slide-129
SLIDE 129

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

F.R., Hollender, Sotiraki and Zampetakis. 
 A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting (SODA 2020).

Necklace Splitting with p thieves is in PPA-p, for p a prime power. What about hardness?

F.R., Hollender, Sotiraki and Zampetakis. 
 Consensus Halving: Does It Ever Get Easier?. (EC 2020).

slide-130
SLIDE 130

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

F.R., Hollender, Sotiraki and Zampetakis. 
 A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting (SODA 2020).

Necklace Splitting with p thieves is in PPA-p, for p a prime power. What about hardness?

F.R., Hollender, Sotiraki and Zampetakis. 
 Consensus Halving: Does It Ever Get Easier?. (EC 2020).

Some evidence of hardness, but still far from it.

slide-131
SLIDE 131

NECKLACE SPLITTING WITH MANY THIEVES

For k=2, the problem is PPA-complete. What about general k?

F.R., Hollender, Sotiraki and Zampetakis. 
 A Topological Characterization of Modulo-p Arguments and Implications for Necklace Splitting (SODA 2020).

Necklace Splitting with p thieves is in PPA-p, for p a prime power. What about hardness?

F.R., Hollender, Sotiraki and Zampetakis. 
 Consensus Halving: Does It Ever Get Easier?. (EC 2020).

Some evidence of hardness, but still far from it. Biggest open problem: Is Necklace Splitting with p thieves PPA-p complete?