The Complexity of Finding Tangles Oksana Firman, Philipp Kindermann, - PowerPoint PPT Presentation

The Complexity of Finding Tangles Oksana Firman, Philipp Kindermann, Alexander Wolff, Johannes Zink Julius-Maximilians-Universit¨ at W¨ urzburg, Germany Alexander Ravsky Stefan Felsner Pidstryhach Institute for Applied Problems TU Berlin, of Mechanics and Mathematics, Germany National Academy of Sciences of Ukraine, Lviv, Ukraine

Introduction Given an ordered set of n y -monotone wires

Introduction 1 ≤ i < j ≤ n Given an ordered set swap ij of n y -monotone wires i j

Introduction 1 ≤ i < j ≤ n Given an ordered set swap ij of n y -monotone wires disjoint swaps

Introduction 1 ≤ i < j ≤ n Given an ordered set swap ij of n y -monotone wires disjoint swaps adjacent permutations

Introduction 1 ≤ i < j ≤ n Given an ordered set swap ij of n y -monotone wires disjoint swaps adjacent permutations multiple swaps

Introduction 1 ≤ i < j ≤ n Given an ordered set swap ij of n y -monotone wires disjoint swaps π 1 adjacent π 2 permutations π 3 multiple swaps π 4 tangle T of height h ( T ) π 5 π 6

Introduction 1 ≤ i < j ≤ n Given an ordered set swap ij of n y -monotone wires · · · disjoint swaps 1 2 n π 1 adjacent π 2 permutations π 3 multiple swaps π 4 tangle T of height h ( T ) π 5 π 6

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 adjacent π 2 permutations π 3 multiple swaps π 4 tangle T of height h ( T ) π 5 π 6

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 2 π 4 tangle T of 1 height h ( T ) π 5 1 π 6

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 2 π 4 tangle T of 1 height h ( T ) π 5 1 π 6 Tangle T realizes list L .

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 2 π 4 tangle T of 1 height h ( T ) π 5 1 π 6 Tangle T realizes list L .

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 2 π 4 tangle T of 1 height h ( T ) π 5 1 not feasible π 6 Tangle T realizes list L .

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 2 π 4 tangle T of 1 height h ( T ) π 5 1 π 6 Tangle T realizes list L . A list L of swaps is feasible if there exists a tangle that realizes L . There may be multiple tangles realizing the same list of swaps.

Introduction 1 ≤ i < j ≤ n Given an ordered set . . . and given a list L swap ij of n y -monotone wires of swaps · · · disjoint swaps 1 2 n π 1 as a multiset ( ℓ ij ) adjacent 1 π 2 permutations 3 π 3 multiple swaps 1 2 π 4 tangle T of 1 height h ( T ) π 5 1 π 6 Tangle T realizes list L . A list L of swaps is feasible if there exists a tangle that realizes L . There may be multiple tangles realizing the same list of swaps.

Related Work • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for finding optimal-height tangles

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for ? Complexity ? finding optimal-height tangles

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for ? Complexity ? finding optimal-height tangles • Sado and Igarashi : A function for evaluating the computing time of a bubbling system. TCS 1987 Given: initial and final permutations

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for ? Complexity ? finding optimal-height tangles • Sado and Igarashi : A function for evaluating the computing time of a bubbling system. TCS 1987 Given: initial and final permutations • Bereg et al. : Drawing Permutations with Few Corners. GD 2013 Objective: minimize the number of bends

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for ? Complexity ? finding optimal-height tangles • Sado and Igarashi : A function for evaluating the computing time of a bubbling system. TCS 1987 Given: initial and final permutations • Bereg et al. : Drawing Permutations with Few Corners. GD 2013 Objective: minimize the number of bends • FKRWZ : Computing optimal-height tangles faster. GD 2019

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for ? Complexity ? finding optimal-height tangles • Sado and Igarashi : A function for evaluating the computing time of a bubbling system. TCS 1987 Given: initial and final permutations • Bereg et al. : Drawing Permutations with Few Corners. GD 2013 Objective: minimize the number of bends • FKRWZ : Computing optimal-height tangles faster. GD 2019 Faster exp.-time algorithm for finding optimal-height tangles

list Related Work � • Olszewski et al. : Visualizing the template of a chaotic attractor. GD 2018 Exp.-time algorithm for ? Complexity ? finding optimal-height tangles • Sado and Igarashi : A function for evaluating the computing time of a bubbling system. TCS 1987 Given: initial and final permutations • Bereg et al. : Drawing Permutations with Few Corners. GD 2013 Objective: minimize the number of bends • FKRWZ : Computing optimal-height tangles faster. GD 2019 Faster exp.-time algorithm for Finding optimal-height finding optimal-height tangles tangles is NP-hard

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard.

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT ( F ∨ F ∨ F )

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT ( F ∨ F ∨ F )

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT ( F ∨ F ∨ F ) ( T ∨ T ∨ T )

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT ( F ∨ F ∨ F ) ( T ∨ T ∨ T )

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT ( F ∨ F ∨ F ) ( T ∨ T ∨ T ) negative literals

Contribution Theorem. Deciding whether a given list of swaps is feasible is NP-hard. Proof. Reduction from Positive Not-All-Equal 3-SAT

Idea • Two wires build 4 loops that we consider

Idea λ λ ′ • Two wires build 4 loops that we consider λ ′ λ

Idea λ λ ′ • Two wires build 4 loops that we consider λ ′ λ

Idea λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , the other two false λ ′ λ

Idea λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , T the other two false T λ ′ λ

Idea λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , T the other two false T λ ′ λ

Idea λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , T the other two false T F F λ ′ λ

Idea λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , T the other two false • For each clause, there is a wire with an arm in each of the 4 loops. T F F λ ′ λ

Idea c i λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , T the other two false • For each clause, there is a wire with an arm in each of the 4 loops. T F F c i λ ′ λ

Idea c i λ λ ′ • Two wires build 4 loops that we consider • Two loops represent true , T the other two false • For each clause, there is a wire with an arm in each of the 4 loops. T F F c i λ ′ λ