Reconfiguration of Common Independent Sets of Matroids Moritz M - PowerPoint PPT Presentation

Reconfiguration of Common Independent Sets of Matroids Moritz M¨ uhlenthaler TU Dortmund Aussois, 2018 Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 1 / 21

Reconfiguration of Common Independent Sets of Matroids Moritz M¨ uhlenthaler TU Dortmund Aussois, 2018 Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 2 / 21

Reconfiguration Problems transformation step: rotate a face of the cube question: can we reach the target configuration, where each face has a single color? ≈ 51 · 10 19 configurations! (rules out exploration/enumeration. . . ) Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 3 / 21

Reconfiguration Graphs adjacency of configurations yields graph structure, the reconfiguration graph Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 4 / 21

Reconfiguration Graphs adjacency of configurations yields graph structure, the reconfiguration graph in the following: configurations are solutions of an instance of some combinatorial problem Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 4 / 21

Reconfiguration Graphs adjacency of configurations yields graph structure, the reconfiguration graph in the following: configurations are solutions of an instance of some combinatorial problem Given an instance of some search problem P , the problem P Reconfiguration asks for the existence of an st -path in the reconfiguration graph. Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 4 / 21

Reconfiguration Graphs adjacency of configurations yields graph structure, the reconfiguration graph in the following: configurations are solutions of an instance of some combinatorial problem Given an instance of some search problem P , the problem P Reconfiguration asks for the existence of an st -path in the reconfiguration graph. △ The size of a reconfiguration graph is generally ! exponential in the size of the instance. Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 4 / 21

Complexity of Reconfiguration many problems P exhibit the following pattern P ∈ P ⇒ P Reconfiguration ∈ P P is NP-hard ⇒ P Reconfiguration is PSPACE-compl. Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 5 / 21

Complexity of Reconfiguration many problems P exhibit the following pattern P ∈ P ⇒ P Reconfiguration ∈ P P is NP-hard ⇒ P Reconfiguration is PSPACE-compl. exceptions are known, for example ◮ 3-Coloring Reconfiguration is in P [Cereceda et al., 2011] ◮ Shortest Path Reconfiguration is PSPACE-complete [Bonsma, 2013] complexity may depend on choice of the adjacency relation Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 5 / 21

Tools and Techniques PSPACE-hardness non-deterministic constraint logic (NCL) 1 reductions preserving reachability 2 polynomial-time algorithms dynamic programming 1 kernelization 2 alternating paths algorithms 3 Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 6 / 21

Matroids and Reconfiguration (1) Why matroids are natural structures for reconfiguration We can characterize feasible solutions of combinatorial problems using common independent sets of matroids and use exchange properties for adjacency. Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 7 / 21

Matroids and Reconfiguration (1) Why matroids are natural structures for reconfiguration We can characterize feasible solutions of combinatorial problems using common independent sets of matroids and use exchange properties for adjacency. known results ◮ basis graph characterization [Maurer, 1972] ◮ reconfiguration of two weighted bases always possible such that the weight does not exeed that of the heavier one [Ito et al., 2011] ◮ reconfiguration of ordered bases [Lubiw and Pathak, 2016] Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 7 / 21

Matroids and Reconfiguration (2) Feasible solutions of many combinatorial problems can be characterized by common independent sets of ℓ ≥ 1 matroids: ≤ 1 ≤ 1 ≤ 1 ≤ 1 ≤ 1 ≤ 2 ≤ 1 ≤ 1 ≤ 1 Bipartite Matching Graph Orientation Colorful Spanning Tree Directed Hamilton Path Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 8 / 21

Common Independent Set Reconfiguration ℓ - Common Independent Set (CIS) Reconfiguration input Independence oracles for matroids M 1 , M 2 , . . . , M ℓ , each on ground-set X , S , T ∈ X such that S and T independent in each matroid, number k ∈ N question Is T reachable from S in the reconfiguration graph G k ( M 1 , M 2 , . . . , M ℓ ) = ( V , E )? V := { A ⊆ E | A independent in M 1 , M 2 , . . . , M ℓ , | A | ≥ k − 1 } E := { AB | A , B ∈ V , | A △ B | = 1 } Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 9 / 21

Results Theorem (M., 2017) 2- CIS Reconfiguration admits a polynomial-time algorithm. Theorem (M., 2017) For ℓ ≥ 3 , ℓ - CIS Reconfiguration is PSPACE -complete, even for a very restricted class of matroids. very restricted class: 1-uniform partition matroids having blocks of size at most two Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 10 / 21

2- CIS Reconfiguration (1) variant of the alternating paths technique two common independent sets are connected iff their symmetric difference contains no “frozen” structure Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 11 / 21

2- CIS Reconfiguration (1) variant of the alternating paths technique two common independent sets are connected iff their symmetric difference contains no “frozen” structure obstructions to reconfiguration ≤ 2 ≤ 1 ≤ 1 ≤ 2 ≤ 2 ≤ 1 ≤ 1 M 1 partition matroid M 1 graphic matroid M 2 partition matroid M 2 partition matroid Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 11 / 21

2- CIS Reconfiguration (2) obstructions are (essentially) chordless cycles in the exchange graph ≤ 2 a b a f c d e b e f ≤ 2 ≤ 2 c d I \ J J \ I arcs in the exchange graph y x iff I − x + y independent in M 1 y x iff I − x + y independent in M 2 Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 12 / 21

3- CIS Reconfiguration (1) Theorem For ℓ ≥ 3 , ℓ - CIS Reconfiguration is PSPACE -complete, even for 1-uniform partition matroids having blocks of size at most two. M 2 M 1 M 3 Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 13 / 21

3- CIS Reconfiguration (2) [Gopalan et al., 2009] 3-SAT Reconf. [Ito et al., 2011] Stable Set Reconf. [M., 2017] 3-CIS Reconfiguration new reduction required Construct partition matroids from 3-edge-coloring of the stable set instance graph. Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 14 / 21

3- CIS Reconfiguration (3) 3-SAT Reconfiguration − → Stable Set Reconf. example ϕ = ( x 1 ∨ x 2 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 2 ∨ x 3 ) [Ito et al., 2011] x 1 x 1 x 2 x 2 x 3 x 3 x 2 x 1 x 3 x 2 x 1 x 2 x 3 Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 15 / 21

3- CIS Reconfiguration (3) 3-SAT Reconfiguration − → Stable Set Reconf. example ϕ = ( x 1 ∨ x 2 ) ∧ ( x 1 ∨ x 2 ∨ x 3 ) ∧ ( x 2 ∨ x 3 ) [Ito et al., 2011] x 1 x 1 x 2 x 2 x 3 x 3 x 2 x 1 x 3 x 2 x 1 x 2 x 3 max. stable set correspond to satisfying assignments subgraph induced by a literal is a complete bipartite graph Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 15 / 21

3- CIS Reconfiguration (4) 3-SAT Reconfiguration − → Stable Set Reconf. replace each subgraph induced by a literal by the following gadget x x x . . . . . . x x x Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 16 / 21

3- CIS Reconfiguration (5) 3-SAT Reconfiguration − → Stable Set Reconf. x x x = True x x x x x = False x x Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 17 / 21

3- CIS Reconfiguration (6) Stable Set Reconf. − → 3- CIS Reconfiguration ¬ x 2 ¬ x 2 x 2 ¬ x 1 x 3 x 1 ¬ x 3 construct partition matroids M 1 , M 2 , M 3 from the colored edges Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 18 / 21

NAE 3SAT Reconfiguration Corollary (M., 2017) Monotone 4-Occ NAE 3-SAT Reconfiguration is PSPACE -complete. x x x x x T T T T T T T T T T T T p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 F F F NAE 3-SAT version of the path gadget Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 19 / 21

Conclusions and Future Directions solution graphs of combinatorial problems are interesting objects matroids capture solution graphs of many such problems reachability in the solutions graph. . . ◮ can be decided in polynomial time for common independent sets of two matroids ◮ is PSPACE-complete for common independent sets of three of more matroids What is the reconfiguration complexity for related structures? Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 20 / 21

Thank you! Moritz M¨ uhlenthaler (TU Dortmund) Aussois, 2018 21 / 21