Document

**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We investigate generalizations of the graph theoretic notions of effective resistance and capacitance to simplicial complexes and prove analogs of formulas known in the case of graphs. In graphs the effective resistance between two vertices is O(n); however, we show that in a simplicial complex the effective resistance of a null-homologous cycle may be exponential. This is caused by relative torsion in the simplicial complex. We provide upper bounds on both effective resistance and capacitance that are polynomial in the number of simplices as well as the maximum cardinality of the torsion subgroup of a relative homology group denoted 𝒯_{max}(𝒦). We generalize the quantum algorithm deciding st-connectivity in a graph and obtain an algorithm deciding whether or not a (d-1)-dimensional cycle γ is null-homologous in a d-dimensional simplicial complex 𝒦. The quantum algorithm has query complexity parameterized by the effective resistance and capacitance of γ. Using our upper bounds we find that the query complexity is O (n^{5/2}⋅ d^{1/2} ⋅ 𝒯_{max}(𝒦)²). Under the assumptions that γ is the boundary of a d-simplex (which may or may not be included in the complex) and that 𝒦 is relative torsion-free, we match the O(n^{3/2}) query complexity obtained for st-connectivity. These assumptions always hold in the case of st-connectivity. We provide an implementation of the algorithm whose running time is polynomial in the size of the complex and the relative torsion. Finally, we prove a duality theorem relating effective resistance and capacitance when 𝒦 is d-dimensional and admits an embedding into ℝ^{d+1}.

Mitchell Black and William Maxwell. Effective Resistance and Capacitance in Simplicial Complexes and a Quantum Algorithm. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 31:1-31:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{black_et_al:LIPIcs.ISAAC.2021.31, author = {Black, Mitchell and Maxwell, William}, title = {{Effective Resistance and Capacitance in Simplicial Complexes and a Quantum Algorithm}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {31:1--31:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.31}, URN = {urn:nbn:de:0030-drops-154641}, doi = {10.4230/LIPIcs.ISAAC.2021.31}, annote = {Keywords: Simplicial complexes, quantum computing} }

Document

**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We consider high dimensional variants of the maximum flow and minimum cut problems in the setting of simplicial complexes and provide both algorithmic and hardness results. By viewing flows and cuts topologically in terms of the simplicial (co)boundary operator we can state these problems as linear programs and show that they are dual to one another. Unlike graphs, complexes with integral capacity constraints may have fractional max-flows. We show that computing a maximum integral flow is NP-hard. Moreover, we give a combinatorial definition of a simplicial cut that seems more natural in the context of optimization problems and show that computing such a cut is NP-hard. However, we provide conditions on the simplicial complex for when the cut found by the linear program is a combinatorial cut. For d-dimensional simplicial complexes embedded into ℝ^{d+1} we provide algorithms operating on the dual graph: computing a maximum flow is dual to computing a shortest path and computing a minimum cut is dual to computing a minimum cost circulation. Finally, we investigate the Ford-Fulkerson algorithm on simplicial complexes, prove its correctness, and provide a heuristic which guarantees it to halt.

William Maxwell and Amir Nayyeri. Generalized Max-Flows and Min-Cuts in Simplicial Complexes. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 69:1-69:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{maxwell_et_al:LIPIcs.ESA.2021.69, author = {Maxwell, William and Nayyeri, Amir}, title = {{Generalized Max-Flows and Min-Cuts in Simplicial Complexes}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {69:1--69:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.69}, URN = {urn:nbn:de:0030-drops-146509}, doi = {10.4230/LIPIcs.ESA.2021.69}, annote = {Keywords: Max-flow min-cut, simplicial complexes, algebraic topology} }

Document

**Published in:** LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

The objective of the well-known Towers of Hanoi puzzle is to move a set of disks one at a time from one of a set of pegs to another, while keeping the disks sorted on each peg. We propose an adversarial variation in which the first player forbids a set of states in the puzzle, and the second player must then convert one randomly-selected state to another without passing through forbidden states. Analyzing this version raises the question of the treewidth of Hanoi graphs. We find this number exactly for three-peg puzzles and provide nearly-tight asymptotic bounds for larger numbers of pegs.

David Eppstein, Daniel Frishberg, and William Maxwell. On the Treewidth of Hanoi Graphs. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{eppstein_et_al:LIPIcs.FUN.2021.13, author = {Eppstein, David and Frishberg, Daniel and Maxwell, William}, title = {{On the Treewidth of Hanoi Graphs}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {13:1--13:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-145-0}, ISSN = {1868-8969}, year = {2020}, volume = {157}, editor = {Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.13}, URN = {urn:nbn:de:0030-drops-127741}, doi = {10.4230/LIPIcs.FUN.2021.13}, annote = {Keywords: Hanoi graph, Treewidth, Graph separators, Kneser graph, Vertex expansion, Haven, Tensor product} }

Document

**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

We study the problem of low-stretch spanning trees in graphs of bounded width: bandwidth, cutwidth, and treewidth. We show that any simple connected graph G with a linear arrangement of bandwidth b can be embedded into a distribution T of spanning trees such that the expected stretch of each edge of G is O(b²). Our proof implies a linear time algorithm for sampling from T. Therefore, we have a linear time algorithm that finds a spanning tree of G with average stretch O(b²) with high probability. We also describe a deterministic linear-time algorithm for computing a spanning tree of G with average stretch O(b³). For graphs of cutwidth c, we construct a spanning tree with stretch O(c²) in linear time. Finally, when G has treewidth k we provide a dynamic programming algorithm computing a minimum stretch spanning tree of G that runs in polynomial time with respect to the number of vertices of G.

Glencora Borradaile, Erin Wolf Chambers, David Eppstein, William Maxwell, and Amir Nayyeri. Low-Stretch Spanning Trees of Graphs with Bounded Width. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{borradaile_et_al:LIPIcs.SWAT.2020.15, author = {Borradaile, Glencora and Chambers, Erin Wolf and Eppstein, David and Maxwell, William and Nayyeri, Amir}, title = {{Low-Stretch Spanning Trees of Graphs with Bounded Width}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {15:1--15:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.15}, URN = {urn:nbn:de:0030-drops-122622}, doi = {10.4230/LIPIcs.SWAT.2020.15}, annote = {Keywords: Treewidth, low-stretch spanning tree, fundamental cycle basis} }

Document

**Published in:** LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

We study two optimization problems on simplicial complexes with homology over ℤ₂, the minimum bounded chain problem: given a d-dimensional complex 𝒦 embedded in ℝ^(d+1) and a null-homologous (d-1)-cycle C in 𝒦, find the minimum d-chain with boundary C, and the minimum homologous chain problem: given a (d+1)-manifold ℳ and a d-chain D in ℳ, find the minimum d-chain homologous to D. We show strong hardness results for both problems even for small values of d; d = 2 for the former problem, and d=1 for the latter problem. We show that both problems are APX-hard, and hard to approximate within any constant factor assuming the unique games conjecture. On the positive side, we show that both problems are fixed-parameter tractable with respect to the size of the optimal solution. Moreover, we provide an O(√{log β_d})-approximation algorithm for the minimum bounded chain problem where β_d is the dth Betti number of 𝒦. Finally, we provide an O(√{log n_{d+1}})-approximation algorithm for the minimum homologous chain problem where n_{d+1} is the number of (d+1)-simplices in ℳ.

Glencora Borradaile, William Maxwell, and Amir Nayyeri. Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{borradaile_et_al:LIPIcs.SoCG.2020.21, author = {Borradaile, Glencora and Maxwell, William and Nayyeri, Amir}, title = {{Minimum Bounded Chains and Minimum Homologous Chains in Embedded Simplicial Complexes}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.21}, URN = {urn:nbn:de:0030-drops-121799}, doi = {10.4230/LIPIcs.SoCG.2020.21}, annote = {Keywords: computational topology, algorithmic complexity, simplicial complexes} }