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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We study the minimum weight basis problem on matroid when elements' weights are uncertain. For each element we only know a set of possible values (an uncertainty area) that contains its real weight. In some cases there exist bases that are uniformly optimal, that is, they are minimum weight bases for every possible weight function obeying the uncertainty areas. In other cases, computing such a basis is not possible unless we perform some queries for the exact value of some elements.
Our main result is a polynomial time algorithm for the following problem. Given a matroid with uncertainty areas and a query cost function on its elements, find the set of elements of minimum total cost that we need to simultaneously query such that, no matter their revelation, the resulting instance admits a uniformly optimal base. We also provide combinatorial characterizations of all uniformly optimal bases, when one exists; and of all sets of queries that can be performed so that after revealing the corresponding weights the resulting instance admits a uniformly optimal base.

Arturo I. Merino and José A. Soto. The Minimum Cost Query Problem on Matroids with Uncertainty Areas. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 83:1-83:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{merino_et_al:LIPIcs.ICALP.2019.83, author = {Merino, Arturo I. and Soto, Jos\'{e} A.}, title = {{The Minimum Cost Query Problem on Matroids with Uncertainty Areas}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {83:1--83:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.83}, URN = {urn:nbn:de:0030-drops-106592}, doi = {10.4230/LIPIcs.ICALP.2019.83}, annote = {Keywords: Minimum spanning tree, matroids, uncertainty, queries} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

We say that a Hamilton cycle C = (x₁,…,x_n) in a graph G is k-symmetric, if the mapping x_i ↦ x_{i+n/k} for all i = 1,…,n, where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x₁,…,x_n equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360^∘/k wedge of the drawing. We refer to the maximum k for which there exists a k-symmetric Hamilton cycle in G as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases we determine their Hamilton compression exactly, and in other cases we provide close lower and upper bounds. The cycles we construct have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.

Petr Gregor, Arturo Merino, and Torsten Mütze. The Hamilton Compression of Highly Symmetric Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gregor_et_al:LIPIcs.MFCS.2022.54, author = {Gregor, Petr and Merino, Arturo and M\"{u}tze, Torsten}, title = {{The Hamilton Compression of Highly Symmetric Graphs}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {54:1--54:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.54}, URN = {urn:nbn:de:0030-drops-168529}, doi = {10.4230/LIPIcs.MFCS.2022.54}, annote = {Keywords: Hamilton cycle, Gray code, hypercube, permutahedron, Johnson graph, Cayley graph, abelian group, vertex-transitive} }

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**Published in:** LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

You provide us with a matroid and an initial base. We say that a subset of the bases "belongs to us" if we can visit each one via a sequence of base exchanges starting from the initial base. It is well-known that "All your base are belong to us". We refine this classic result by showing that it can be done by a simple greedy algorithm. For example, the spanning trees of a graph can be generated by edge exchanges using the following greedy rule: Minimize the larger label of an edge that enters or exits the current spanning tree and which creates a spanning tree that is new (i.e., hasn't been visited already). Amazingly, this works for any graph, for any labeling of its edges, for any initial spanning tree, and regardless of how you choose the edge with the smaller label in each exchange. Furthermore, by maintaining a small amount of information, we can generate each successive spanning tree without storing the previous trees.
In general, for any matroid, we can greedily compute a listing of all its bases matroid such that consecutive bases differ by a base exchange. Our base exchange Gray codes apply a prefix-exchange on a prefix-minor of the matroid, and we can generate these orders using "history-free" iterative algorithms. More specifically, we store O(m) bits of data, and use O(m) time per base assuming O(1) time independence and coindependence oracles.
Our work generalizes and extends a number of previous results. For example, the bases of the uniform matroid are combinations, and they belong to us using homogeneous transpositions via an Eades-McKay style order. Similarly, the spanning trees of fan graphs belong to us via face pivot Gray codes, which extends recent results of Cameron, Grubb, and Sawada [Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan, COCOON 2021].

Arturo Merino, Torsten Mütze, and Aaron Williams. All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 22:1-22:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{merino_et_al:LIPIcs.FUN.2022.22, author = {Merino, Arturo and M\"{u}tze, Torsten and Williams, Aaron}, title = {{All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {22:1--22:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-232-7}, ISSN = {1868-8969}, year = {2022}, volume = {226}, editor = {Fraigniaud, Pierre and Uno, Yushi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.22}, URN = {urn:nbn:de:0030-drops-159928}, doi = {10.4230/LIPIcs.FUN.2022.22}, annote = {Keywords: Matroids, base exchange, Gray codes, combinatorial generation, greedy algorithms, spanning trees} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Given integers k ≥ 2 and a_1,…,a_k ≥ 1, let a: = (a_1,…,a_k) and n: = a_1+⋯+a_k. An a-multiset permutation is a string of length n that contains exactly a_i symbols i for each i = 1,…,k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a_1 = ⋯ = a_k = 1) can be generated by star transpositions, while combinations (k = 2) can be generated by these operations if and only if they are balanced (a_1 = a_2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Δ(a): = n-2max{a_1,…,a_k} that allows us to distinguish three different regimes for this problem. We show that if Δ(a) < 0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Δ(a) > 0, assuming that they exist for the case Δ(a) = 0. For the case Δ(a) = 0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.

Petr Gregor, Torsten Mütze, and Arturo Merino. Star Transposition Gray Codes for Multiset Permutations. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gregor_et_al:LIPIcs.STACS.2022.34, author = {Gregor, Petr and M\"{u}tze, Torsten and Merino, Arturo}, title = {{Star Transposition Gray Codes for Multiset Permutations}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.34}, URN = {urn:nbn:de:0030-drops-158448}, doi = {10.4230/LIPIcs.STACS.2022.34}, annote = {Keywords: Gray code, permutation, combination, transposition, Hamilton cycle} }

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**Published in:** LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.

Arturo Merino and Torsten Mütze. Efficient Generation of Rectangulations via Permutation Languages. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 54:1-54:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{merino_et_al:LIPIcs.SoCG.2021.54, author = {Merino, Arturo and M\"{u}tze, Torsten}, title = {{Efficient Generation of Rectangulations via Permutation Languages}}, booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)}, pages = {54:1--54:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-184-9}, ISSN = {1868-8969}, year = {2021}, volume = {189}, editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.54}, URN = {urn:nbn:de:0030-drops-138534}, doi = {10.4230/LIPIcs.SoCG.2021.54}, annote = {Keywords: Exhaustive generation, Gray code, flip graph, polytope, generic rectangulation, diagonal rectangulation, cartogram, floorplan, permutation pattern} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study the two-dimensional geometric knapsack problem for convex polygons. Given a set of weighted convex polygons and a square knapsack, the goal is to select the most profitable subset of the given polygons that fits non-overlappingly into the knapsack. We allow to rotate the polygons by arbitrary angles. We present a quasi-polynomial time O(1)-approximation algorithm for the general case and a polynomial time O(1)-approximation algorithm if all input polygons are triangles, both assuming polynomially bounded integral input data. Also, we give a quasi-polynomial time algorithm that computes a solution of optimal weight under resource augmentation, i.e., we allow to increase the size of the knapsack by a factor of 1+δ for some δ > 0 but compare ourselves with the optimal solution for the original knapsack. To the best of our knowledge, these are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in which the input polygons can be rotated by arbitrary angles.

Arturo Merino and Andreas Wiese. On the Two-Dimensional Knapsack Problem for Convex Polygons. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 84:1-84:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{merino_et_al:LIPIcs.ICALP.2020.84, author = {Merino, Arturo and Wiese, Andreas}, title = {{On the Two-Dimensional Knapsack Problem for Convex Polygons}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {84:1--84:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.84}, URN = {urn:nbn:de:0030-drops-124916}, doi = {10.4230/LIPIcs.ICALP.2020.84}, annote = {Keywords: Approximation algorithms, geometric knapsack problem, polygons, rotation} }

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