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DOI: 10.4230/LIPIcs.FUN.2026.11

An Almost-Optimal Upper Bound on the Push Number of the Torus Puzzle

Authors: Matteo Caporrella and Stefano Leucci

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract

We study the Torus Puzzle, a solitaire game in which the elements of an input m × n matrix need to be rearranged into a target configuration via a sequence of unit rotations (i.e., circular shifts) of rows and/or columns. Amano et al. proposed a more permissive variant of the above puzzle, where each row and column rotation can shift the involved elements by any amount of positions. The number of rotations needed to solve the original and the permissive variants of the puzzle are respectively known as the push number and the drag number, where the latter is always smaller than or equal to the former and admits an existential lower bound of Ω(mn). While this lower bound is matched by an O(mn) upper bound, the push number is not so well understood. Indeed, to the best of our knowledge, only an O(mn ⋅ max{m, n}) upper bound is currently known. In this paper, we provide an algorithm that solves the Torus Puzzle using O(mn ⋅ log max {m, n}) unit rotations in a model that is more restricted than that of the original puzzle. This implies a corresponding upper bound on the push number and reduces the gap between the known upper and lower bounds from Θ(max{m,n}) to Θ(log max{m, n}).

Cite as

Matteo Caporrella and Stefano Leucci. An Almost-Optimal Upper Bound on the Push Number of the Torus Puzzle. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 11:1-11:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{caporrella_et_al:LIPIcs.FUN.2026.11,
  author =	{Caporrella, Matteo and Leucci, Stefano},
  title =	{{An Almost-Optimal Upper Bound on the Push Number of the Torus Puzzle}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{11:1--11:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.11},
  URN =		{urn:nbn:de:0030-drops-257307},
  doi =		{10.4230/LIPIcs.FUN.2026.11},
  annote =	{Keywords: Torus puzzle, Push number, Permutation puzzles}
}

Document

DOI: 10.4230/LIPIcs.FUN.2026.26

Hexasort - the Complexity of Stacking Colors on Graphs

Authors: Linus Klocker and Simon Dominik Fink

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract

Many popular puzzle and matching games have been analyzed through the lens of computational complexity. Prominent examples include Sudoku [Takayuki Yato and Takahiro Seta, 2003], Candy Crush [Luciano Gualà et al., 2014], and Flood-It [Fellows et al., 2018]. A common theme among these widely played games is that their generalized decision versions are NP-hard, which is often thought of as a source of their inherent difficulty and addictive appeal to human players. In this paper, we study a popular single-player stacking game commonly known as Hexasort. The game can be modelled as placing colored stacks onto the vertices of a graph, where adjacent stacks of the same color merge and vanish according to deterministic rules. We prove that Hexasort is NP-hard, even when restricted to single-color stacks and progressively more constrained classes of graphs, culminating in strong NP-hardness on trees of either bounded height or degree. Towards fixed-parameter tractable algorithms, we identify settings in which the problem becomes polynomial-time solvable and present dynamic programming algorithms.

Cite as

Linus Klocker and Simon Dominik Fink. Hexasort - the Complexity of Stacking Colors on Graphs. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{klocker_et_al:LIPIcs.FUN.2026.26,
  author =	{Klocker, Linus and Fink, Simon Dominik},
  title =	{{Hexasort - the Complexity of Stacking Colors on Graphs}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{26:1--26:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.26},
  URN =		{urn:nbn:de:0030-drops-257457},
  doi =		{10.4230/LIPIcs.FUN.2026.26},
  annote =	{Keywords: Hexasort, offline color stacking on graphs, NP-complete, polynomial-time solvable, dynamic programming}
}

Document

DOI: 10.4230/LIPIcs.FUN.2026.27

Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard

Authors: Kolja Kühn and Wendy Yi

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract

We extend the study of the 2-Solo Chess problem which was first introduced by Aravind, Misra, and Mittal in 2022. 2-Solo Chess is a single-player variant of chess in which the player must clear the board via captures such that only one piece remains, with each piece capturing at most twice. It is known that the problem is solvable in polynomial time for instances containing only pawns, while it becomes NP-complete for instances restricted to rooks, bishops, or queens. In this work, we complete the complexity classification by proving that 2-Solo Chess is NP-complete if the instance contains only knights or only kings.

Cite as

Kolja Kühn and Wendy Yi. Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kuhn_et_al:LIPIcs.FUN.2026.27,
  author =	{K\"{u}hn, Kolja and Yi, Wendy},
  title =	{{Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{27:1--27:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.27},
  URN =		{urn:nbn:de:0030-drops-257464},
  doi =		{10.4230/LIPIcs.FUN.2026.27},
  annote =	{Keywords: Solo chess, puzzle games, board games, NP-completeness}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.27

Fault-Tolerant Approximate Distance Oracles with a Source Set

Authors: Dipan Dey and Telikepalli Kavitha

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

Our input is an undirected weighted graph G = (V,E) on n vertices along with a source set S ⊆ V. The problem is to preprocess G and build a compact data structure such that upon query Qu(s,v,f) where (s,v) ∈ S×V and f is any faulty edge, we can quickly find a good estimate (i.e., within a small multiplicative stretch) of the s-v distance in G-f. We use a fault-tolerant ST-distance oracle from the work of Bilò et al. (STACS 2018) to construct an S×V approximate distance oracle or sourcewise approximate distance oracle of size Õ(|S|n + n^{3/2}) with multiplicative stretch at most 5. We construct another fault-tolerant sourcewise approximate distance oracle of size Õ(|S|n + n^{4/3}) with multiplicative stretch at most 13. Both the oracles have O(1) query answering time.

Cite as

Dipan Dey and Telikepalli Kavitha. Fault-Tolerant Approximate Distance Oracles with a Source Set. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.FSTTCS.2025.27,
  author =	{Dey, Dipan and Kavitha, Telikepalli},
  title =	{{Fault-Tolerant Approximate Distance Oracles with a Source Set}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.27},
  URN =		{urn:nbn:de:0030-drops-251081},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.27},
  annote =	{Keywords: Weighted graphs, approximate distances, fault-tolerant data structures}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2025.14

On the (In)Approximability of the Monitoring Edge Geodetic Set Problem

Authors: Davide Bilò, Giordano Colli, Luca Forlizzi, and Stefano Leucci

Published in: LIPIcs, Volume 359, 36th International Symposium on Algorithms and Computation (ISAAC 2025)

Abstract

We study the minimum Monitoring Edge Geodetic Set (MEG-Set) problem introduced in [Foucaud et al., CALDAM'23]: given a graph G, we say that an edge is monitored by a pair u,v of vertices if all shortest paths between u and v traverse e; the goal is to find a subset M of vertices of G such that each edge of G is monitored by at least one pair of vertices in M, and |M| is minimized. In this paper, we prove that all polynomial-time approximation algorithms for the minimum MEG-Set problem must have an approximation ratio of Ω(log n), unless 𝖯 = NP. To the best of our knowledge, this is the first non-constant inapproximability result known for this problem. We also strengthen the known NP-hardness of the problem on 2-apex graphs by showing that the same result holds for 1-apex graphs. This leaves open the question of determining whether the problem remains NP-hard on planar (i.e., 0-apex) graphs. On the positive side, we design an algorithm that computes good approximate solutions for hereditary graph classes that admit efficiently computable balanced separators of truly sublinear size. This immediately yields polynomial-time approximation algorithms achieving an approximation ratio of O(n^{1/4} √{log n}) on planar graphs, graphs with bounded genus, and k-apex graphs with k = O(n^{1/4}). On graphs with bounded treewidth, we obtain an approximation ratio of O(log^{3/2} n). This compares favorably with the best-known approximation algorithm for general graphs, which achieves an approximation ratio of O(√{n log n}) via a simple reduction to the Set Cover problem.

Cite as

Davide Bilò, Giordano Colli, Luca Forlizzi, and Stefano Leucci. On the (In)Approximability of the Monitoring Edge Geodetic Set Problem. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2025.14,
  author =	{Bil\`{o}, Davide and Colli, Giordano and Forlizzi, Luca and Leucci, Stefano},
  title =	{{On the (In)Approximability of the Monitoring Edge Geodetic Set Problem}},
  booktitle =	{36th International Symposium on Algorithms and Computation (ISAAC 2025)},
  pages =	{14:1--14:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-408-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{359},
  editor =	{Chen, Ho-Lin and Hon, Wing-Kai and Tsai, Meng-Tsung},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.14},
  URN =		{urn:nbn:de:0030-drops-249226},
  doi =		{10.4230/LIPIcs.ISAAC.2025.14},
  annote =	{Keywords: Monitoring Edge Geodetic Set, Inapproximability, Approximation Algorithms}
}

Document

DOI: 10.4230/LIPIcs.WADS.2025.24

Computational Geometry with Probabilistically Noisy Primitive Operations

Authors: David Eppstein, Michael T. Goodrich, and Vinesh Sridhar

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract

Much prior work has been done on designing computational geometry algorithms that handle input degeneracies, data imprecision, and arithmetic round-off errors. We take a new approach, inspired by the noisy sorting literature, and study computational geometry algorithms subject to noisy Boolean primitive operations in which, e.g., the comparison "is point q above line 𝓁?" returns the wrong answer with some fixed probability. We propose a novel technique called path-guided pushdown random walks that generalizes the results of noisy sorting. We apply this technique to solve point-location, plane-sweep, convex hulls in 2D and 3D, and Delaunay triangulations for noisy primitives in optimal time with high probability.

Cite as

David Eppstein, Michael T. Goodrich, and Vinesh Sridhar. Computational Geometry with Probabilistically Noisy Primitive Operations. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eppstein_et_al:LIPIcs.WADS.2025.24,
  author =	{Eppstein, David and Goodrich, Michael T. and Sridhar, Vinesh},
  title =	{{Computational Geometry with Probabilistically Noisy Primitive Operations}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.24},
  URN =		{urn:nbn:de:0030-drops-242552},
  doi =		{10.4230/LIPIcs.WADS.2025.24},
  annote =	{Keywords: Computational geometry, noisy comparisons, random walks}
}

Document

DOI: 10.4230/LIPIcs.MFCS.2025.75

Temporal Graph Realization with Bounded Stretch

Authors: George B. Mertzios, Hendrik Molter, Nils Morawietz, and Paul G. Spirakis

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract

A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first Δ time steps, and then it reappears recurrently every Δ time steps, where Δ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are "stretched", compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph G, the task is to assign to each edge one time-label between 1 and Δ such that, in the resulting periodic temporal graph with period Δ, the duration of the fastest temporal path from any vertex u to any other vertex v is at most α times the distance between u and v in G. Here, the value of α measures how much the shortest paths are allowed to be stretched once we assign the periodic time-labels. Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the radius-algorithm) which always guarantees an approximation strictly smaller than Δ, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most k edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to k.

Cite as

George B. Mertzios, Hendrik Molter, Nils Morawietz, and Paul G. Spirakis. Temporal Graph Realization with Bounded Stretch. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 75:1-75:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mertzios_et_al:LIPIcs.MFCS.2025.75,
  author =	{Mertzios, George B. and Molter, Hendrik and Morawietz, Nils and Spirakis, Paul G.},
  title =	{{Temporal Graph Realization with Bounded Stretch}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{75:1--75:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.75},
  URN =		{urn:nbn:de:0030-drops-241829},
  doi =		{10.4230/LIPIcs.MFCS.2025.75},
  annote =	{Keywords: Temporal graph, periodic temporal labeling, fastest temporal path, graph realization, temporal connectivity, stretch}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.100

Robust-Sorting and Applications to Ulam-Median

Authors: Ragesh Jaiswal, Amit Kumar, and Jatin Yadav

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract

Sorting is one of the most basic primitives in many algorithms and data analysis tasks. Comparison-based sorting algorithms, like quick-sort and merge-sort, are known to be optimal when the outcome of each comparison is error-free. However, many real-world sorting applications operate in scenarios where the outcome of each comparison can be noisy. In this work, we explore settings where a bounded number of comparisons are potentially corrupted by erroneous agents, resulting in arbitrary, adversarial outcomes. We model the sorting problem as a query-limited tournament graph where edges involving erroneous nodes may yield arbitrary results. Our primary contribution is a randomized algorithm inspired by quick-sort that, in expectation, produces an ordering close to the true total order while only querying Õ(n) edges. We achieve a distance from the target order π within (3 + ε)|B|, where B is the set of erroneous nodes, balancing the competing objectives of minimizing both query complexity and misalignment with π. Our algorithm needs to carefully balance two aspects - identify a pivot that partitions the vertex set evenly and ensure that this partition is "truthful" and yet query as few "triangles" in the graph G as possible. Since the nodes in B can potentially hide in an intricate manner, our algorithm requires several technical steps that ensure that progress is made in each recursive step. Additionally, we demonstrate significant implications for the Ulam-k-Median problem. This is a classical clustering problem where the metric is defined on the set of permutations on a set of d elements. Chakraborty, Das, and Krauthgamer gave a (2-ε) FPT approximation algorithm for this problem, where the running time is super-linear in both n and d. We give the first (2-ε) FPT linear time approximation algorithm for this problem. Our main technical result gives a strengthening of the results in Chakraborty et al. by showing that a good 1-median solution can be obtained from a constant-size random sample of the input. We use our robust sorting framework to find a good solution from such a random sample. We feel that the notion of robust sorting should have applications in several such settings.

Cite as

Ragesh Jaiswal, Amit Kumar, and Jatin Yadav. Robust-Sorting and Applications to Ulam-Median. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 100:1-100:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jaiswal_et_al:LIPIcs.ICALP.2025.100,
  author =	{Jaiswal, Ragesh and Kumar, Amit and Yadav, Jatin},
  title =	{{Robust-Sorting and Applications to Ulam-Median}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{100:1--100:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.100},
  URN =		{urn:nbn:de:0030-drops-234774},
  doi =		{10.4230/LIPIcs.ICALP.2025.100},
  annote =	{Keywords: Sorting, clustering, query complexity}
}

Document

DOI: 10.4230/LIPIcs.SAND.2025.6

Dismountability in Temporal Cliques Revisited

Authors: Daniele Carnevale, Arnaud Casteigts, and Timothée Corsini

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract

A temporal graph is a graph whose edges are available only at certain points in time. It is temporally connected if the nodes can reach each other by paths that traverse the edges chronologically (temporal paths). Unlike static graphs, temporal graphs do not always admit small subsets of edges that preserve connectivity (temporal spanners) - there exist temporal graphs with Θ(n²) edges, all of which are critical. In the case of temporal cliques (the underlying graph is complete), spanners of size O(nlog n) are guaranteed. The original proof of this result by Casteigts et al. [ICALP 2019] combines a number of techniques, one of which is called dismountability. In a recent work, Angrick et al. [ESA 2024] simplified the proof and showed, among other things, that a one-sided version of dismountability can replace elegantly the second part of the proof. In this paper, we revisit methodically the dismountability principle. We start by characterizing the structure that a temporal clique must have if it is non 1-hop dismountable, then neither 1-hop nor 2-hop (i.e. non {1,2}-hop) dismountable, and finally non {1,2,3}-hop dismountable. It turns out that if a clique is k-hop dismountable for any other k, then it must also be {1,2,3}-hop dismountable, thus no additional structure can be obtained beyond this point. Interestingly, excluding 1-hop and 2-hop dismountability is already sufficient for reducing the spanner problem from cliques to extremally matched bicliques, where the O(nlog n) result is subsequently obtained. Put together with the strategy of Angrick et al., this entire result can now be recovered using only dismountability. An interesting by-product of our analysis is that any minimal counter-example to the existence of 4n spanners must satisfy the properties of non {1,2,3}-hop dismountable cliques. In the second part, we discuss further connections between dismountability and another technique called pivotability. In particular, we show that if a temporal clique is recursively k-hop dismountable, then it is also pivotable (and thus admits a 2n spanner, whatever k). We also study a family of labelings called full-range that forces both dismountability and pivotability. The latter gives some evidence that large lifetimes could be exploited more generally for the construction of spanners.

Cite as

Daniele Carnevale, Arnaud Casteigts, and Timothée Corsini. Dismountability in Temporal Cliques Revisited. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carnevale_et_al:LIPIcs.SAND.2025.6,
  author =	{Carnevale, Daniele and Casteigts, Arnaud and Corsini, Timoth\'{e}e},
  title =	{{Dismountability in Temporal Cliques Revisited}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.6},
  URN =		{urn:nbn:de:0030-drops-230591},
  doi =		{10.4230/LIPIcs.SAND.2025.6},
  annote =	{Keywords: Dynamic networks, Temporal graphs, Reachability, Dismountability, Pivotability, Temporal spanners, Full-range graphs}
}

Document

DOI: 10.4230/LIPIcs.SAND.2025.3

Temporal Connectivity Augmentation

Authors: Thomas Bellitto, Jules Bouton Popper, and Bruno Escoffier

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract

Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More precisely, we tackle the problem of finding, among a set of proposed temporal edges, the smallest subset such that its addition makes the graph temporally connected (TCA). We study the complexity of this problem and variants, under restricted lifespan of the graph, i.e. the maximum time step in the graph. Our main result on TCA is that for any fixed lifespan at least 2, it is NP-complete in both the strict and non-strict setting. We additionally provide a set of restrictions in the non-strict setting which makes the problem solvable in polynomial time and design an algorithm achieving this complexity. Interestingly, we prove that the source variant (making a given vertex a source in the augmented graph) is as difficult as TCA. On the opposite, we prove that the version where a list of connectivity demands has to be satisfied is solvable in polynomial time, when the size of the list is fixed. Finally, we highlight a variant of the previous case for which even with two pairs the problem is already NP-hard.

Cite as

Thomas Bellitto, Jules Bouton Popper, and Bruno Escoffier. Temporal Connectivity Augmentation. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bellitto_et_al:LIPIcs.SAND.2025.3,
  author =	{Bellitto, Thomas and Popper, Jules Bouton and Escoffier, Bruno},
  title =	{{Temporal Connectivity Augmentation}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{3:1--3:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.3},
  URN =		{urn:nbn:de:0030-drops-230565},
  doi =		{10.4230/LIPIcs.SAND.2025.3},
  annote =	{Keywords: Temporal graph, temporal connectivity}
}

Document

DOI: 10.4230/LIPIcs.SAND.2025.9

Spanner Enumeration for Temporal Graphs

Authors: Kazuhiro Kurita, Andrea Marino, Jason Schoeters, and Takeaki Uno

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract

A spanner of a temporal graph is a subset of edges that preserves connectivity over time between vertices. A minimal spanner is one in which no additional edges can be removed without breaking this connectivity. Our focus is on enumerating minimal spanners for a given temporal graph. We explore several variations of this problem based on the type of connectivity that must be maintained, ranging from one-to-all connectivity to one-to-all-to-one, many-to-all, and finally all-to-all connectivity. We establish that these problems become progressively harder: (i) We present a polynomial-delay enumeration algorithm for one-to-all connectivity; (ii) We prove Dual-hardness for both one-to-all-to-one and many-to-all connectivity, even in the restricted case of two-to-all; (iii) Finally, for all-to-all connectivity, we show that enumeration cannot be performed in output-polynomial time unless P = NP.

Cite as

Kazuhiro Kurita, Andrea Marino, Jason Schoeters, and Takeaki Uno. Spanner Enumeration for Temporal Graphs. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kurita_et_al:LIPIcs.SAND.2025.9,
  author =	{Kurita, Kazuhiro and Marino, Andrea and Schoeters, Jason and Uno, Takeaki},
  title =	{{Spanner Enumeration for Temporal Graphs}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{9:1--9:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.9},
  URN =		{urn:nbn:de:0030-drops-230621},
  doi =		{10.4230/LIPIcs.SAND.2025.9},
  annote =	{Keywords: temporal graphs, temporal spanners, one-to-all connectivity, all-to-all connectivity enumeration, NP-completeness, Dual-hardness, binary partition tree, flashlight search, polynomial delay}
}

Document

DOI: 10.4230/LIPIcs.ISAAC.2024.11

Temporal Queries for Dynamic Temporal Forests

Authors: Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Alessandro Straziota

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract

In a temporal forest each edge has an associated set of time labels that specify the time instants in which the edges are available. A temporal path from vertex u to vertex v in the forest is a selection of a label for each edge in the unique path from u to v, assuming it exists, such that the labels selected for any two consecutive edges are non-decreasing. We design linear-size data structures that maintain a temporal forest of rooted trees under addition and deletion of both edge labels and singleton vertices, insertion of root-to-node edges, and removal of edges with no labels. Such data structures can answer temporal reachability, earliest arrival, and latest departure queries. All queries and updates are handled in polylogarithmic worst-case time. Our results can be adapted to deal with latencies. More precisely, all the worst-case time bounds are asymptotically unaffected when latencies are uniform. For arbitrary latencies, the update time becomes amortized in the incremental case where only label additions and edge/singleton insertions are allowed as well as in the decremental case in which only label deletions and edge/singleton removals are allowed. To the best of our knowledge, the only previously known data structure supporting temporal reachability queries is due to Brito, Albertini, Casteigts, and Travençolo [Social Network Analysis and Mining, 2021], which can handle general temporal graphs, answers queries in logarithmic time in the worst case, but requires an amortized update time that is quadratic in the number of vertices, up to polylogarithmic factors.

Cite as

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Alessandro Straziota. Temporal Queries for Dynamic Temporal Forests. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2024.11,
  author =	{Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Straziota, Alessandro},
  title =	{{Temporal Queries for Dynamic Temporal Forests}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.11},
  URN =		{urn:nbn:de:0030-drops-221382},
  doi =		{10.4230/LIPIcs.ISAAC.2024.11},
  annote =	{Keywords: temporal graphs, temporal reachability, earliest arrival, latest departure, dynamic forests}
}

Document

DOI: 10.4230/LIPIcs.ESA.2024.25

Graph Spanners for Group Steiner Distances

Authors: Davide Bilò, Luciano Gualà, Stefano Leucci, and Alessandro Straziota

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract

A spanner is a sparse subgraph of a given graph G which preserves distances, measured w.r.t. some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t. the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t. the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.

Cite as

Davide Bilò, Luciano Gualà, Stefano Leucci, and Alessandro Straziota. Graph Spanners for Group Steiner Distances. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2024.25,
  author =	{Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Straziota, Alessandro},
  title =	{{Graph Spanners for Group Steiner Distances}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.25},
  URN =		{urn:nbn:de:0030-drops-210968},
  doi =		{10.4230/LIPIcs.ESA.2024.25},
  annote =	{Keywords: Network sparsification, Graph spanners, Group Steiner tree, Distance oracles}
}

Document

DOI: 10.4230/LIPIcs.FUN.2024.4

Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard

Authors: Davide Bilò, Luca Di Donato, Luciano Gualà, and Stefano Leucci

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract

We study the Solo-Chess problem which has been introduced in [Aravind et al., FUN 2022]. This is a single-player variant of chess in which the player must clear all but one piece from the board via a sequence captures while ensuring that the number of captures performed by each piece does not exceed the piece’s budget. The time complexity of finding a winning sequence of captures has already been pinpointed for several combination of piece types and initial budgets. We contribute to a better understanding of the computational landscape of Solo-Chess by closing two problems left open in [Aravind et al., FUN 2022]. Namely, we show that Solo-Chess is hard even when all pieces are restricted to be only rooks with budget exactly 2, or only knights with budget exactly 11.

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Davide Bilò, Luca Di Donato, Luciano Gualà, and Stefano Leucci. Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2024.4,
  author =	{Bil\`{o}, Davide and Di Donato, Luca and Gual\`{a}, Luciano and Leucci, Stefano},
  title =	{{Uniform-Budget Solo Chess with Only Rooks or Only Knights Is Hard}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{4:1--4:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.4},
  URN =		{urn:nbn:de:0030-drops-199121},
  doi =		{10.4230/LIPIcs.FUN.2024.4},
  annote =	{Keywords: solo chess, puzzle games, board games, NP-completeness}
}

Document

DOI: 10.4230/LIPIcs.FUN.2024.5

Swapping Mixed-Up Beers to Keep Them Cool

Authors: Davide Bilò, Maurizio Fiusco, Luciano Gualà, and Stefano Leucci

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract

There was a mix-up in Escher’s bar and n customers sitting at the same table have each received a beer ordered by somebody else in the party. The drinks can be rearranged by swapping them in pairs, but the eccentric table shape only allows drinks to be exchanged between people sitting on opposite sides of the table. We study the problem of finding the minimum number of swaps needed so that each customer receives its desired beer before it gets warm. Formally, we consider the Colored Token Swapping problem on complete bipartite graphs. This problem is known to be solvable in polynomial time when all ordered drinks are different [Yamanaka et al., FUN 2014], but no results are known for the more general case in which multiple people in the party can order the same beer. We prove that Colored Token Swapping on complete bipartite graphs is NP-hard and that it is fixed-parameter tractable when parameterized by the number of distinct types of beer served by the bar.

Cite as

Davide Bilò, Maurizio Fiusco, Luciano Gualà, and Stefano Leucci. Swapping Mixed-Up Beers to Keep Them Cool. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2024.5,
  author =	{Bil\`{o}, Davide and Fiusco, Maurizio and Gual\`{a}, Luciano and Leucci, Stefano},
  title =	{{Swapping Mixed-Up Beers to Keep Them Cool}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.5},
  URN =		{urn:nbn:de:0030-drops-199132},
  doi =		{10.4230/LIPIcs.FUN.2024.5},
  annote =	{Keywords: Colored Token Swapping, Complete Bipartite Graphs, Labeled Token Swapping, FPT Algorithms, NP-Hardness}
}

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