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

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.

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} }

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

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.

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} }

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

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.

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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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the problem of estimating the ST-diameter of a graph that is subject to a bounded number of edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a given graph G, two sets of vertices S,T, and positive integer f. When queried with a set F of at most f edges, the oracle returns an estimate D̂ of the ST-diameter diam(G-F,S,T), the maximum distance between vertices in S and T in G-F. The oracle has stretch σ ⩾ 1 if diam(G-F,S,T) ⩽ D̂ ⩽ σ diam(G-F,S,T). If S and T both contain all vertices, the data structure is called an f-edge fault-tolerant diameter oracle (f-FDO). An f-edge fault-tolerant distance sensitivity oracles (f-DSO) estimates the pairwise graph distances under up to f failures.
We design new f-FDOs and f-FDO-STs by reducing their construction to that of all-pairs and single-source f-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature.
We also provide an information-theoretic lower bound on the space requirement of approximate f-FDOs. We show that there exists a family of graphs for which any f-FDO with sensitivity f ⩾ 2 and stretch less than 5/3 requires Ω(n^{3/2}) bits of space, regardless of the query time.

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, Simon Krogmann, and Martin Schirneck. Fault-Tolerant ST-Diameter Oracles. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2023.24, author = {Bil\`{o}, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Krogmann, Simon and Schirneck, Martin}, title = {{Fault-Tolerant ST-Diameter Oracles}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {24:1--24:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel 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.2023.24}, URN = {urn:nbn:de:0030-drops-180762}, doi = {10.4230/LIPIcs.ICALP.2023.24}, annote = {Keywords: diameter oracles, distance sensitivity oracles, space lower bounds, fault-tolerant data structures} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

A temporal graph is an undirected graph G = (V,E) along with a function λ : E → ℕ^+ that assigns a time-label to each edge in E. A path in G such that the traversed time-labels are non-decreasing is called a temporal path. Accordingly, the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal α-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of α. The size of H is measured as the number of its edges.
In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2k-1)-spanner with Õ(kn^{1+1/k}) edges, where k > 1 is an integer parameter of choice. Choosing k = ⌊log n⌋, we obtain a temporal O(log n)-spanner with Õ(n) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity.
We then turn our attention to general temporal graphs. Since Ω(n²) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that Õ(n/log(1+ε)) edges suffice to obtain a stretch of (1+ε), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω(n²) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of Õ(n² / β) on the size of any temporal β-additive spanner, which we show to be tight up to polylogarithmic factors.
Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Sparse Temporal Spanners with Low Stretch. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2022.19, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Rossi, Mirko}, title = {{Sparse Temporal Spanners with Low Stretch}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.19}, URN = {urn:nbn:de:0030-drops-169575}, doi = {10.4230/LIPIcs.ESA.2022.19}, annote = {Keywords: temporal spanners, temporal graphs, graph sparsification, approximate distances} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. [ITCS 2017] initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs.
Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen [ICALP 2019] as well as Karthik and Parter [SODA 2021], we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter and eccentricity oracles and then show its versatility by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren [JCSS 2022] and the Single-Source Replacement Path algorithm of Chechik and Magen [ICALP 2020]. This way, we obtain the first deterministic distance sensitivity oracle with subcubic preprocessing time.

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2022.22, author = {Bil\`{o}, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, title = {{Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {22:1--22:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.22}, URN = {urn:nbn:de:0030-drops-163633}, doi = {10.4230/LIPIcs.ICALP.2022.22}, annote = {Keywords: derandomization, diameter, eccentricity, fault-tolerant data structure, sensitivity oracle, space lower bound} }

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

Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem.
We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {12:1--12:21}, 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.2022.12}, URN = {urn:nbn:de:0030-drops-158221}, doi = {10.4230/LIPIcs.STACS.2022.12}, annote = {Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch.
We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds.
Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover.
Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2022.23, author = {Bil\`{o}, Davide and Casel, Katrin and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Lagodzinski, J.A. Gregor and Schirneck, Martin and Wietheger, Simon}, title = {{Fixed-Parameter Sensitivity Oracles}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.23}, URN = {urn:nbn:de:0030-drops-156196}, doi = {10.4230/LIPIcs.ITCS.2022.23}, annote = {Keywords: data structures, distance preservers, distance sensitivity oracles, fault tolerance, fixed-parameter tractability, k-path, vertex cover} }

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

Given a graph with a distinguished source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute and output, for any target vertex t and edge e, the length d(s,t,e) of a shortest path from s to t that avoids a failing edge e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a compact data structure that answers queries of the form (t,e) by returning the distance d(s,t,e). We show how to deterministically compress the output of the SSRP problem on n-vertex, m-edge graphs with integer edge weights in the range [1,M] into a Single-Source DSO that has size O(M^{1/2} n^{3/2}) and query time Õ(1). We prove that the space requirement is optimal (up to the word size). Our techniques can also handle vertex failures within the same bounds.
Chechik and Cohen [SODA 2019] presented a combinatorial, randomized Õ(m√n+n²) time SSRP algorithm for undirected and unweighted graphs. We derandomize their algorithm with the same asymptotic running time and apply our compression to obtain a deterministic Single-Source DSO with Õ(m√n+n²) preprocessing time, O(n^{3/2}) space, and Õ(1) query time. Our combinatorial Single-Source DSO has near-optimal space, preprocessing and query time for unweighted graphs, improving the preprocessing time by a √n-factor compared to previous results with o(n²) space.
Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized Õ(Mn^ω) time SSRP algorithm for (undirected and directed) graphs with integer edge weights in the range [1,M], where ω < 2.373 is the matrix multiplication exponent. We derandomize it for undirected graphs and apply our compression to obtain an algebraic Single-Source DSO with Õ(Mn^ω) preprocessing time, O(M^{1/2} n^{3/2}) space, and Õ(1) query time. This improves the preprocessing time of algebraic Single-Source DSOs by polynomial factors compared to previous o(n²)-space oracles.
We also present further improvements of our Single-Source DSOs. We show that the query time can be reduced to a constant at the cost of increasing the size of the oracle to O(M^{1/3} n^{5/3}) and that all our oracles can be made path-reporting. On sparse graphs with m = O(n^{5/4-ε}/M^{7/4}) edges, for any constant ε > 0, we reduce the preprocessing to randomized Õ(M^{7/8} m^{1/2} n^{11/8}) = O(n^{2-ε/2}) time. To the best of our knowledge, this is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2021.18, author = {Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, title = {{Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {18:1--18:17}, 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.2021.18}, URN = {urn:nbn:de:0030-drops-145999}, doi = {10.4230/LIPIcs.ESA.2021.18}, annote = {Keywords: derandomization, distance sensitivity oracle, single-source replacement paths, space lower bound} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We design f-edge fault-tolerant diameter oracles (f-FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f, we preprocess the graph and construct a data structure that, when queried with a set F of edges, where |F| ⩽ f, returns the diameter of G-F. An f-FDO has stretch σ ⩾ 1 if the returned value D^ satisfies diam(G-F) ⩽ D^ ⩽ σ diam(G-F).
For the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1+ε), constant query time, space O(m), and a combinatorial preprocessing time of Õ(mn + n^{1.5} √{Dm/ε}), where D is the diameter.
We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of Õ(mn + n²/ε), which is better for constant ε > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of Õ(n^{2.5794} + n²/ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1+ε)-approximate FDO is near-optimal in all parameters.
In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f-FDO with stretch (f+2), query time O(f²log²{n}), Õ(fn) space, and preprocessing time Õ(fm). We complement this with a lower bound excluding any finite stretch in o(fn) space.
Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f-FDO with preprocessing time mn^{1+o(1)}, query time n^o(1), and space n^{2+o(1)}. When using fast matrix multiplication instead, the preprocessing time can be improved to n^{ω+o(1)}, where ω < 2.373 is the matrix multiplication exponent.

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Space-Efficient Fault-Tolerant Diameter Oracles. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2021.18, author = {Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, title = {{Space-Efficient Fault-Tolerant Diameter Oracles}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.18}, URN = {urn:nbn:de:0030-drops-144581}, doi = {10.4230/LIPIcs.MFCS.2021.18}, annote = {Keywords: derandomization, diameter, distance sensitivity oracle, fault-tolerant data structure, space lower bound} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

How do rational agents self-organize when trying to connect to a common target? We study this question with a simple tree formation game which is related to the well-known fair single-source connection game by Anshelevich et al. (FOCS'04) and selfish spanning tree games by Gourvès and Monnot (WINE'08). In our game agents correspond to nodes in a network that activate a single outgoing edge to connect to the common target node (possibly via other nodes). Agents pay for their path to the common target, and edge costs are shared fairly among all agents using an edge. The main novelty of our model is dynamic edge costs that depend on the in-degree of the respective endpoint. This reflects that connecting to popular nodes that have increased internal coordination costs is more expensive since they can charge higher prices for their routing service.
In contrast to related models, we show that equilibria are not guaranteed to exist, but we prove the existence for infinitely many numbers of agents. Moreover, we analyze the structure of equilibrium trees and employ these insights to prove a constant upper bound on the Price of Anarchy as well as non-trivial lower bounds on both the Price of Anarchy and the Price of Stability. We also show that in comparison with the social optimum tree the overall cost of an equilibrium tree is more fairly shared among the agents. Thus, we prove that self-organization of rational agents yields on average only slightly higher cost per agent compared to the centralized optimum, and at the same time, it induces a more fair cost distribution. Moreover, equilibrium trees achieve a beneficial trade-off between a low height and low maximum degree, and hence these trees might be of independent interest from a combinatorics point-of-view. We conclude with a discussion of promising extensions of our model.

Davide Bilò, Tobias Friedrich, Pascal Lenzner, Anna Melnichenko, and Louise Molitor. Fair Tree Connection Games with Topology-Dependent Edge Cost. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.FSTTCS.2020.15, author = {Bil\`{o}, Davide and Friedrich, Tobias and Lenzner, Pascal and Melnichenko, Anna and Molitor, Louise}, title = {{Fair Tree Connection Games with Topology-Dependent Edge Cost}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.15}, URN = {urn:nbn:de:0030-drops-132562}, doi = {10.4230/LIPIcs.FSTTCS.2020.15}, annote = {Keywords: Network Design Games, Spanning Tree Games, Fair Cost Sharing, Price of Anarchy, Nash Equilibrium, Algorithmic Game Theory, Combinatorics} }

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

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house.
The garden consists of n bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the Bamboo Garden Trimming (BGT) problem, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan, i.e., the maximum height ever reached by a bamboo.
Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than x. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of O(log n) and 4 for the best choice of x = 2, respectively. We prove the first constant upper bound of 9 for Reduce-Max and improve the one for Reduce-Fastest(x) to (3+√5)/2 < 2.62 for x = 1+1/√5.
Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x).

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Giacomo Scornavacca. Cutting Bamboo down to Size. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2021.5, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Scornavacca, Giacomo}, title = {{Cutting Bamboo down to Size}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {5:1--5:18}, 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.2021.5}, URN = {urn:nbn:de:0030-drops-127663}, doi = {10.4230/LIPIcs.FUN.2021.5}, annote = {Keywords: bamboo garden trimming, trimming oracles, approximation algorithms, pinwheel scheduling} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Residential segregation is a wide-spread phenomenon that can be observed in almost every major city. In these urban areas residents with different racial or socioeconomic background tend to form homogeneous clusters. Schelling’s famous agent-based model for residential segregation explains how such clusters can form even if all agents are tolerant, i.e., if they agree to live in mixed neighborhoods. For segregation to occur, all it needs is a slight bias towards agents preferring similar neighbors. Very recently, Schelling’s model has been investigated from a game-theoretic point of view with selfish agents that strategically select their residential location. In these games, agents can improve on their current location by performing a location swap with another agent who is willing to swap.
We significantly deepen these investigations by studying the influence of the underlying topology modeling the residential area on the existence of equilibria, the Price of Anarchy and on the dynamic properties of the resulting strategic multi-agent system. Moreover, as a new conceptual contribution, we also consider the influence of locality, i.e., if the location swaps are restricted to swaps of neighboring agents. We give improved almost tight bounds on the Price of Anarchy for arbitrary underlying graphs and we present (almost) tight bounds for regular graphs, paths and cycles. Moreover, we give almost tight bounds for grids, which are commonly used in empirical studies. For grids we also show that locality has a severe impact on the game dynamics.

Davide Bilò, Vittorio Bilò, Pascal Lenzner, and Louise Molitor. Topological Influence and Locality in Swap Schelling Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2020.15, author = {Bil\`{o}, Davide and Bil\`{o}, Vittorio and Lenzner, Pascal and Molitor, Louise}, title = {{Topological Influence and Locality in Swap Schelling Games}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.15}, URN = {urn:nbn:de:0030-drops-126841}, doi = {10.4230/LIPIcs.MFCS.2020.15}, annote = {Keywords: Residential Segregation, Schelling’s Segregation Model, Non-cooperative Games, Price of Anarchy, Game Dynamics} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Given a 2-edge connected, unweighted, and undirected graph G with n vertices and m edges, a sigma-tree spanner is a spanning tree T of G in which the ratio between the distance in T of any pair of vertices and the corresponding distance in G is upper bounded by sigma. The minimum value of sigma for which T is a sigma-tree spanner of G is also called the stretch factor of T. We address the fault-tolerant scenario in which each edge e of a given tree spanner may temporarily fail and has to be replaced by a best swap edge, i.e. an edge that reconnects T-e at a minimum stretch factor. More precisely, we design an O(n^2) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n^2 log^4 n) time and O(n^2+m log^2n) space algorithm was known for edge-weighted graphs [Bilò et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a o(n^2) time and space algorithm would be considered a breakthrough.

Davide Bilò and Kleitos Papadopoulos. A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 7:1-7:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2018.7, author = {Bil\`{o}, Davide and Papadopoulos, Kleitos}, title = {{A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {7:1--7:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.7}, URN = {urn:nbn:de:0030-drops-99557}, doi = {10.4230/LIPIcs.ISAAC.2018.7}, annote = {Keywords: Transient edge failure, best swap edges, tree spanner} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We consider the problem of augmenting an n-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of vertices u and v, reports the weight of the shortcut (u,v) in constant time. Previously, the problem was solved in O(n^2 log^3 n) time for general weights [Oh and Ahn, ISAAC 2016], in O(n^2 log n) time for trees embedded in a metric space [Große et al., https://arxiv.org/abs/1607.05547], and in O(n log n) time for paths embedded in a metric space [Wang, WADS 2017]. Furthermore, a (1+epsilon)-approximation algorithm running in O(n+1/epsilon^3) has been designed for paths embedded in R^d, for constant values of d [Große et al., ICALP 2015].
The contribution of this paper is twofold: we address the problem for trees (not only paths) and we also improve upon all known results. More precisely, we design a time-optimal O(n^2) time algorithm for general weights. Moreover, for trees embedded in a metric space, we design (i) an exact O(n log n) time algorithm and (ii) a (1+epsilon)-approximation algorithm that runs in O(n+ epsilon^{-1}log epsilon^{-1}) time.

Davide Bilò. Almost Optimal Algorithms for Diameter-Optimally Augmenting Trees. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo:LIPIcs.ISAAC.2018.40, author = {Bil\`{o}, Davide}, title = {{Almost Optimal Algorithms for Diameter-Optimally Augmenting Trees}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {40:1--40:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.40}, URN = {urn:nbn:de:0030-drops-99888}, doi = {10.4230/LIPIcs.ISAAC.2018.40}, annote = {Keywords: Graph diameter, augmentation problem, trees, time-efficient algorithms} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Reoptimization is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the modified instance.
We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decade. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP.

Davide Bilò. New algorithms for Steiner tree reoptimization. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo:LIPIcs.ICALP.2018.19, author = {Bil\`{o}, Davide}, title = {{New algorithms for Steiner tree reoptimization}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.19}, URN = {urn:nbn:de:0030-drops-90234}, doi = {10.4230/LIPIcs.ICALP.2018.19}, annote = {Keywords: Steiner tree problem, reoptimization, approximation algorithms} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Neeldhara Misra. On the Complexity of Two Dots for Narrow Boards and Few Colors. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.7, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Misra, Neeldhara}, title = {{On the Complexity of Two Dots for Narrow Boards and Few Colors}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.7}, URN = {urn:nbn:de:0030-drops-87988}, doi = {10.4230/LIPIcs.FUN.2018.7}, annote = {Keywords: puzzle, NP-complete, perfect information, combinatorial game theory} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors.

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.8, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.8}, URN = {urn:nbn:de:0030-drops-87994}, doi = {10.4230/LIPIcs.FUN.2018.8}, annote = {Keywords: peg duotaire, pspace-completeness, peg solitaire, two-player games} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \in S x T, where S, T \subseteq V(G), and every e \in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)).
More precisely, we provide the following results:
(1) DSO:
For every S,T \subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\sqrt{|S||T|}) and query time of
O(\sqrt{|S||T|}). At the expense of having quasi-polynomial query time,
the size of the oracle can be improved to tilde{O}(n|S|+|T|\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings.
(2) FTP:
We show the construction of a path-reporting DSO of size tilde{O}(n^{4/3}(|S||T|)^{1/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors).
(3) Routing and Labeling Schemes:
For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.

Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.13, author = {Bil\`{o}, Davide and Choudhary, Keerti and Gual\`{a}, Luciano and Leucci, Stefano and Parter, Merav and Proietti, Guido}, title = {{Efficient Oracles and Routing Schemes for Replacement Paths}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.13}, URN = {urn:nbn:de:0030-drops-85249}, doi = {10.4230/LIPIcs.STACS.2018.13}, annote = {Keywords: Fault tolerant, Shortest path, Oracle, Routing} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC'03] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees.
We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

Davide Bilò and Pascal Lenzner. On the Tree Conjecture for the Network Creation Game. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.14, author = {Bil\`{o}, Davide and Lenzner, Pascal}, title = {{On the Tree Conjecture for the Network Creation Game}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.14}, URN = {urn:nbn:de:0030-drops-85092}, doi = {10.4230/LIPIcs.STACS.2018.14}, annote = {Keywords: Algorithmic Game Theory, Network Creation Game, Price of Anarchy, Quality of Nash Equilibria} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

A tree sigma-spanner of a positively real-weighted n-vertex and m-edge undirected graph G is a spanning tree T of G which approximately preserves (i.e., up to a multiplicative stretch factor sigma) distances in G.
Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in T, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge -- a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in G, but also in all the scenarios in which an edge of T has failed. For this problem we provide a very efficient solution, running in O(n^2 log^4 n) time, which drastically improves (almost by a quadratic factor in n in dense graphs!) on the previous known best result.

Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2017.14, author = {Bil\`{o}, Davide and Colella, Feliciano and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.14}, URN = {urn:nbn:de:0030-drops-82663}, doi = {10.4230/LIPIcs.ISAAC.2017.14}, annote = {Keywords: Transient edge failure, Swap algorithm, Tree spanner} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0<epsilon<1, another sensitivity oracle having size O(n*1/epsilon*log(1/epsilon)), and is able to report a (1+epsilon)-approximate distance from s to any vertex of G in O(log(n)*1/epsilon*log(1/epsilon)) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.

Davide Bilo, Luciano Guala, Stefano Leucci, and Guido Proietti. Compact and Fast Sensitivity Oracles for Single-Source Distances. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2016.13, author = {Bilo, Davide and Guala, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Compact and Fast Sensitivity Oracles for Single-Source Distances}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.13}, URN = {urn:nbn:de:0030-drops-63640}, doi = {10.4230/LIPIcs.ESA.2016.13}, annote = {Keywords: fault-tolerant shortest-path tree, approximate distance, distance sensitivity oracle} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer f >= 1, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) sigma-approximate single-source shortest-path tree (sigma-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched at most by a factor of sigma. To this respect, we provide an algorithm that efficiently computes an f-EFT (2|F|+1)-ASPT of size O(f n). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1), plus an additive term of (f+1)*log(n).
Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle (SSDO), that can be built in ~{O}(f m) time, has size O(fn *log^2(n)), and is able to report, after the failure of the edge set F, in O(|F|^2 * log^2(n)) time a (2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G after that a batch of k simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(m * log^3(n)) time a sensitivity oracle of size O(m * log^2(n)), that reports in O(k^2 * log^2(n)) time the (at most 2k) edges either exiting from or entering into the MSF. As a result of independent interest, it is worth noticing that our MSF oracle can be employed to handle arbitrary sequences of o(sqrt[4]{n}/log(n)) (non-simultaneous) updates with a worst-case time per update of o(sqrt{n}). Thus, for relatively short sequences of updates, our oracle should be preferred w.r.t. the best-known (in a worst-case sense) MSF fully-dynamic algorithm, requiring O(sqrt{n}) time per update.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2016.18, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.18}, URN = {urn:nbn:de:0030-drops-57196}, doi = {10.4230/LIPIcs.STACS.2016.18}, annote = {Keywords: fault-tolerant shortest-path tree, distance oracle, minimum spanning tree} }

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

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.

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} }

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

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.

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} }

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

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.

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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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the problem of estimating the ST-diameter of a graph that is subject to a bounded number of edge failures. An f-edge fault-tolerant ST-diameter oracle (f-FDO-ST) is a data structure that preprocesses a given graph G, two sets of vertices S,T, and positive integer f. When queried with a set F of at most f edges, the oracle returns an estimate D̂ of the ST-diameter diam(G-F,S,T), the maximum distance between vertices in S and T in G-F. The oracle has stretch σ ⩾ 1 if diam(G-F,S,T) ⩽ D̂ ⩽ σ diam(G-F,S,T). If S and T both contain all vertices, the data structure is called an f-edge fault-tolerant diameter oracle (f-FDO). An f-edge fault-tolerant distance sensitivity oracles (f-DSO) estimates the pairwise graph distances under up to f failures.
We design new f-FDOs and f-FDO-STs by reducing their construction to that of all-pairs and single-source f-DSOs. We obtain several new tradeoffs between the size of the data structure, stretch guarantee, query and preprocessing times for diameter oracles by combining our black-box reductions with known results from the literature.
We also provide an information-theoretic lower bound on the space requirement of approximate f-FDOs. We show that there exists a family of graphs for which any f-FDO with sensitivity f ⩾ 2 and stretch less than 5/3 requires Ω(n^{3/2}) bits of space, regardless of the query time.

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, Simon Krogmann, and Martin Schirneck. Fault-Tolerant ST-Diameter Oracles. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2023.24, author = {Bil\`{o}, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Krogmann, Simon and Schirneck, Martin}, title = {{Fault-Tolerant ST-Diameter Oracles}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {24:1--24:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel 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.2023.24}, URN = {urn:nbn:de:0030-drops-180762}, doi = {10.4230/LIPIcs.ICALP.2023.24}, annote = {Keywords: diameter oracles, distance sensitivity oracles, space lower bounds, fault-tolerant data structures} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

A temporal graph is an undirected graph G = (V,E) along with a function λ : E → ℕ^+ that assigns a time-label to each edge in E. A path in G such that the traversed time-labels are non-decreasing is called a temporal path. Accordingly, the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v. A temporal α-spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V, up to a multiplicative stretch factor of α. The size of H is measured as the number of its edges.
In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2k-1)-spanner with Õ(kn^{1+1/k}) edges, where k > 1 is an integer parameter of choice. Choosing k = ⌊log n⌋, we obtain a temporal O(log n)-spanner with Õ(n) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity.
We then turn our attention to general temporal graphs. Since Ω(n²) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP'16], we focus on approximating distances from a single source. We show that Õ(n/log(1+ε)) edges suffice to obtain a stretch of (1+ε), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω(n²) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of Õ(n² / β) on the size of any temporal β-additive spanner, which we show to be tight up to polylogarithmic factors.
Finally, we investigate how the lifetime of G, i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Sparse Temporal Spanners with Low Stretch. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2022.19, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Rossi, Mirko}, title = {{Sparse Temporal Spanners with Low Stretch}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {19:1--19:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.19}, URN = {urn:nbn:de:0030-drops-169575}, doi = {10.4230/LIPIcs.ESA.2022.19}, annote = {Keywords: temporal spanners, temporal graphs, graph sparsification, approximate distances} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We construct data structures for extremal and pairwise distances in directed graphs in the presence of transient edge failures. Henzinger et al. [ITCS 2017] initiated the study of fault-tolerant (sensitivity) oracles for the diameter and vertex eccentricities. We extend this with a special focus on space efficiency. We present several new data structures, among them the first fault-tolerant eccentricity oracle for dual failures in subcubic space. We further prove lower bounds that show limits to approximation vs. space and diameter vs. space trade-offs for fault-tolerant oracles. They highlight key differences between data structures for undirected and directed graphs.
Initially, our oracles are randomized leaning on a sampling technique frequently used in sensitivity analysis. Building on the work of Alon, Chechik, and Cohen [ICALP 2019] as well as Karthik and Parter [SODA 2021], we develop a hierarchical framework to derandomize fault-tolerant data structures. We first apply it to our own diameter and eccentricity oracles and then show its versatility by derandomizing algorithms from the literature: the distance sensitivity oracle of Ren [JCSS 2022] and the Single-Source Replacement Path algorithm of Chechik and Magen [ICALP 2020]. This way, we obtain the first deterministic distance sensitivity oracle with subcubic preprocessing time.

Davide Bilò, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ICALP.2022.22, author = {Bil\`{o}, Davide and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, title = {{Deterministic Sensitivity Oracles for Diameter, Eccentricities and All Pairs Distances}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {22:1--22:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.22}, URN = {urn:nbn:de:0030-drops-163633}, doi = {10.4230/LIPIcs.ICALP.2022.22}, annote = {Keywords: derandomization, diameter, eccentricity, fault-tolerant data structure, sensitivity oracle, space lower bound} }

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

Let G be a directed graph with n vertices, m edges, and non-negative edge costs. Given G, a fixed source vertex s, and a positive integer p, we consider the problem of computing, for each vertex t≠ s, p edge-disjoint paths of minimum total cost from s to t in G. Suurballe and Tarjan [Networks, 1984] solved the above problem for p = 2 by designing a O(m+nlog n) time algorithm which also computes a sparse single-source 2-multipath preserver, i.e., a subgraph containing 2 edge-disjoint paths of minimum total cost from s to every other vertex of G. The case p ≥ 3 was left as an open problem.
We study the general problem (p ≥ 2) and prove that any graph admits a sparse single-source p-multipath preserver with p(n-1) edges. This size is optimal since the in-degree of each non-root vertex v must be at least p. Moreover, we design an algorithm that requires O(pn² (p + log n)) time to compute both p edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source p-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves n-1 single-pair instances using the well-known successive shortest paths algorithm by a factor of Θ(m/(np)) and is asymptotically near optimal if p = O(1) and m = Θ(n²). Our results extend naturally to the case of p vertex-disjoint paths.

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2022.12, author = {Bil\`{o}, Davide and D'Angelo, Gianlorenzo and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{Single-Source Shortest p-Disjoint Paths: Fast Computation and Sparse Preservers}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {12:1--12:21}, 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.2022.12}, URN = {urn:nbn:de:0030-drops-158221}, doi = {10.4230/LIPIcs.STACS.2022.12}, annote = {Keywords: multipath spanners, graph sparsification, edge-disjoint paths, min-cost flow} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

We combine ideas from distance sensitivity oracles (DSOs) and fixed-parameter tractability (FPT) to design sensitivity oracles for FPT graph problems. An oracle with sensitivity f for an FPT problem Π on a graph G with parameter k preprocesses G in time O(g(f,k) ⋅ poly(n)). When queried with a set F of at most f edges of G, the oracle reports the answer to the Π - with the same parameter k - on the graph G-F, i.e., G deprived of F. The oracle should answer queries in a time that is significantly faster than merely running the best-known FPT algorithm on G-F from scratch.
We design sensitivity oracles for the k-Path and the k-Vertex Cover problem. Our first oracle for k-Path has size O(k^{f+1}) and query time O(f min{f, log(f) + k}). We use a technique inspired by the work of Weimann and Yuster [FOCS 2010, TALG 2013] on distance sensitivity problems to reduce the space to O(({f+k}/f)^f ({f+k}/k)^k fk⋅log(n)) at the expense of increasing the query time to O(({f+k}/f)^f ({f+k}/k)^k f min{f,k}⋅log(n)). Both oracles can be modified to handle vertex-failures, but we need to replace k with 2k in all the claimed bounds.
Regarding k-Vertex Cover, we design three oracles offering different trade-offs between the size and the query time. The first oracle takes O(3^{f+k}) space and has O(2^f) query time, the second one has a size of O(2^{f+k²+k}) and a query time of O(f+k²); finally, the third one takes O(fk+k²) space and can be queried in time O(1.2738^k + f). All our oracles are computable in time (at most) proportional to their size and the time needed to detect a k-path or k-vertex cover, respectively. We also provide an interesting connection between k-Vertex Cover and the fault-tolerant shortest path problem, by giving a DSO of size O(poly(f,k) ⋅ n) with query time in O(poly(f,k)), where k is the size of a vertex cover.
Following our line of research connecting fault-tolerant FPT and shortest paths problems, we introduce parameterization to the computation of distance preservers. We study the problem, given a directed unweighted graph with a fixed source s and parameters f and k, to construct a polynomial-sized oracle that efficiently reports, for any target vertex v and set F of at most f edges, whether the distance from s to v increases at most by an additive term of k in G-F. The oracle size is O(2^k k²⋅n), while the time needed to answer a query is O(2^k f^ω k^ω), where ω < 2.373 is the matrix multiplication exponent. The second problem we study is about the construction of bounded-stretch fault-tolerant preservers. We construct a subgraph with O(2^{fk+f+k} k ⋅ n) edges that preserves those s-v-distances that do not increase by more than k upon failure of F. This improves significantly over the Õ(f n^{2-1/(2^f)}) bound in the unparameterized case by Bodwin et al. [ICALP 2017].

Davide Bilò, Katrin Casel, Keerti Choudhary, Sarel Cohen, Tobias Friedrich, J.A. Gregor Lagodzinski, Martin Schirneck, and Simon Wietheger. Fixed-Parameter Sensitivity Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bilo_et_al:LIPIcs.ITCS.2022.23, author = {Bil\`{o}, Davide and Casel, Katrin and Choudhary, Keerti and Cohen, Sarel and Friedrich, Tobias and Lagodzinski, J.A. Gregor and Schirneck, Martin and Wietheger, Simon}, title = {{Fixed-Parameter Sensitivity Oracles}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {23:1--23:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.23}, URN = {urn:nbn:de:0030-drops-156196}, doi = {10.4230/LIPIcs.ITCS.2022.23}, annote = {Keywords: data structures, distance preservers, distance sensitivity oracles, fault tolerance, fixed-parameter tractability, k-path, vertex cover} }

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

Given a graph with a distinguished source vertex s, the Single Source Replacement Paths (SSRP) problem is to compute and output, for any target vertex t and edge e, the length d(s,t,e) of a shortest path from s to t that avoids a failing edge e. A Single-Source Distance Sensitivity Oracle (Single-Source DSO) is a compact data structure that answers queries of the form (t,e) by returning the distance d(s,t,e). We show how to deterministically compress the output of the SSRP problem on n-vertex, m-edge graphs with integer edge weights in the range [1,M] into a Single-Source DSO that has size O(M^{1/2} n^{3/2}) and query time Õ(1). We prove that the space requirement is optimal (up to the word size). Our techniques can also handle vertex failures within the same bounds.
Chechik and Cohen [SODA 2019] presented a combinatorial, randomized Õ(m√n+n²) time SSRP algorithm for undirected and unweighted graphs. We derandomize their algorithm with the same asymptotic running time and apply our compression to obtain a deterministic Single-Source DSO with Õ(m√n+n²) preprocessing time, O(n^{3/2}) space, and Õ(1) query time. Our combinatorial Single-Source DSO has near-optimal space, preprocessing and query time for unweighted graphs, improving the preprocessing time by a √n-factor compared to previous results with o(n²) space.
Grandoni and Vassilevska Williams [FOCS 2012, TALG 2020] gave an algebraic, randomized Õ(Mn^ω) time SSRP algorithm for (undirected and directed) graphs with integer edge weights in the range [1,M], where ω < 2.373 is the matrix multiplication exponent. We derandomize it for undirected graphs and apply our compression to obtain an algebraic Single-Source DSO with Õ(Mn^ω) preprocessing time, O(M^{1/2} n^{3/2}) space, and Õ(1) query time. This improves the preprocessing time of algebraic Single-Source DSOs by polynomial factors compared to previous o(n²)-space oracles.
We also present further improvements of our Single-Source DSOs. We show that the query time can be reduced to a constant at the cost of increasing the size of the oracle to O(M^{1/3} n^{5/3}) and that all our oracles can be made path-reporting. On sparse graphs with m = O(n^{5/4-ε}/M^{7/4}) edges, for any constant ε > 0, we reduce the preprocessing to randomized Õ(M^{7/8} m^{1/2} n^{11/8}) = O(n^{2-ε/2}) time. To the best of our knowledge, this is the first truly subquadratic time algorithm for building Single-Source DSOs on sparse graphs.

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2021.18, author = {Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, title = {{Near-Optimal Deterministic Single-Source Distance Sensitivity Oracles}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {18:1--18:17}, 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.2021.18}, URN = {urn:nbn:de:0030-drops-145999}, doi = {10.4230/LIPIcs.ESA.2021.18}, annote = {Keywords: derandomization, distance sensitivity oracle, single-source replacement paths, space lower bound} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

We design f-edge fault-tolerant diameter oracles (f-FDO, or simply FDO if f = 1). For a given directed or undirected and possibly edge-weighted graph G with n vertices and m edges and a positive integer f, we preprocess the graph and construct a data structure that, when queried with a set F of edges, where |F| ⩽ f, returns the diameter of G-F. An f-FDO has stretch σ ⩾ 1 if the returned value D^ satisfies diam(G-F) ⩽ D^ ⩽ σ diam(G-F).
For the case of a single edge failure (f = 1) in an unweighted directed graph, there exists an approximate FDO by Henzinger et al. [ITCS 2017] with stretch (1+ε), constant query time, space O(m), and a combinatorial preprocessing time of Õ(mn + n^{1.5} √{Dm/ε}), where D is the diameter.
We present an FDO for directed graphs with the same stretch, query time, and space. It has a preprocessing time of Õ(mn + n²/ε), which is better for constant ε > 0. The preprocessing time nearly matches a conditional lower bound for combinatorial algorithms, also by Henzinger et al. With fast matrix multiplication, we achieve a preprocessing time of Õ(n^{2.5794} + n²/ε). We further prove an information-theoretic lower bound showing that any FDO with stretch better than 3/2 requires Ω(m) bits of space. Thus, for constant 0 < ε < 3/2, our combinatorial (1+ε)-approximate FDO is near-optimal in all parameters.
In the case of multiple edge failures (f > 1) in undirected graphs with non-negative edge weights, we give an f-FDO with stretch (f+2), query time O(f²log²{n}), Õ(fn) space, and preprocessing time Õ(fm). We complement this with a lower bound excluding any finite stretch in o(fn) space.
Many real-world networks have polylogarithmic diameter. We show that for those graphs and up to f = o(log n/ log log n) failures one can swap approximation for query time and space. We present an exact combinatorial f-FDO with preprocessing time mn^{1+o(1)}, query time n^o(1), and space n^{2+o(1)}. When using fast matrix multiplication instead, the preprocessing time can be improved to n^{ω+o(1)}, where ω < 2.373 is the matrix multiplication exponent.

Davide Bilò, Sarel Cohen, Tobias Friedrich, and Martin Schirneck. Space-Efficient Fault-Tolerant Diameter Oracles. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2021.18, author = {Bil\`{o}, Davide and Cohen, Sarel and Friedrich, Tobias and Schirneck, Martin}, title = {{Space-Efficient Fault-Tolerant Diameter Oracles}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.18}, URN = {urn:nbn:de:0030-drops-144581}, doi = {10.4230/LIPIcs.MFCS.2021.18}, annote = {Keywords: derandomization, diameter, distance sensitivity oracle, fault-tolerant data structure, space lower bound} }

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**Published in:** LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

How do rational agents self-organize when trying to connect to a common target? We study this question with a simple tree formation game which is related to the well-known fair single-source connection game by Anshelevich et al. (FOCS'04) and selfish spanning tree games by Gourvès and Monnot (WINE'08). In our game agents correspond to nodes in a network that activate a single outgoing edge to connect to the common target node (possibly via other nodes). Agents pay for their path to the common target, and edge costs are shared fairly among all agents using an edge. The main novelty of our model is dynamic edge costs that depend on the in-degree of the respective endpoint. This reflects that connecting to popular nodes that have increased internal coordination costs is more expensive since they can charge higher prices for their routing service.
In contrast to related models, we show that equilibria are not guaranteed to exist, but we prove the existence for infinitely many numbers of agents. Moreover, we analyze the structure of equilibrium trees and employ these insights to prove a constant upper bound on the Price of Anarchy as well as non-trivial lower bounds on both the Price of Anarchy and the Price of Stability. We also show that in comparison with the social optimum tree the overall cost of an equilibrium tree is more fairly shared among the agents. Thus, we prove that self-organization of rational agents yields on average only slightly higher cost per agent compared to the centralized optimum, and at the same time, it induces a more fair cost distribution. Moreover, equilibrium trees achieve a beneficial trade-off between a low height and low maximum degree, and hence these trees might be of independent interest from a combinatorics point-of-view. We conclude with a discussion of promising extensions of our model.

Davide Bilò, Tobias Friedrich, Pascal Lenzner, Anna Melnichenko, and Louise Molitor. Fair Tree Connection Games with Topology-Dependent Edge Cost. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.FSTTCS.2020.15, author = {Bil\`{o}, Davide and Friedrich, Tobias and Lenzner, Pascal and Melnichenko, Anna and Molitor, Louise}, title = {{Fair Tree Connection Games with Topology-Dependent Edge Cost}}, booktitle = {40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-174-0}, ISSN = {1868-8969}, year = {2020}, volume = {182}, editor = {Saxena, Nitin and Simon, Sunil}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.15}, URN = {urn:nbn:de:0030-drops-132562}, doi = {10.4230/LIPIcs.FSTTCS.2020.15}, annote = {Keywords: Network Design Games, Spanning Tree Games, Fair Cost Sharing, Price of Anarchy, Nash Equilibrium, Algorithmic Game Theory, Combinatorics} }

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

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house.
The garden consists of n bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the Bamboo Garden Trimming (BGT) problem, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the makespan, i.e., the maximum height ever reached by a bamboo.
Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than x. It is known that Reduce-Max and Reduce-Fastest(x) achieve a makespan of O(log n) and 4 for the best choice of x = 2, respectively. We prove the first constant upper bound of 9 for Reduce-Max and improve the one for Reduce-Fastest(x) to (3+√5)/2 < 2.62 for x = 1+1/√5.
Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x).

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Giacomo Scornavacca. Cutting Bamboo down to Size. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2021.5, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Scornavacca, Giacomo}, title = {{Cutting Bamboo down to Size}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {5:1--5:18}, 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.2021.5}, URN = {urn:nbn:de:0030-drops-127663}, doi = {10.4230/LIPIcs.FUN.2021.5}, annote = {Keywords: bamboo garden trimming, trimming oracles, approximation algorithms, pinwheel scheduling} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Residential segregation is a wide-spread phenomenon that can be observed in almost every major city. In these urban areas residents with different racial or socioeconomic background tend to form homogeneous clusters. Schelling’s famous agent-based model for residential segregation explains how such clusters can form even if all agents are tolerant, i.e., if they agree to live in mixed neighborhoods. For segregation to occur, all it needs is a slight bias towards agents preferring similar neighbors. Very recently, Schelling’s model has been investigated from a game-theoretic point of view with selfish agents that strategically select their residential location. In these games, agents can improve on their current location by performing a location swap with another agent who is willing to swap.
We significantly deepen these investigations by studying the influence of the underlying topology modeling the residential area on the existence of equilibria, the Price of Anarchy and on the dynamic properties of the resulting strategic multi-agent system. Moreover, as a new conceptual contribution, we also consider the influence of locality, i.e., if the location swaps are restricted to swaps of neighboring agents. We give improved almost tight bounds on the Price of Anarchy for arbitrary underlying graphs and we present (almost) tight bounds for regular graphs, paths and cycles. Moreover, we give almost tight bounds for grids, which are commonly used in empirical studies. For grids we also show that locality has a severe impact on the game dynamics.

Davide Bilò, Vittorio Bilò, Pascal Lenzner, and Louise Molitor. Topological Influence and Locality in Swap Schelling Games. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bilo_et_al:LIPIcs.MFCS.2020.15, author = {Bil\`{o}, Davide and Bil\`{o}, Vittorio and Lenzner, Pascal and Molitor, Louise}, title = {{Topological Influence and Locality in Swap Schelling Games}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.15}, URN = {urn:nbn:de:0030-drops-126841}, doi = {10.4230/LIPIcs.MFCS.2020.15}, annote = {Keywords: Residential Segregation, Schelling’s Segregation Model, Non-cooperative Games, Price of Anarchy, Game Dynamics} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Given a 2-edge connected, unweighted, and undirected graph G with n vertices and m edges, a sigma-tree spanner is a spanning tree T of G in which the ratio between the distance in T of any pair of vertices and the corresponding distance in G is upper bounded by sigma. The minimum value of sigma for which T is a sigma-tree spanner of G is also called the stretch factor of T. We address the fault-tolerant scenario in which each edge e of a given tree spanner may temporarily fail and has to be replaced by a best swap edge, i.e. an edge that reconnects T-e at a minimum stretch factor. More precisely, we design an O(n^2) time and space algorithm that computes a best swap edge of every tree edge. Previously, an O(n^2 log^4 n) time and O(n^2+m log^2n) space algorithm was known for edge-weighted graphs [Bilò et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a o(n^2) time and space algorithm would be considered a breakthrough.

Davide Bilò and Kleitos Papadopoulos. A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 7:1-7:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2018.7, author = {Bil\`{o}, Davide and Papadopoulos, Kleitos}, title = {{A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {7:1--7:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.7}, URN = {urn:nbn:de:0030-drops-99557}, doi = {10.4230/LIPIcs.ISAAC.2018.7}, annote = {Keywords: Transient edge failure, best swap edges, tree spanner} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

We consider the problem of augmenting an n-vertex tree with one shortcut in order to minimize the diameter of the resulting graph. The tree is embedded in an unknown space and we have access to an oracle that, when queried on a pair of vertices u and v, reports the weight of the shortcut (u,v) in constant time. Previously, the problem was solved in O(n^2 log^3 n) time for general weights [Oh and Ahn, ISAAC 2016], in O(n^2 log n) time for trees embedded in a metric space [Große et al., https://arxiv.org/abs/1607.05547], and in O(n log n) time for paths embedded in a metric space [Wang, WADS 2017]. Furthermore, a (1+epsilon)-approximation algorithm running in O(n+1/epsilon^3) has been designed for paths embedded in R^d, for constant values of d [Große et al., ICALP 2015].
The contribution of this paper is twofold: we address the problem for trees (not only paths) and we also improve upon all known results. More precisely, we design a time-optimal O(n^2) time algorithm for general weights. Moreover, for trees embedded in a metric space, we design (i) an exact O(n log n) time algorithm and (ii) a (1+epsilon)-approximation algorithm that runs in O(n+ epsilon^{-1}log epsilon^{-1}) time.

Davide Bilò. Almost Optimal Algorithms for Diameter-Optimally Augmenting Trees. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 40:1-40:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo:LIPIcs.ISAAC.2018.40, author = {Bil\`{o}, Davide}, title = {{Almost Optimal Algorithms for Diameter-Optimally Augmenting Trees}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {40:1--40:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.40}, URN = {urn:nbn:de:0030-drops-99888}, doi = {10.4230/LIPIcs.ISAAC.2018.40}, annote = {Keywords: Graph diameter, augmentation problem, trees, time-efficient algorithms} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Reoptimization is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the modified instance.
We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decade. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP.

Davide Bilò. New algorithms for Steiner tree reoptimization. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo:LIPIcs.ICALP.2018.19, author = {Bil\`{o}, Davide}, title = {{New algorithms for Steiner tree reoptimization}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.19}, URN = {urn:nbn:de:0030-drops-90234}, doi = {10.4230/LIPIcs.ICALP.2018.19}, annote = {Keywords: Steiner tree problem, reoptimization, approximation algorithms} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Neeldhara Misra. On the Complexity of Two Dots for Narrow Boards and Few Colors. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.7, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Misra, Neeldhara}, title = {{On the Complexity of Two Dots for Narrow Boards and Few Colors}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {7:1--7:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.7}, URN = {urn:nbn:de:0030-drops-87988}, doi = {10.4230/LIPIcs.FUN.2018.7}, annote = {Keywords: puzzle, NP-complete, perfect information, combinatorial game theory} }

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**Published in:** LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors.

Davide Bilò, Luciano Gualà, Stefano Leucci, Guido Proietti, and Mirko Rossi. On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.FUN.2018.8, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido and Rossi, Mirko}, title = {{On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {8:1--8:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-067-5}, ISSN = {1868-8969}, year = {2018}, volume = {100}, editor = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.8}, URN = {urn:nbn:de:0030-drops-87994}, doi = {10.4230/LIPIcs.FUN.2018.8}, annote = {Keywords: peg duotaire, pspace-completeness, peg solitaire, two-player games} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Real life graphs and networks are prone to failure of nodes (vertices) and links (edges). In particular, for a pair of nodes s and t and a failing edge e in an n-vertex unweighted graph G=(V(G),E(G)), the replacement path pi_{G-e}(s,t) is a shortest s-t path that avoids e. In this paper we present several efficient constructions that, for every (s,t) \in S x T, where S, T \subseteq V(G), and every e \in E(G), maintain the collection of all pi_{G-e}(s,t), either implicitly (i.e., through compact data structures a.k.a. distance sensitivity oracles (DSO)), or explicitly (i.e., through sparse subgraphs a.k.a. fault-tolerant preservers (FTP)).
More precisely, we provide the following results:
(1) DSO:
For every S,T \subseteq V(G), we construct a DSO for maintaining S x T distances under single edge (or vertex) faults. This DSO has size tilde{O}(n\sqrt{|S||T|}) and query time of
O(\sqrt{|S||T|}). At the expense of having quasi-polynomial query time,
the size of the oracle can be improved to tilde{O}(n|S|+|T|\sqrt{|S|n}), which is optimal for |T| = Omega(sqrt{n|S|}). When |T| = Omega(n^frac{3}{4} |S|^frac{1}{4}), the construction can be further refined in order to get a polynomial query time. We also consider the approximate additive setting, and show a family of DSOs that exhibits a tradeoff between the additive stretch and the size of the oracle. Finally, for the meaningful single-source case, the above result is complemented by a lower bound conditioned on the Set-Intersection conjecture. This lower bound establishes a separation between the oracle and the subgraph settings.
(2) FTP:
We show the construction of a path-reporting DSO of size tilde{O}(n^{4/3}(|S||T|)^{1/3}) reporting pi_{G-e}(s,t) in O(|pi_{G-e}(s,t)|+(n|S||T|)^{1/3}) time. Such a DSO can be transformed into a FTP having the same size, and moreover it can be elaborated in order to make it optimal (up to a poly-logarithmic factor) both in space and query time for the special case in which T=V(G). Our FTP improves over previous constructions when |T|=O(sqrt{|S|n}) (up to inverse poly-logarithmic factors).
(3) Routing and Labeling Schemes:
For the well-studied single-source setting, we present a novel routing scheme, that allows to route messages on pi_{G-e}(s,t) by using edge labels and routing tables of size tilde{O}(\sqrt{n}), and a header message of poly-logarithmic size. We also present a labeling scheme for the setting which is optimal in space up to constant factors.

Davide Bilò, Keerti Choudhary, Luciano Gualà, Stefano Leucci, Merav Parter, and Guido Proietti. Efficient Oracles and Routing Schemes for Replacement Paths. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.13, author = {Bil\`{o}, Davide and Choudhary, Keerti and Gual\`{a}, Luciano and Leucci, Stefano and Parter, Merav and Proietti, Guido}, title = {{Efficient Oracles and Routing Schemes for Replacement Paths}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {13:1--13:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.13}, URN = {urn:nbn:de:0030-drops-85249}, doi = {10.4230/LIPIcs.STACS.2018.13}, annote = {Keywords: Fault tolerant, Shortest path, Oracle, Routing} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Selfish Network Creation focuses on modeling real world networks from a game-theoretic point of view. One of the classic models by Fabrikant et al.[PODC'03] is the network creation game, where agents correspond to nodes in a network which buy incident edges for the price of alpha per edge to minimize their total distance to all other nodes. The model is well-studied but still has intriguing open problems. The most famous conjectures state that the price of anarchy is constant for all alpha and that for alpha >= n all equilibrium networks are trees.
We introduce a novel technique for analyzing stable networks for high edge-price alpha and employ it to improve on the best known bounds for both conjectures. In particular we show that for alpha > 4n-13 all equilibrium networks must be trees, which implies a constant price of anarchy for this range of alpha. Moreover, we also improve the constant upper bound on the price of anarchy for equilibrium trees.

Davide Bilò and Pascal Lenzner. On the Tree Conjecture for the Network Creation Game. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2018.14, author = {Bil\`{o}, Davide and Lenzner, Pascal}, title = {{On the Tree Conjecture for the Network Creation Game}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.14}, URN = {urn:nbn:de:0030-drops-85092}, doi = {10.4230/LIPIcs.STACS.2018.14}, annote = {Keywords: Algorithmic Game Theory, Network Creation Game, Price of Anarchy, Quality of Nash Equilibria} }

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**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

A tree sigma-spanner of a positively real-weighted n-vertex and m-edge undirected graph G is a spanning tree T of G which approximately preserves (i.e., up to a multiplicative stretch factor sigma) distances in G.
Tree spanners with provably good stretch factors find applications in communication networks, distributed systems, and network design. However, finding an optimal or even a good tree spanner is a very hard computational task. Thus, if one has to face a transient edge failure in T, the overall effort that has to be afforded to rebuild a new tree spanner (i.e., computational costs, set-up of new links, updating of the routing tables, etc.) can be rather prohibitive. To circumvent this drawback, an effective alternative is that of associating with each tree edge a best possible (in terms of resulting stretch) swap edge -- a well-established approach in the literature for several other tree topologies. Correspondingly, the problem of computing all the best swap edges of a tree spanner is a challenging algorithmic problem, since solving it efficiently means to exploit the structure of shortest paths not only in G, but also in all the scenarios in which an edge of T has failed. For this problem we provide a very efficient solution, running in O(n^2 log^4 n) time, which drastically improves (almost by a quadratic factor in n in dense graphs!) on the previous known best result.

Davide Bilò, Feliciano Colella, Luciano Gualà, Stefano Leucci, and Guido Proietti. An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bilo_et_al:LIPIcs.ISAAC.2017.14, author = {Bil\`{o}, Davide and Colella, Feliciano and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{An Improved Algorithm for Computing All the Best Swap Edges of a Tree Spanner}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {14:1--14:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.14}, URN = {urn:nbn:de:0030-drops-82663}, doi = {10.4230/LIPIcs.ISAAC.2017.14}, annote = {Keywords: Transient edge failure, Swap algorithm, Tree spanner} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Let s denote a distinguished source vertex of a non-negatively real weighted and undirected graph G with n vertices and m edges. In this paper we present two efficient single-source approximate-distance sensitivity oracles, namely compact data structures which are able to quickly report an approximate (by a multiplicative stretch factor) distance from s to any node of G following the failure of any edge in G. More precisely, we first present a sensitivity oracle of size O(n) which is able to report 2-approximate distances from the source in O(1) time. Then, we further develop our construction by building, for any 0<epsilon<1, another sensitivity oracle having size O(n*1/epsilon*log(1/epsilon)), and is able to report a (1+epsilon)-approximate distance from s to any vertex of G in O(log(n)*1/epsilon*log(1/epsilon)) time. Thus, this latter oracle is essentially optimal as far as size and stretch are concerned, and it only asks for a logarithmic query time. Finally, our results are complemented with a space lower bound for the related class of single-source additively-stretched sensitivity oracles, which is helpful to realize the hardness of designing compact oracles of this type.

Davide Bilo, Luciano Guala, Stefano Leucci, and Guido Proietti. Compact and Fast Sensitivity Oracles for Single-Source Distances. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.ESA.2016.13, author = {Bilo, Davide and Guala, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Compact and Fast Sensitivity Oracles for Single-Source Distances}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.13}, URN = {urn:nbn:de:0030-drops-63640}, doi = {10.4230/LIPIcs.ESA.2016.13}, annote = {Keywords: fault-tolerant shortest-path tree, approximate distance, distance sensitivity oracle} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer f >= 1, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) sigma-approximate single-source shortest-path tree (sigma-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched at most by a factor of sigma. To this respect, we provide an algorithm that efficiently computes an f-EFT (2|F|+1)-ASPT of size O(f n). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of 3(f+1), plus an additive term of (f+1)*log(n).
Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle (SSDO), that can be built in ~{O}(f m) time, has size O(fn *log^2(n)), and is able to report, after the failure of the edge set F, in O(|F|^2 * log^2(n)) time a (2|F|+1)-approximate distance from the source to any node, and a corresponding approximate path in the same amount of time plus the path's size. Such an oracle is obtained by handling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G after that a batch of k simultaneous edge modifications (i.e., edge insertions, deletions and weight changes) is performed. For this problem, we build in O(m * log^3(n)) time a sensitivity oracle of size O(m * log^2(n)), that reports in O(k^2 * log^2(n)) time the (at most 2k) edges either exiting from or entering into the MSF. As a result of independent interest, it is worth noticing that our MSF oracle can be employed to handle arbitrary sequences of o(sqrt[4]{n}/log(n)) (non-simultaneous) updates with a worst-case time per update of o(sqrt{n}). Thus, for relatively short sequences of updates, our oracle should be preferred w.r.t. the best-known (in a worst-case sense) MSF fully-dynamic algorithm, requiring O(sqrt{n}) time per update.

Davide Bilò, Luciano Gualà, Stefano Leucci, and Guido Proietti. Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bilo_et_al:LIPIcs.STACS.2016.18, author = {Bil\`{o}, Davide and Gual\`{a}, Luciano and Leucci, Stefano and Proietti, Guido}, title = {{Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {18:1--18:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.18}, URN = {urn:nbn:de:0030-drops-57196}, doi = {10.4230/LIPIcs.STACS.2016.18}, annote = {Keywords: fault-tolerant shortest-path tree, distance oracle, minimum spanning tree} }

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