eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
1
1080
10.4230/LIPIcs.ISAAC.2022
article
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
Bae, Sang Won
1
https://orcid.org/0000-0002-8802-4247
Park, Heejin
2
https://orcid.org/0000-0002-8608-5994
Kyonggi University, Suwon, Korea
Hanyang University, Seoul, Korea
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022/LIPIcs.ISAAC.2022.pdf
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
0:i
0:xx
10.4230/LIPIcs.ISAAC.2022.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Bae, Sang Won
1
https://orcid.org/0000-0002-8802-4247
Park, Heejin
2
https://orcid.org/0000-0002-8608-5994
Kyonggi University, Suwon, Korea
Hanyang University, Seoul, Korea
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.0/LIPIcs.ISAAC.2022.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
1:1
1:1
10.4230/LIPIcs.ISAAC.2022.1
article
Succinct Representations of Graphs (Invited Talk)
Sadakane, Kunihiko
1
Graduate School of Information Science and Technology, The University of Tokyo, Japan
We consider the problem of finding succinct representations of graphs, that is, encodings using asymptotically the minimum number of bits which support queries on the graphs efficiently. For a special class of graphs, there exist many theoretical results and practical implementations on ordered trees. On the other hand, for wider classes of graphs, though there are many results on counting the number of non-isomorphic graphs belonging to a graph class, there were few number of results on their succinct representations until recently.
In this talk, we review some recent results on succinct representations of graphs such as interval, permutation, circle, circular-arc, trapezoid, circle-trapezoid, k-polygon, circle-polygon, cograph, separable, ptolemaic, distance hereditary, clique width k, block, cactus, series-parallel, planar, tree width k, path, boxicity k, chordal bipartite, strongly chordal, chordal graphs, etc.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.1/LIPIcs.ISAAC.2022.1.pdf
Graph Enumeration
Succinct Data Structure
Compression
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
2:1
2:1
10.4230/LIPIcs.ISAAC.2022.2
article
The Tragedy of Being Almost but Not Quite Planar (Invited Talk)
Erickson, Jeff
1
https://orcid.org/0000-0002-5253-2282
University of Illinois Urbana-Champaign, IL, USA
Planar graphs have been fertile grounds for algorithms research for decades, both because they model several types of real-world networks, and because they admit simpler and and faster algorithms than arbitrary graphs. Many important structural properties of planar graphs extend naturally to graphs that embed on more complex surfaces. As a result, efficient algorithms for planar graphs often extend naturally to higher-genus surface graphs with little or no modification.
I will describe a few of my favorite exceptions to this rule - classical problems that admit simple, efficient, and practical algorithms for planar graphs, but where algorithms for graphs on other surfaces are significantly slower and/or more complex.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.2/LIPIcs.ISAAC.2022.2.pdf
planar graphs
surface graphs
algorithms
open problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
3:1
3:13
10.4230/LIPIcs.ISAAC.2022.3
article
A Local Search Algorithm for the Min-Sum Submodular Cover Problem
Hellerstein, Lisa
1
https://orcid.org/0000-0002-3743-7965
Lidbetter, Thomas
2
3
https://orcid.org/0000-0001-6111-2899
Witter, R. Teal
1
https://orcid.org/0000-0003-3096-3767
Department of Computer Science and Engineering, New York University, Brooklyn, NY, USA
Department of Engineering Systems and Environment, University of Virginia, Charlottesville, VA, USA
Department of Management Science and Information Systems, Rutgers Business School, Newark, NJ, USA
We consider the problem of solving the Min-Sum Submodular Cover problem using local search. The Min-Sum Submodular Cover problem generalizes the NP-complete Min-Sum Set Cover problem, replacing the input set cover instance with a monotone submodular set function. A simple greedy algorithm achieves an approximation factor of 4, which is tight unless P=NP [Streeter and Golovin, NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show that, using simple initialization, a straightforward local search algorithm achieves a (4+ε)-approximate solution in time O(n³log(n/ε)), provided that the monotone submodular set function is also second-order supermodular. Second-order supermodularity has been shown to hold for a number of submodular functions of practical interest, including functions associated with set cover, matching, and facility location. We present experiments on two special cases of Min-Sum Submodular Cover and find that the local search algorithm can outperform the greedy algorithm on small data sets.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.3/LIPIcs.ISAAC.2022.3.pdf
Local search
submodularity
second-order supermodularity
min-sum set cover
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
4:1
4:19
10.4230/LIPIcs.ISAAC.2022.4
article
Algorithms for Coloring Reconfiguration Under Recolorability Digraphs
Fujii, Soichiro
1
2
https://orcid.org/0000-0002-4109-3472
Iwamasa, Yuni
3
https://orcid.org/0000-0002-6794-3543
Kimura, Kei
4
https://orcid.org/0000-0002-0560-5127
Suzuki, Akira
5
https://orcid.org/0000-0002-5212-0202
Research Institute for Mathematical Sciences, Kyoto University, Japan
School of Mathematical and Physical Sciences, Macquarie University, Sydney, Australia
Graduate School of Informatics, Kyoto University, Japan
Faculty of Information Science and Electrical Engineering, Kyushu University, Fukuoka, Japan
Graduate School of Information Sciences, Tohoku University, Sendai, Japan
In the k-Recoloring problem, we are given two (vertex-)colorings of a graph using k colors, and asked to transform one into the other by recoloring only one vertex at a time, while at all times maintaining a proper k-coloring. This problem is known to be solvable in polynomial time if k ≤ 3, and is PSPACE-complete if k ≥ 4. In this paper, we consider a (directed) recolorability constraint on the k colors, which forbids some pairs of colors to be recolored directly. The recolorability constraint is given in terms of a digraph R, whose vertices correspond to the colors and whose arcs represent the pairs of colors that can be recolored directly. We provide algorithms for the problem based on the structure of recolorability constraints R, showing that the problem is solvable in linear time when R is a directed cycle or is in a class of multitrees.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.4/LIPIcs.ISAAC.2022.4.pdf
combinatorial reconfiguration
graph coloring
recolorability
recoloring
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
5:1
5:17
10.4230/LIPIcs.ISAAC.2022.5
article
Algorithms for Landmark Hub Labeling
Storandt, Sabine
1
Universität Konstanz, Germany
Landmark-based routing and Hub Labeling (HL) are shortest path planning techniques, both of which rely on storing shortest path distances between selected pairs of nodes in a preprocessing phase to accelerate query answering. In Landmark-based routing, stored distances to landmark nodes are used to obtain distance lower bounds that guide A* search from node s to node t. With HL, tight upper bounds for shortest path distances between any s-t-pair can be interfered from their stored node labels, making HL an efficient distance oracle. However, for shortest path retrieval, the oracle has to be called once per edge in said path. Furthermore, HL often suffers from a large space consumption as many node pair distances have to be stored in the labels to allow for correct query answering. In this paper, we propose a novel technique, called Landmark Hub Labeling (LHL), which integrates the landmark concept into HL. We prove better worst-case path retrieval times for LHL in case it is path-consistent (a new labeling property we introduce). Moreover, we design efficient (approximation) algorithms that produce path-consistent LHL with small label size and provide parametrized upper bounds, depending on the highway dimension h or the geodesic transversal number gt of the graph. Finally, we show that the space consumption of LHL is smaller than that of (hierarchical) HL, both in theory and in experiments on real-world road networks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.5/LIPIcs.ISAAC.2022.5.pdf
Hub Labeling
Landmark
Geodesic
Hitting Set
Highway Dimension
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
6:1
6:14
10.4230/LIPIcs.ISAAC.2022.6
article
An Optimal Oracle Separation of Classical and Quantum Hybrid Schemes
Hasegawa, Atsuya
1
Le Gall, François
2
Graduate School of Information Science and Technology, The University of Tokyo, Japan
Graduate School of Mathematics, Nagoya University, Japan
Recently, Chia, Chung and Lai (STOC 2020) and Coudron and Menda (STOC 2020) have shown that there exists an oracle 𝒪 such that BQP^𝒪 ≠ (BPP^BQNC)^𝒪 ∪ (BQNC^BPP)^𝒪. In fact, Chia et al. proved a stronger statement: for any depth parameter d, there exists an oracle that separates quantum depth d and 2d+1, when polynomial-time classical computation is allowed. This implies that relative to an oracle, doubling quantum depth gives classical and quantum hybrid schemes more computational power.
In this paper, we show that for any depth parameter d, there exists an oracle that separates quantum depth d and d+1, when polynomial-time classical computation is allowed. This gives an optimal oracle separation of classical and quantum hybrid schemes. To prove our result, we consider d-Bijective Shuffling Simon’s Problem (which is a variant of d-Shuffling Simon’s Problem considered by Chia et al.) and an oracle inspired by an "in-place" permutation oracle.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.6/LIPIcs.ISAAC.2022.6.pdf
small-depth quantum circuit
hybrid quantum computer
oracle separation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
7:1
7:15
10.4230/LIPIcs.ISAAC.2022.7
article
Approximating the Minimum Logarithmic Arrangement Problem
Mestre, Julián
1
2
https://orcid.org/0000-0003-4948-2998
Pupyrev, Sergey
1
https://orcid.org/0000-0003-4089-673X
Meta Platforms Inc., USA
School of Computer Science, The University of Sydney, Australia
We study a graph reordering problem motivated by compressing massive graphs such as social networks and inverted indexes. Given a graph, G = (V, E), the Minimum Logarithmic Arrangement problem is to find a permutation, π, of the vertices that minimizes ∑_{(u, v) ∈ E} (1 + ⌊ lg |π(u) - π(v)| ⌋).
This objective has been shown to be a good measure of how many bits are needed to encode the graph if the adjacency list of each vertex is encoded using relative positions of two consecutive neighbors under the π order in the list rather than using absolute indices or node identifiers, which requires at least lg n bits per edge.
We show the first non-trivial approximation factor for this problem by giving a polynomial time 𝒪(log k)-approximation algorithm for graphs with treewidth k.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.7/LIPIcs.ISAAC.2022.7.pdf
approximation algorithms
graph compression
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
8:1
8:17
10.4230/LIPIcs.ISAAC.2022.8
article
Bi-Criteria Approximation Algorithms for Bounded-Degree Subset TSP
Friggstad, Zachary
1
Mousavi, Ramin
1
Department of Computing Science, University of Alberta, Edmonton, Canada
We initiate the study of the Bounded-Degree Subset Traveling Salesman problem (BDSTSP) in which we are given a graph G = (V,E) with edge cost c_e ≥ 0 on each edge e, degree bounds b_v ≥ 0 on each vertex v ∈ V and a subset of terminals X ⊆ V. The goal is to find a minimum-cost closed walk that spans all terminals and visits each vertex v ∈ V at most b_v/2 times. In this paper, we study bi-criteria approximations that find tours whose cost is within a constant-factor of the optimum tour length while violating the bounds b_v at each vertex by additive quantities.
If X = V, an adaptation of the Christofides-Serdyukov algorithm yields a (3/2, +4)-approximation, that is the tour passes through each vertex at most b_v/2+2 times (the degree of v in the multiset of edges on the tour being at most b_v + 4). This is enabled through known results in bounded-degree Steiner trees and integrality of the bounded-degree Y-join polytope. The general case X ≠ V is more challenging since we cannot pass to the metric completion on X. However, it is at least simple to get a (5/3, +4)-bicriteria approximation by using ideas similar to Hoogeveen’s TSP-Path algorithm.
Our main result is an improved approximation with marginally worse violations of the vertex bounds: a (13/8, +6)-approximation. We obtain this primarily through adapting the bounded-degree Steiner tree approximation to ensure certain "dangerous" nodes always have even degree in the resulting tree which allows us to bound the cost of the resulting degree-bounded Y-join. We also recover a (3/2, +8)-approximation for this general case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.8/LIPIcs.ISAAC.2022.8.pdf
Linear programming
approximation algorithms
combinatorial optimization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
9:1
9:19
10.4230/LIPIcs.ISAAC.2022.9
article
Budgeted Out-Tree Maximization with Submodular Prizes
D'Angelo, Gianlorenzo
1
https://orcid.org/0000-0003-0377-7037
Delfaraz, Esmaeil
1
Gilbert, Hugo
2
https://orcid.org/0000-0001-9729-2959
Gran Sasso Science Institute, L'Aquila, Italy
Université Paris-Dauphine, Université PSL, CNRS, LAMSADE, 75016 Paris, France
We consider a variant of the prize collecting Steiner tree problem in which we are given a directed graph D = (V,A), a monotone submodular prize function p:2^V → ℝ^+ ∪ {0}, a cost function c:V → ℤ^+, a root vertex r ∈ V, and a budget B. The aim is to find an out-subtree T of D rooted at r that costs at most B and maximizes the prize function. We call this problem Directed Rooted Submodular Tree (DRST).
For the case of undirected graphs and additive prize functions, Moss and Rabani [SIAM J. Comput. 2007] gave an algorithm that guarantees an O(log|V|)-approximation factor if a violation by a factor 2 of the budget constraint is allowed. Bateni et al. [SIAM J. Comput. 2018] improved the budget violation factor to 1+ε at the cost of an additional approximation factor of O(1/ε²), for any ε ∈ (0,1]. For directed graphs, Ghuge and Nagarajan [SODA 2020] gave an optimal quasi-polynomial time O({log n'}/{log log n'})-approximation algorithm, where n' is the number of vertices in an optimal solution, for the case in which the costs are associated to the edges.
In this paper, we give a polynomial time algorithm for DRST that guarantees an approximation factor of O(√B/ε³) at the cost of a budget violation of a factor 1+ε, for any ε ∈ (0,1]. The same result holds for the edge-cost case, to the best of our knowledge this is the first polynomial time approximation algorithm for this case. We further show that the unrooted version of DRST can be approximated to a factor of O(√B) without budget violation, which is an improvement over the factor O(Δ √B) given in [Kuo et al. IEEE/ACM Trans. Netw. 2015] for the undirected and unrooted case, where Δ is the maximum degree of the graph. Finally, we provide some new/improved approximation bounds for several related problems, including the additive-prize version of DRST, the maximum budgeted connected set cover problem, and the budgeted sensor cover problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.9/LIPIcs.ISAAC.2022.9.pdf
Prize Collecting Steiner Tree
Directed graphs
Approximation Algorithms
Budgeted Problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
10:1
10:14
10.4230/LIPIcs.ISAAC.2022.10
article
Clustering with Faulty Centers
Fox, Kyle
1
Huang, Hongyao
1
Raichel, Benjamin
1
University of Texas at Dallas, TX, USA
In this paper we introduce and formally study the problem of k-clustering with faulty centers. Specifically, we study the faulty versions of k-center, k-median, and k-means clustering, where centers have some probability of not existing, as opposed to prior work where clients had some probability of not existing. For all three problems we provide fixed parameter tractable algorithms, in the parameters k, d, and ε, that (1+ε)-approximate the minimum expected cost solutions for points in d dimensional Euclidean space. For Faulty k-center we additionally provide a 5-approximation for general metrics. Significantly, all of our algorithms have a small dependence on n. Specifically, our Faulty k-center algorithms have only linear dependence on n, while for our algorithms for Faulty k-median and Faulty k-means the dependence is still only n^(1 + o(1)).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.10/LIPIcs.ISAAC.2022.10.pdf
clustering
approximation
probabilistic input
uncertain input
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
11:1
11:17
10.4230/LIPIcs.ISAAC.2022.11
article
Combinatorial and Algorithmic Aspects of Monadic Stability
Dreier, Jan
1
https://orcid.org/0000-0002-2662-5303
Mählmann, Nikolas
2
https://orcid.org/0000-0003-3657-7736
Mouawad, Amer E.
3
2
https://orcid.org/0000-0003-2481-4968
Siebertz, Sebastian
2
https://orcid.org/0000-0002-6347-1198
Vigny, Alexandre
2
https://orcid.org/0000-0002-4298-8876
TU Wien, Austria
Universität Bremen, Germany
American University of Beirut, Lebanon
Nowhere dense classes of graphs are classes of sparse graphs with rich structural and algorithmic properties, however, they fail to capture even simple classes of dense graphs. Monadically stable classes, originating from model theory, generalize nowhere dense classes and close them under transductions, i.e. transformations defined by colorings and simple first-order interpretations. In this work we aim to extend some combinatorial and algorithmic properties of nowhere dense classes to monadically stable classes of finite graphs. We prove the following results.
- For every monadically stable class C and fixed integer s ≥ 3, the Ramsey numbers R_C(s,t) are bounded from above by 𝒪(t^{s-1-δ}) for some δ > 0, improving the bound R(s,t) ∈ 𝒪(t^{s-1}/(log t)^{s-1}) known for the class of all graphs and the bounds known for k-stable graphs when s ≤ k.
- For every monadically stable class C and every integer r, there exists δ > 0 such that every graph G ∈ C that contains an r-subdivision of the biclique K_{t,t} as a subgraph also contains K_{t^δ,t^δ} as a subgraph. This generalizes earlier results for nowhere dense graph classes.
- We obtain a stronger regularity lemma for monadically stable classes of graphs.
- Finally, we show that we can compute polynomial kernels for the independent set and dominating set problems in powers of nowhere dense classes. Formerly, only fixed-parameter tractable algorithms were known for these problems on powers of nowhere dense classes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.11/LIPIcs.ISAAC.2022.11.pdf
Monadic Stability
Structural Graph Theory
Ramsey Numbers
Regularity
Kernels
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
12:1
12:17
10.4230/LIPIcs.ISAAC.2022.12
article
Complexity and Algorithms for ISOMETRIC PATH COVER on Chordal Graphs and Beyond
Chakraborty, Dibyayan
1
Dailly, Antoine
2
Das, Sandip
3
Foucaud, Florent
4
5
https://orcid.org/0000-0001-8198-693X
Gahlawat, Harmender
6
3
https://orcid.org/0000-0001-7663-6265
Ghosh, Subir Kumar
7
Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
G-SCOP, Univ. Grenoble-Alpes, Grenoble, France
Indian Statistical Institute, Kolkata, India
Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans Cedex 2, France
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Ramakrishna Mission Vivekananda Educational and Research Institute, Kolkata, India
A path is isometric if it is a shortest path between its endpoints. In this article, we consider the graph covering problem Isometric Path Cover, where we want to cover all the vertices of the graph using a minimum-size set of isometric paths. Although this problem has been considered from a structural point of view (in particular, regarding applications to pursuit-evasion games), it is little studied from the algorithmic perspective. We consider Isometric Path Cover on chordal graphs, and show that the problem is NP-hard for this class. On the positive side, for chordal graphs, we design a 4-approximation algorithm and an FPT algorithm for the parameter solution size. The approximation algorithm is based on a reduction to the classic path covering problem on a suitable directed acyclic graph obtained from a breadth first search traversal of the graph. The approximation ratio of our algorithm is 3 for interval graphs and 2 for proper interval graphs. Moreover, we extend the analysis of our approximation algorithm to k-chordal graphs (graphs whose induced cycles have length at most k) by showing that it has an approximation ratio of k+7 for such graphs, and to graphs of treelength at most 𝓁, where the approximation ratio is at most 6𝓁+2.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.12/LIPIcs.ISAAC.2022.12.pdf
Shortest paths
Isometric path cover
Chordal graph
Interval graph
AT-free graph
Approximation algorithm
FPT algorithm
Treewidth
Chordality
Treelength
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
13:1
13:13
10.4230/LIPIcs.ISAAC.2022.13
article
Computation of Cycle Bases in Surface Embedded Graphs
Fox, Kyle
1
Stanley, Thomas
2
University of Texas at Dallas, Richardson, TX, USA
Unaffiliated, Dallas, TX, USA
We present an O(n³ g²log g + m) + Õ(n^{ω + 1}) time deterministic algorithm to find the minimum cycle basis of a directed graph embedded on an orientable surface of genus g. This result improves upon the previous fastest known running time of O(m³n + m²n² log n) applicable to general directed graphs.
While an O(n^ω + 2^{2g}n² + m) time deterministic algorithm was known for undirected graphs, the use of the underlying field ℚ in the directed case (as opposed to ℤ₂ for the undirected case) presents extra challenges. It turns out that some of our new observations are useful for both variants of the problem, so we present an O(n^ω + n² g² log g + m) time deterministic algorithm for undirected graphs as well.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.13/LIPIcs.ISAAC.2022.13.pdf
cycle basis
surface embedded graphs
homology
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
14:1
14:16
10.4230/LIPIcs.ISAAC.2022.14
article
Computing Homomorphisms in Hereditary Graph Classes: The Peculiar Case of the 5-Wheel and Graphs with No Long Claws
Dębski, Michał
1
Lonc, Zbigniew
1
Okrasa, Karolina
1
2
https://orcid.org/0000-0003-1414-3507
Piecyk, Marta
1
https://orcid.org/0000-0001-9162-8300
Rzążewski, Paweł
1
2
https://orcid.org/0000-0001-7696-3848
Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
Institute of Informatics, University of Warsaw, Poland
For graphs G and H, an H-coloring of G is an edge-preserving mapping from V(G) to V(H). In the H-Coloring problem the graph H is fixed and we ask whether an instance graph G admits an H-coloring. A generalization of this problem is H-ColoringExt, where some vertices of G are already mapped to vertices of H and we ask if this partial mapping can be extended to an H-coloring.
We study the complexity of variants of H-Coloring in F-free graphs, i.e., graphs excluding a fixed graph F as an induced subgraph. For integers a,b,c ⩾ 1, by S_{a,b,c} we denote the graph obtained by identifying one endvertex of three paths on a+1, b+1, and c+1 vertices, respectively. For odd k ⩾ 5, by W_k we denote the graph obtained from the k-cycle by adding a universal vertex.
As our main algorithmic result we show that W_5-ColoringExt is polynomial-time solvable in S_{2,1,1}-free graphs. This result exhibits an interesting non-monotonicity of H-ColoringExt with respect to taking induced subgraphs of H. Indeed, W_5 contains a triangle, and K_3-Coloring, i.e., classical 3-coloring, is NP-hard already in claw-free (i.e., S_{1,1,1}-free) graphs. Our algorithm is based on two main observations:
1) W_5-ColoringExt in S_{2,1,1}-free graphs can be in polynomial time reduced to a variant of the problem of finding an independent set intersecting all triangles, and
2) the latter problem can be solved in polynomial time in S_{2,1,1}-free graphs.
We complement this algorithmic result with several negative ones. In particular, we show that W_5-Coloring is NP-hard in P_t-free graphs for some constant t and W_5-ColoringExt is NP-hard in S_{3,3,3}-free graphs of bounded degree. This is again uncommon, as usually problems that are NP-hard in S_{a,b,c}-free graphs for some constant a,b,c are already hard in claw-free graphs
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.14/LIPIcs.ISAAC.2022.14.pdf
graph homomorphism
forbidden induced subgraphs
precoloring extension
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
15:1
15:15
10.4230/LIPIcs.ISAAC.2022.15
article
Computing Palindromes on a Trie in Linear Time
Mieno, Takuya
1
https://orcid.org/0000-0003-2922-9434
Funakoshi, Mitsuru
2
3
https://orcid.org/0000-0002-2547-1509
Inenaga, Shunsuke
2
4
https://orcid.org/0000-0002-1833-010X
Department of Computer and Network Engineering, University of Electro-Communications, Tokyo, Japan
Department of Informatics, Kyushu University, Fukuoka, Japan
Japan Society for the Promotion of Science, Tokyo, Japan
PRESTO, Japan Science and Technology Agency, Tokyo, Japan
A trie 𝒯 is a rooted tree such that each edge is labeled by a single character from the alphabet, and the labels of out-going edges from the same node are mutually distinct. Given a trie 𝒯 with n edges, we show how to compute all distinct palindromes and all maximal palindromes on 𝒯 in O(n) time, in the case of integer alphabets of size polynomial in n. This improves the state-of-the-art O(n log h)-time algorithms by Funakoshi et al. [PSC 2019], where h is the height of 𝒯. Using our new algorithms, the eertree with suffix links for a given trie 𝒯 can readily be obtained in O(n) time. Further, our trie-based O(n)-space data structure allows us to report all distinct palindromes and maximal palindromes in a query string represented in the trie 𝒯, in output optimal time. This is an improvement over an existing (naïve) solution that precomputes and stores all distinct palindromes and maximal palindromes for each and every string in the trie 𝒯 separately, using a total O(n²) preprocessing time and space, and reports them in output optimal time upon query.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.15/LIPIcs.ISAAC.2022.15.pdf
palindromes
suffix trees
tries
labeled trees
eertrees
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
16:1
16:18
10.4230/LIPIcs.ISAAC.2022.16
article
Distortion-Oblivious Algorithms for Scheduling on Multiple Machines
Azar, Yossi
1
Peretz, Eldad
1
Touitou, Noam
1
2
Tel Aviv University, Israel
Amazon, Tel Aviv, Israel
We consider the classic online problem of scheduling on multiple machines to minimize total flow time and total stretch where the input consists of estimates on the processing time provided for each job once released. The performance of such algorithms should depend on μ, the error in the estimates of the processing time for that instance (such an algorithm is called a distortion oblivious algorithm). Previously, a distortion oblivious algorithm to minimize flow time was provided only for a single machine. In this paper we extend the work to multiple machines and also consider the total stretch objective. In particular, we design a non-migrative distortion oblivious algorithm to minimize total flow time with a competitive ratio of O(μ log P), where P is the ratio between the maximum to minimum processing time. We show that with immediate-dispatching one cannot achieve a competitive ratio which is a function of μ and P; moreover, a competitive ratio which is sub-polynomial in the number of jobs is also impossible. We also present the first distortion-oblivious algorithm for minimizing the stretch time, both on a single and on multiple machines. The competitive ratio of these algorithms are O(μ²) which is optimal as we also prove a matching Ω(μ²) lower bound.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.16/LIPIcs.ISAAC.2022.16.pdf
Online
Scheduling
Predictions
Stretch
Flow Time
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
17:1
17:15
10.4230/LIPIcs.ISAAC.2022.17
article
Efficiently Reconfiguring a Connected Swarm of Labeled Robots
Fekete, Sándor P.
1
https://orcid.org/0000-0002-9062-4241
Kramer, Peter
1
https://orcid.org/0000-0001-9635-5890
Rieck, Christian
1
https://orcid.org/0000-0003-0846-5163
Scheffer, Christian
2
https://orcid.org/0000-0002-3471-2706
Schmidt, Arne
1
https://orcid.org/0000-0001-8950-3963
Department of Computer Science, TU Braunschweig, Germany
Faculty of Electrical Engineering and Computer Science, Bochum University of Applied Sciences, Germany
When considering motion planning for a swarm of n labeled robots, we need to rearrange a given start configuration into a desired target configuration via a sequence of parallel, continuous, collision-free robot motions. The objective is to reach the new configuration in a minimum amount of time; an important constraint is to keep the swarm connected at all times. Problems of this type have been considered before, with recent notable results achieving constant stretch for not necessarily connected reconfiguration: If mapping the start configuration to the target configuration requires a maximum Manhattan distance of d, the total duration of an overall schedule can be bounded to 𝒪(d), which is optimal up to constant factors. However, constant stretch could only be achieved if disconnected reconfiguration is allowed, or for scaled configurations (which arise by increasing all dimensions of a given object by the same multiplicative factor) of unlabeled robots.
We resolve these major open problems by (1) establishing a lower bound of Ω(√n) for connected, labeled reconfiguration and, most importantly, by (2) proving that for scaled arrangements, constant stretch for connected reconfiguration can be achieved. In addition, we show that (3) it is NP-hard to decide whether a makespan of 2 can be achieved, while it is possible to check in polynomial time whether a makespan of 1 can be achieved.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.17/LIPIcs.ISAAC.2022.17.pdf
Motion planning
parallel motion
bounded stretch
makespan
connectivity
swarm robotics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
18:1
18:18
10.4230/LIPIcs.ISAAC.2022.18
article
Entropy Matters: Understanding Performance of Sparse Random Embeddings
Skorski, Maciej
1
University of Luxembourg, Luxembourg
This work shows how the performance of sparse random embeddings depends on the Renyi entropy-like property of data, improving upon recent works from NIPS'18 and NIPS'19.
While the prior works relied on involved combinatorics, the novel approach is simpler and modular. As the building blocks, it develops the following probabilistic facts of general interest:
b) a comparison inequality between the linear and quadratic chaos
c) a comparison inequality between heterogenic and homogenic linear chaos
d) a simpler proof of Latala’s strong result on estimating distributions of IID sums
e) sharp bounds for binomial moments in all parameter regimes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.18/LIPIcs.ISAAC.2022.18.pdf
Random Embeddings
Sparse Projections
Renyi Entropy
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
19:1
19:16
10.4230/LIPIcs.ISAAC.2022.19
article
Evacuation from a Disk for Robots with Asymmetric Communication
Georgiou, Konstantinos
1
Giachoudis, Nikos
1
https://orcid.org/0000-0002-5037-4575
Kranakis, Evangelos
2
https://orcid.org/0000-0002-8959-4428
Department of Mathematics, Toronto Metropolitan University, Canada
School of Computer Science, Carleton University, Ottawa, Canada
We consider evacuation of two robots from an Exit placed at an unknown location on the perimeter of a unit (radius) disk. The robots can move with max speed 1 and start at the center of the disk at the same time. We consider a new communication model, known as the SR model, in which the robots have communication faults as follows: one of the robots is a Sender and can only send wirelessly at any distance, while the other is a Receiver in that it can only receive wirelessly from any distance. The communication status of each robot is known to the other robot. In addition, both robots can exchange messages when they are co-located, which is known as Face-to-Face (F2F) model.
There have been several studies in the literature concerning the evacuation time when both robots may employ either F2F or Wireless (WiFi) communication. The SR communication model diverges from these two in that the two robots themselves have differing communication capabilities. We study the evacuation time, namely the time it takes until the last robot reaches the Exit, and show that the evacuation time in the SR model is strictly between the F2F and the WiFi models. The main part of our technical contribution is also an evacuation algorithm in which two cooperating robots accomplish the task in worst-case time at most π+2. Interesting features of the proposed algorithm are the asymmetry inherent in the resulting trajectories, as well as that the robots do not move at full speed for the entire duration of their trajectories.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.19/LIPIcs.ISAAC.2022.19.pdf
Communication
Cycle
Evacuation
Receiver
Sender
Mobile Agents
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
20:1
20:15
10.4230/LIPIcs.ISAAC.2022.20
article
Extended MSO Model Checking via Small Vertex Integrity
Gima, Tatsuya
1
https://orcid.org/0000-0003-2815-5699
Otachi, Yota
1
https://orcid.org/0000-0002-0087-853X
Nagoya University, Japan
We study the model checking problem of an extended MSO with local and global cardinality constraints, called MSO^GL_Lin, introduced recently by Knop, Koutecký, Masařík, and Toufar [Log. Methods Comput. Sci., 15(4), 2019]. We show that the problem is fixed-parameter tractable parameterized by vertex integrity, where vertex integrity is a graph parameter standing between vertex cover number and treedepth. Our result thus narrows the gap between the fixed-parameter tractability parameterized by vertex cover number and the W[1]-hardness parameterized by treedepth.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.20/LIPIcs.ISAAC.2022.20.pdf
vertex integrity
monadic second-order logic
cardinality constraint
fixed-parameter tractability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
21:1
21:13
10.4230/LIPIcs.ISAAC.2022.21
article
External-Memory Dictionaries with Worst-Case Update Cost
Das, Rathish
1
Iacono, John
2
3
Nekrich, Yakov
4
University of Liverpool, UK
Université libre de Bruxelles, Belgium
New York University, USA
Michigan Technological University, Houghton, MI, USA
The B^ε-tree [Brodal and Fagerberg 2003] is a simple I/O-efficient external-memory-model data structure that supports updates orders of magnitude faster than B-tree with a query performance comparable to the B-tree: for any positive constant ε < 1 insertions and deletions take O(1/B^(1-ε) log_B N) time (rather than O(log_BN) time for the classic B-tree), queries take O(log_B N) time and range queries returning k items take O(log_B N + k/B) time. Although the B^ε-tree has an optimal update/query tradeoff, the runtimes are amortized. Another structure, the write-optimized skip list, introduced by Bender et al. [PODS 2017], has the same performance as the B^ε-tree but with runtimes that are randomized rather than amortized. In this paper, we present a variant of the B^ε-tree with deterministic worst-case running times that are identical to the original’s amortized running times.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.21/LIPIcs.ISAAC.2022.21.pdf
Data Structures
External Memory
Buffer Tree
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
22:1
22:16
10.4230/LIPIcs.ISAAC.2022.22
article
Finding Matching Cuts in H-Free Graphs
Lucke, Felicia
1
https://orcid.org/0000-0002-9860-2928
Paulusma, Daniël
2
https://orcid.org/0000-0001-5945-9287
Ries, Bernard
1
https://orcid.org/0000-0003-4395-5547
Department of Informatics, University of Fribourg, Switzerland
Department of Computer Science, Durham University, UK
The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r > 0 such that the first variant is NP-complete for P_r-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.22/LIPIcs.ISAAC.2022.22.pdf
matching cut
perfect matching
H-free graph
computational complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
23:1
23:15
10.4230/LIPIcs.ISAAC.2022.23
article
Graph Product Structure for h-Framed Graphs
Bekos, Michael A.
1
https://orcid.org/0000-0002-3414-7444
Da Lozzo, Giordano
2
https://orcid.org/0000-0003-2396-5174
Hliněný, Petr
3
https://orcid.org/0000-0003-2125-1514
Kaufmann, Michael
4
https://orcid.org/0000-0001-9186-3538
University of Ioannina, Greece
Roma Tre University, Rome, Italy
Masaryk University, Brno, Czech Republic
Universität Tübingen, Germany
Graph product structure theory expresses certain graphs as subgraphs of the strong product of much simpler graphs. In particular, an elegant formulation for the corresponding structural theorems involves the strong product of a path and of a bounded treewidth graph, and allows to lift combinatorial results for bounded treewidth graphs to graph classes for which the product structure holds, such as to planar graphs [Dujmović et al., J. ACM, 67(4), 22:1-38, 2020].
In this paper, we join the search for extensions of this powerful tool beyond planarity by considering the h-framed graphs, a graph class that includes 1-planar, optimal 2-planar, and k-map graphs (for appropriate values of h). We establish a graph product structure theorem for h-framed graphs stating that the graphs in this class are subgraphs of the strong product of a path, of a planar graph of treewidth at most 3, and of a clique of size 3⌊ h/2 ⌋+⌊ h/3 ⌋-1. This allows us to improve over the previous structural theorems for 1-planar and k-map graphs. Our results constitute significant progress over the previous bounds on the queue number, non-repetitive chromatic number, and p-centered chromatic number of these graph classes, e.g., we lower the currently best upper bound on the queue number of 1-planar graphs and k-map graphs from 115 to 82 and from ⌊ 33/2(k+3 ⌊ k/2⌋ -3)⌋ to ⌊ 33/2 (3⌊ k/2 ⌋+⌊ k/3 ⌋-1) ⌋, respectively. We also employ the product structure machinery to improve the current upper bounds on the twin-width of 1-planar graphs from O(1) to 80. All our structural results are constructive and yield efficient algorithms to obtain the corresponding decompositions.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.23/LIPIcs.ISAAC.2022.23.pdf
Graph product structure theory
h-framed graphs
k-map graphs
queue number
twin-width
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
24:1
24:15
10.4230/LIPIcs.ISAAC.2022.24
article
Hardness of Approximation for H-Free Edge Modification Problems: Towards a Dichotomy
Belova, Tatiana
1
Bliznets, Ivan
1
St. Petersburg Department of Steklov Mathematical Institute of the RAS, Russia
For a fixed graph H, the H-free Edge Deletion/Completion/Editing problem asks to delete/add/edit the minimum possible number of edges in G to get a graph that does not contain an induced subgraph isomorphic to the graph H. In this work, we investigate H-free Edge Deletion/Completion/Editing problems in terms of the hardness of their approximation. We formulate a conjecture according to which all the graphs with at least five vertices can be classified into several groups of graphs with specific structural properties depending on the hardness of approximation for the corresponding H-free Edge Deletion/Completion/Editing problems. Also, we make significant progress in proving that conjecture by showing that it is sufficient to resolve it only for a finite set of graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.24/LIPIcs.ISAAC.2022.24.pdf
Parameterized complexity
Hardness of approximation
Edge modification problems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
25:1
25:15
10.4230/LIPIcs.ISAAC.2022.25
article
Hierarchical Categories in Colored Searching
Afshani, Peyman
1
Killmann, Rasmus
1
Larsen, Kasper Green
1
Aarhus University, Denmark
In colored range counting (CRC), the input is a set of points where each point is assigned a "color" (or a "category") and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data.
However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exists or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through reductions from the orthogonal vectors problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.25/LIPIcs.ISAAC.2022.25.pdf
Categorical Data
Computational Geometry
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
26:1
26:12
10.4230/LIPIcs.ISAAC.2022.26
article
How to Base Security on the Perfect/Statistical Binding Property of Quantum Bit Commitment?
Fang, Junbin
1
Unruh, Dominique
2
Yan, Jun
1
Zhou, Dehua
1
Jinan University, Guangzhou, China
University of Tartu, Estonia
The concept of quantum bit commitment was introduced in the early 1980s for the purpose of basing bit commitments solely on principles of quantum theory. Unfortunately, such unconditional quantum bit commitments still turn out to be impossible. As a compromise like in classical cryptography, Dumais et al. [Paul Dumais et al., 2000] introduce the conditional quantum bit commitments that additionally rely on complexity assumptions. However, in contrast to classical bit commitments which are widely used in classical cryptography, up until now there is relatively little work towards studying the application of quantum bit commitments in quantum cryptography. This may be partly due to the well-known weakness of the general quantum binding that comes from the possible superposition attack of the sender of quantum commitments, making it unclear whether quantum commitments could be useful in quantum cryptography.
In this work, following Yan et al. [Jun Yan et al., 2015] we continue studying using (canonical non-interactive) perfectly/statistically-binding quantum bit commitments as the drop-in replacement of classical bit commitments in some well-known constructions. Specifically, we show that the (quantum) security can still be established for zero-knowledge proof, oblivious transfer, and proof-of-knowledge. In spite of this, we stress that the corresponding security analyses are by no means trivial extensions of their classical analyses; new techniques are needed to handle possible superposition attacks by the cheating sender of quantum bit commitments.
Since (canonical non-interactive) statistically-binding quantum bit commitments can be constructed from quantum-secure one-way functions, we hope using them (as opposed to classical commitments) in cryptographic constructions can reduce the round complexity and weaken the complexity assumption simultaneously.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.26/LIPIcs.ISAAC.2022.26.pdf
Quantum bit commitment
quantum zero-knowledge
quantum proof-of-knowledge
quantum oblivious transfer
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
27:1
27:19
10.4230/LIPIcs.ISAAC.2022.27
article
Improved Compression of the Okamura-Seymour Metric
Mozes, Shay
1
https://orcid.org/0000-0001-9262-1821
Wallheimer, Nathan
2
https://orcid.org/0000-0001-7147-2855
Weimann, Oren
2
https://orcid.org/0000-0002-4510-7552
Reichman University, Herzliya, Israel
University of Haifa, Israel
Let G = (V,E) be an undirected unweighted planar graph. Let S = {s_1,…,s_k} be the vertices of some face in G and let T ⊆ V be an arbitrary set of vertices. The Okamura-Seymour metric compression problem asks to compactly encode the S-to-T distances.
Consider a vector storing the distances from an arbitrary vertex v to all vertices S = {s_1,…,s_k} in their cyclic order. The pattern of v is obtained by taking the difference between every pair of consecutive values of this vector. In STOC'19, Li and Parter used a VC-dimension argument to show that in planar graphs, the number of distinct patterns, denoted p_#, is only O(k³). This resulted in a simple Õ(min{k⁴+|T|, k⋅|T|}) space compression of the Okamura-Seymour metric.
We give an alternative proof of the p_# = O(k³) bound that exploits planarity beyond the VC-dimension argument. Namely, our proof relies on cut-cycle duality, as well as on the fact that distances among vertices of S are bounded by k. Our method implies the following:
(1) An Õ(p_#+k+|T|) space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to Õ(min{k³+|T|, k⋅|T|}).
(2) An optimal Õ(k+|T|) space compression of the Okamura-Seymour metric, in the case where the vertices of T induce a connected component in G.
(3) A tight bound of p_# = Θ(k²) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing p_# = O(k³).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.27/LIPIcs.ISAAC.2022.27.pdf
Shortest paths
planar graphs
metric compression
distance oracles
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
28:1
28:16
10.4230/LIPIcs.ISAAC.2022.28
article
Improving the Bounds of the Online Dynamic Power Management Problem
Liang, Ya-Chun
1
Iwama, Kazuo
1
Liao, Chung-Shou
1
Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan
We investigate the power-down mechanism which decides when a machine transitions between states such that the total energy consumption, characterized by execution cost, idle cost and switching cost, is minimized. In contrast to most of the previous studies on the offline model, we focus on the online model in which a sequence of jobs with their release time, execution time and deadline, arrive in an online fashion. More precisely, we exploit a different switching on and off strategy and present an upper bound of 3, and further show a lower bound of 2.1, in a dual-machine model, introduced by Chen et al. in 2014 [STACS 2014: 226-238], both of which beat the currently best result.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.28/LIPIcs.ISAAC.2022.28.pdf
Online algorithm
Energy scheduling
Dynamic power management
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
29:1
29:16
10.4230/LIPIcs.ISAAC.2022.29
article
Integer Complexity and Mixed Binary-Ternary Representation
Amano, Kazuyuki
1
https://orcid.org/0000-0003-2322-6072
Gunma University, Kiryu, Japan
The integer complexity of a natural number n, denoted by ‖n‖, is the smallest number of 1’s needed to express n using an arbitrary combination of addition and multiplication (and parentheses). For example, ‖6‖ = 5 since the expression 6 = (1+1)⋅ (1+1+1) contains five 1’s and there are no such expressions containing at most four 1’s. The investigation of this cute complexity measure was originated by Mahler and Popken in the 1950s. It is easy to see that ‖n‖/(log₃ n) ∈ [3, 3 log₂ 3] (∼ [3,4.755]) for every n, but the distribution of ‖n‖ is largely unknown.
In this work, we focus on the restricted expressions obtained by applying Horner’s schema to a mixed binary-ternary representation of a given number in which we can arrange base-two and base-three digits in an arbitrary order. Let f(n) denote the minimum number of 1’s needed to express n in this way. Apparently, f(n) ≥ ‖n‖ for every n. We extensively investigate on f(n) via the combination of computer experiments and theoretical analysis and obtain the following set of results: (i) Computer experiments supporting the hypothesis that f(n)/log₃ n < 3.483 on average and f(n)/log₃ n < 4.212 for all n, (ii) For almost all natural numbers n, 3.120 < f(n)/log₃ n < 3.587, and (iii) There are infinitely many n’s such that f(n)/log₃ n > 3.934. Several new bounds on the original integer complexity are also presented in the paper.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.29/LIPIcs.ISAAC.2022.29.pdf
Integer complexity
Lower bounds
Upper bounds
Horner’s schema
Computer assisted proof
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
30:1
30:15
10.4230/LIPIcs.ISAAC.2022.30
article
List Locally Surjective Homomorphisms in Hereditary Graph Classes
Dvořák, Pavel
1
2
https://orcid.org/0000-0002-6838-1538
Masařík, Tomáš
3
https://orcid.org/0000-0001-8524-4036
Novotná, Jana
3
4
https://orcid.org/0000-0002-7955-4692
Krawczyk, Monika
5
Rzążewski, Paweł
5
3
https://orcid.org/0000-0001-7696-3848
Żuk, Aneta
5
Tata Institute of Fundamental Research, Mumbai, India
Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland
Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time.
We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family 𝒮 consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ∈ 𝒮 (unless the ETH fails).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.30/LIPIcs.ISAAC.2022.30.pdf
Homomorphism
Hereditary graphs
Subexponential-time algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
31:1
31:20
10.4230/LIPIcs.ISAAC.2022.31
article
Locally Checkable Problems Parameterized by Clique-Width
Baghirova, Narmina
1
Gonzalez, Carolina Lucía
2
https://orcid.org/0000-0002-2398-2145
Ries, Bernard
1
https://orcid.org/0000-0003-4395-5547
Schindl, David
1
https://orcid.org/0000-0002-7009-5530
Department of Informatics, University of Fribourg, Switzerland
Instituto de Investigación en Ciencias de la Computación (ICC), CONICET-University of Buenos Aires, Argentina
We continue the study initiated by Bonomo-Braberman and Gonzalez in 2020 on r-locally checkable problems. We propose a dynamic programming algorithm that takes as input a graph with an associated clique-width expression and solves a 1-locally checkable problem under certain restrictions. We show that it runs in polynomial time in graphs of bounded clique-width, when the number of colors of the locally checkable problem is fixed. Furthermore, we present a first extension of our framework to global properties by taking into account the sizes of the color classes, and consequently enlarge the set of problems solvable in polynomial time with our approach in graphs of bounded clique-width. As examples, we apply this setting to show that, when parameterized by clique-width, the [k]-Roman domination problem is FPT, and the k-community problem, Max PDS and other variants are XP.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.31/LIPIcs.ISAAC.2022.31.pdf
locally checkable problem
clique-width
dynamic programming
coloring
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
32:1
32:12
10.4230/LIPIcs.ISAAC.2022.32
article
Lower Bounds on Retroactive Data Structures
Chung, Lily
1
https://orcid.org/0000-0001-7056-6155
Demaine, Erik D.
1
https://orcid.org/0000-0003-3803-5703
Hendrickson, Dylan
1
https://orcid.org/0000-0002-9967-8799
Lynch, Jayson
2
Massachusetts Institute of Technology, Cambridge, MA, USA
Cheriton School of Computer Science, University of Waterloo, Canada
We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that has a data structure where operations run in O(T(n,m)) time per operation, but any partially retroactive version of that data structure requires T(n,m)⋅m^{1-o(1)} worst-case time per operation, where n is the size of the data structure at any time and m is the number of operations. Any data structure with operations running in O(T(n,m)) time per operation can be converted (via the "rollback method") into a partially retroactive data structure running in O(T(n,m)⋅m) time per operation, so our lower bound is tight up to an m^o(1) factor common in fine-grained complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.32/LIPIcs.ISAAC.2022.32.pdf
Retroactivity
time travel
rollback
fine-grained complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
33:1
33:18
10.4230/LIPIcs.ISAAC.2022.33
article
Minimizing the Maximum Flow Time in the Online Food Delivery Problem
Guo, Xiangyu
1
Luo, Kelin
2
https://orcid.org/0000-0003-2006-0601
Li, Shi
1
Zhang, Yuhao
3
University at Buffalo, NY, USA
Institute of Computer Science, Universität Bonn, Germany
Shanghai Jiao Tong University, China
We study a common delivery problem encountered in nowadays online food-ordering platforms: Customers order dishes online, and the restaurant delivers the food after receiving the order. Specifically, we study a problem where k vehicles of capacity c are serving a set of requests ordering food from one restaurant. After a request arrives, it can be served by a vehicle moving from the restaurant to its delivery location. We are interested in serving all requests while minimizing the maximum flow-time, i.e., the maximum time length a customer waits to receive his/her food after submitting the order.
We show that the problem is hard in both offline and online settings even when k = 1 and c = ∞: There is a hardness of approximation of Ω(n) for the offline problem, and a lower bound of Ω(n) on the competitive ratio of any online algorithm, where n is number of points in the metric.
We circumvent the strong negative results in two directions. Our main result is an O(1)-competitive online algorithm for the uncapacitated (i.e, c = ∞) food delivery problem on tree metrics; we also have negative result showing that the condition c = ∞ is needed. Then we explore the speed-augmentation model where our online algorithm is allowed to use vehicles with faster speed. We show that a moderate speeding factor leads to a constant competitive ratio, and we prove a tight trade-off between the speeding factor and the competitive ratio.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.33/LIPIcs.ISAAC.2022.33.pdf
Online algorithm
Capacitated Vehicle Routing
Flow Time Optimization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
34:1
34:14
10.4230/LIPIcs.ISAAC.2022.34
article
Minimum Link Fencing
Bhore, Sujoy
1
https://orcid.org/0000-0003-0104-1659
Klute, Fabian
2
https://orcid.org/0000-0002-7791-3604
Löffler, Maarten
2
Nöllenburg, Martin
3
https://orcid.org/0000-0003-0454-3937
Terziadis, Soeren
3
https://orcid.org/0000-0001-5161-3841
Villedieu, Anaïs
3
https://orcid.org/0000-0001-6196-8347
Department of Computer Science & Engineering, Indian Institute of Technology Bombay, India
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Algorithms and Complexity Group, TU Wien, Austria
We study a variant of the geometric multicut problem, where we are given a set 𝒫 of colored and pairwise interior-disjoint polygons in the plane. The objective is to compute a set of simple closed polygon boundaries (fences) that separate the polygons in such a way that any two polygons that are enclosed by the same fence have the same color, and the total number of links of all fences is minimized. We call this the minimum link fencing (MLF) problem and consider the natural case of bounded minimum link fencing (BMLF), where 𝒫 contains a polygon Q that is unbounded in all directions and can be seen as an outer polygon. We show that BMLF is NP-hard in general and that it is XP-time solvable when each fence contains at most two polygons and the number of segments per fence is the parameter. Finally, we present an O(n log n)-time algorithm for the case that the convex hull of 𝒫⧵{Q} does not intersect Q.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.34/LIPIcs.ISAAC.2022.34.pdf
computational geometry
polygon nesting
polygon separation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
35:1
35:20
10.4230/LIPIcs.ISAAC.2022.35
article
Multi-Robot Motion Planning for Unit Discs with Revolving Areas
Agarwal, Pankaj K.
1
Geft, Tzvika
2
Halperin, Dan
2
Taylor, Erin
3
Duke University, Durham NC, USA
Tel Aviv University, Israel
Duke University, Durham, NC, USA
We study the problem of motion planning for a collection of n labeled unit disc robots in a polygonal environment. We assume that the robots have revolving areas around their start and final positions: that each start and each final is contained in a radius 2 disc lying in the free space, not necessarily concentric with the start or final position, which is free from other start or final positions. This assumption allows a weakly-monotone motion plan, in which robots move according to an ordering as follows: during the turn of a robot R in the ordering, it moves fully from its start to final position, while other robots do not leave their revolving areas. As R passes through a revolving area, a robot R' that is inside this area may move within the revolving area to avoid a collision. Notwithstanding the existence of a motion plan, we show that minimizing the total traveled distance in this setting, specifically even when the motion plan is restricted to be weakly-monotone, is APX-hard, ruling out any polynomial-time (1+ε)-approximation algorithm.
On the positive side, we present the first constant-factor approximation algorithm for computing a feasible weakly-monotone motion plan. The total distance traveled by the robots is within an O(1) factor of that of the optimal motion plan, which need not be weakly monotone. Our algorithm extends to an online setting in which the polygonal environment is fixed but the initial and final positions of robots are specified in an online manner. Finally, we observe that the overhead in the overall cost that we add while editing the paths to avoid robot-robot collision can vary significantly depending on the ordering we chose. Finding the best ordering in this respect is known to be NP-hard, and we provide a polynomial time O(log n log log n)-approximation algorithm for this problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.35/LIPIcs.ISAAC.2022.35.pdf
motion planning
optimal motion planning
approximation
complexity
NP-hardness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
36:1
36:16
10.4230/LIPIcs.ISAAC.2022.36
article
Nested Active-Time Scheduling
Cao, Nairen
1
Fineman, Jeremy T.
1
Li, Shi
2
Mestre, Julián
3
Russell, Katina
1
Umboh, Seeun William
3
Georgetown University, Washington D.C., USA
University at Buffalo, NY, USA
The University of Sydney, Australia
The active-time scheduling problem considers the problem of scheduling preemptible jobs with windows (release times and deadlines) on a parallel machine that can schedule up to g jobs during each timestep. The goal in the active-time problem is to minimize the number of active steps, i.e., timesteps in which at least one job is scheduled. In this way, the active time models parallel scheduling when there is a fixed cost for turning the machine on at each discrete step.
This paper presents a 9/5-approximation algorithm for a special case of the active-time scheduling problem in which job windows are laminar (nested). This result improves on the previous best 2-approximation for the general case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.36/LIPIcs.ISAAC.2022.36.pdf
Scheduling algorithms
Active time
Approximation algorithm
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
37:1
37:15
10.4230/LIPIcs.ISAAC.2022.37
article
On Algorithmic Self-Assembly of Squares by Co-Transcriptional Folding
Fazekas, Szilárd Zsolt
1
https://orcid.org/0000-0001-5319-0395
Kim, Hwee
2
Matsuoka, Ryuichi
3
Seki, Shinnosuke
3
https://orcid.org/0000-0002-0276-3322
Takeuchi, Hinano
3
Akita University, Japan
Incheon National University, Republic of Korea
The University of Electro-Communications, Tokyo, Japan
Algorithms play a primary role in programming an orchestrated self-assembly of shapes into molecules. In this paper, we study the algorithmic self-assembly of squares by RNA co-transcriptional folding in its oritatami model. We formalize the square self-assembly problem in oritatami and propose a universal oritatami transcript made of 939 types of abstract molecules (beads) and of period 1294 that folds deterministically and co-transcriptionally at delay 3 and maximum arity into the n × n square modulo horizontal and vertical scaling factors for all sufficiently large n’s after building a Θ(log n) width "ruler" that measures n upon the seed of size Θ(log n) on which n is encoded in binary.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.37/LIPIcs.ISAAC.2022.37.pdf
Algorithmic molecular self-assembly
Co-transcriptional folding
Oritatami system
Self-assembly of squares
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
38:1
38:15
10.4230/LIPIcs.ISAAC.2022.38
article
On Constrained Intersection Representations of Graphs and Digraphs
Cicalese, Ferdinando
1
https://orcid.org/0000-0003-1652-0599
Dallard, Clément
2
https://orcid.org/0000-0002-9522-3770
Milanič, Martin
3
https://orcid.org/0000-0002-8222-8097
Department of Computer Science, University of Verona, Italy
Université d'Orléans, INSA Centre Val de Loire, LIFO EA 4022, Orléans, France
FAMNIT and IAM, University of Primorska, Koper, Slovenia
We study the problem of determining minimal directed intersection representations of DAGs in a model introduced by [Kostochka, Liu, Machado, and Milenkovic, ISIT2019]: vertices are assigned color sets, two vertices are connected by an arc if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head, and the goal is to minimize the total number of colors. We show that the problem is polynomially solvable in the class of triangle-free and Hamiltonian DAGs and also disclose the relationship of this problem with several other models of intersection representations of graphs and digraphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.38/LIPIcs.ISAAC.2022.38.pdf
Directed intersection representation
intersection number
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
39:1
39:14
10.4230/LIPIcs.ISAAC.2022.39
article
On Finding Short Reconfiguration Sequences Between Independent Sets
Agrawal, Akanksha
1
Hait, Soumita
2
Mouawad, Amer E.
3
4
https://orcid.org/0000-0003-2481-4968
Indian Institute of Technology Madras, Chennai, India
Indian Institute of Technology, Kharagpur, India
American University of Beirut, Lebanon
Universität Bremen, Germany
Assume we are given a graph G, two independent sets S and T in G of size k ≥ 1, and a positive integer 𝓁 ≥ 1. The goal is to decide whether there exists a sequence ⟨ I₀, I₁, ..., I_𝓁 ⟩ of independent sets such that for all j ∈ {0,…,𝓁-1} the set I_j is an independent set of size k, I₀ = S, I_𝓁 = T, and I_{j+1} is obtained from I_j by a predetermined reconfiguration rule. We consider two reconfiguration rules, namely token sliding and token jumping. Intuitively, we view each independent set as a collection of tokens placed on the vertices of the graph. Then, the Token Sliding Optimization (TSO) problem asks whether there exists a sequence of at most 𝓁 steps that transforms S into T, where at each step we are allowed to slide one token from a vertex to an unoccupied neighboring vertex (while maintaining independence). In the Token Jumping Optimization (TJO) problem, at each step, we are allowed to jump one token from a vertex to any other unoccupied vertex of the graph (as long as we maintain independence). Both TSO and TJO are known to be fixed-parameter tractable when parameterized by 𝓁 on nowhere dense classes of graphs. In this work, we investigate the boundary of tractability for sparse classes of graphs. We show that both problems are fixed-parameter tractable for parameter k + 𝓁 + d on d-degenerate graphs as well as for parameter |M| + 𝓁 + Δ on graphs having a modulator M whose deletion leaves a graph of maximum degree Δ. We complement these result by showing that for parameter 𝓁 alone both problems become W[1]-hard already on 2-degenerate graphs. Our positive result makes use of the notion of independence covering families introduced by Lokshtanov et al. [Daniel Lokshtanov et al., 2020]. Finally, we show as a side result that using such families we can obtain a simpler and unified algorithm for the standard Token Jumping Reachability problem (a.k.a. Token Jumping) parameterized by k on both degenerate and nowhere dense classes of graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.39/LIPIcs.ISAAC.2022.39.pdf
Token sliding
token jumping
fixed-parameter tractability
combinatorial reconfiguration
shortest reconfiguration sequence
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
40:1
40:15
10.4230/LIPIcs.ISAAC.2022.40
article
On Graphs Coverable by k Shortest Paths
Dumas, Maël
1
Foucaud, Florent
2
https://orcid.org/0000-0001-8198-693X
Perez, Anthony
1
Todinca, Ioan
1
Univ. Orléans, INSA Centre Val de Loire, LIFO EA 4022, F-45067 Orléans, France
Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000 Clermont-Ferrand, France
We show that if the edges or vertices of an undirected graph G can be covered by k shortest paths, then the pathwidth of G is upper-bounded by a function of k. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph G and a set of k pairs of vertices called terminals, asks whether G can be covered by k shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph G and a set of k terminals, asks whether there exist binom(k,2) shortest paths, each joining a distinct pair of terminals such that these paths cover G). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter k.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.40/LIPIcs.ISAAC.2022.40.pdf
Shortest paths
covering problems
parameterized complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
41:1
41:16
10.4230/LIPIcs.ISAAC.2022.41
article
On Maximizing Sums of Non-Monotone Submodular and Linear Functions
Qi, Benjamin
1
Massachusetts Institute of Technology, Cambridge, MA, USA
We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by [Bodek and Feldman '22]. In this problem, we are given query access to a non-negative submodular function f: 2^N → ℝ_{≥ 0} and a linear function 𝓁: 2^N → ℝ over the same ground set N, and the objective is to output a set T ⊆ N approximately maximizing the sum f(T)+𝓁(T). Specifically, an algorithm is said to provide an (α,β)-approximation for RegularizedUSM if it outputs a set T such that E[f(T)+𝓁(T)] ≥ max_{S ⊆ N}[α ⋅ f(S)+β⋅ 𝓁(S)]. We also study the setting where S and T are constrained to be independent in a given matroid, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM).
The special case of RegularizedCSM with monotone f has been extensively studied [Sviridenko et al. '17, Feldman '18, Harshaw et al. '19]. On the other hand, we are aware of only one prior work that studies RegularizedCSM with non-monotone f [Lu et al. '21], and that work constrains 𝓁 to be non-positive. In this work, we provide improved (α,β)-approximation algorithms for both {RegularizedUSM} and {RegularizedCSM} with non-monotone f. In particular, we are the first to provide nontrivial (α,β)-approximations for RegularizedCSM where the sign of 𝓁 is unconstrained, and the α we obtain for RegularizedUSM improves over [Bodek and Feldman '22] for all β ∈ (0,1).
In addition to approximation algorithms, we provide improved inapproximability results for all of the aforementioned cases. In particular, we show that the α our algorithm obtains for {RegularizedCSM} with unconstrained 𝓁 is essentially tight for β ≥ e/(e+1). Using similar ideas, we are also able to show 0.478-inapproximability for maximizing a submodular function where S and T are subject to a cardinality constraint, improving a 0.491-inapproximability result due to [Oveis Gharan and Vondrak '10].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.41/LIPIcs.ISAAC.2022.41.pdf
submodular maximization
regularization
continuous greedy
inapproximability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
42:1
42:19
10.4230/LIPIcs.ISAAC.2022.42
article
On Reverse Shortest Paths in Geometric Proximity Graphs
Agarwal, Pankaj K.
1
https://orcid.org/0000-0002-9439-181X
Katz, Matthew J.
2
https://orcid.org/0000-0002-0672-729X
Sharir, Micha
3
https://orcid.org/0000-0002-2541-3763
Department of Computer Science, Duke University, Durham NC, USA
Department of Computer Science, Ben-Gurion University of the Negev, Beer Sheva, Israel
School of Computer Science, Tel Aviv University, Tel Aviv, Israel
Let S be a set of n geometric objects of constant complexity (e.g., points, line segments, disks, ellipses) in ℝ², and let ϱ: S× S → ℝ_{≥ 0} be a distance function on S. For a parameter r ≥ 0, we define the proximity graph G(r) = (S,E) where E = {(e₁,e₂) ∈ S×S ∣ e₁≠e₂, ϱ(e₁,e₂) ≤ r}. Given S, s,t ∈ S, and an integer k ≥ 1, the reverse-shortest-path (RSP) problem asks for computing the smallest value r^* ≥ 0 such that G(r^*) contains a path from s to t of length at most k.
In this paper we present a general randomized technique that solves the RSP problem efficiently for a large family of geometric objects and distance functions. Using standard, and sometimes more involved, semi-algebraic range-searching techniques, we first give an efficient algorithm for the decision problem, namely, given a value r ≥ 0, determine whether G(r) contains a path from s to t of length at most k. Next, we adapt our decision algorithm and combine it with a random-sampling method to compute r^*, by efficiently performing a binary search over an implicit set of O(n²) candidate values that contains r^*.
We illustrate the versatility of our general technique by applying it to a variety of geometric proximity graphs. For example, we obtain (i) an O^*(n^{4/3}) expected-time randomized algorithm (where O^*(⋅) hides polylog(n) factors) for the case where S is a set of pairwise-disjoint line segments in ℝ² and ϱ(e₁,e₂) = min_{x ∈ e₁, y ∈ e₂} ‖x-y‖ (where ‖⋅‖ is the Euclidean distance), and (ii) an O^*(n+m^{4/3}) expected-time randomized algorithm for the case where S is a set of m points lying on an x-monotone polygonal chain T with n vertices, and ϱ(p,q), for p,q ∈ S, is the smallest value h such that the points p' := p+(0,h) and q' := q+(0,h) are visible to each other, i.e., all points on the segment p'q' lie above or on the polygonal chain T.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.42/LIPIcs.ISAAC.2022.42.pdf
Geometric optimization
proximity graphs
semi-algebraic range searching
reverse shortest path
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
43:1
43:13
10.4230/LIPIcs.ISAAC.2022.43
article
On the Complexity of Rainbow Vertex Colouring Diametral Path Graphs
Dyrseth, Jakob
1
Lima, Paloma T.
2
https://orcid.org/0000-0001-9304-4536
University of Bergen, Norway
IT University of Copenhagen, Denmark
Given a graph and a colouring of its vertices, a rainbow vertex path is a path between two vertices such that all the internal nodes of the path are coloured distinctly. A graph is rainbow vertex-connected if between every pair of vertices in the graph there exists a rainbow vertex path. We study the problem of deciding whether a given graph can be coloured using k or less colours such that it is rainbow vertex-connected. Note that every graph G needs at least diam(G)-1 colours to be rainbow vertex connected.
Heggernes et al. [MFCS, 2018] conjectured that if G is a graph in which every induced subgraph has a dominating diametral path, then G can always be rainbow vertex coloured with diam(G)-1 many colours. In this work, we confirm their conjecture for chordal, bipartite and claw-free diametral path graphs. We complement these results by showing the conjecture does not hold if the condition on every induced subgraph is dropped. In fact we show that, in this case, even though diam(G) many colours are always enough, it is NP-complete to determine whether a graph with a dominating diametral path of length three can be rainbow vertex coloured with two colours.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.43/LIPIcs.ISAAC.2022.43.pdf
rainbow vertex colouring
diametral path graphs
interval graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
44:1
44:14
10.4230/LIPIcs.ISAAC.2022.44
article
On the Complexity of Tree Edit Distance with Variables
Akutsu, Tatsuya
1
https://orcid.org/0000-0001-9763-797X
Mori, Tomoya
1
Nakamura, Naotoshi
2
3
4
Kozawa, Satoshi
2
3
Ueno, Yuhei
2
3
5
Sato, Thomas N.
2
3
5
Bioinformatics Center, Institute for Chemical Research, Kyoto University, Japan
The Thomas N. Sato BioMEC-X Laboratories, Advanced Telecommunications Research Institute International (ATR), Kyoto, Japan
Karydo TherapeutiX, Inc., Tokyo, Japan
Interdisciplinary Biology Laboratory (iBLab), Division of Natural Science, Graduate School of Science, Nagoya University, Japan
V-iCliniX Laboratory, Nara Medical University, Japan
In this paper, we propose tree edit distance with variables, which is an extension of the tree edit distance to handle trees with variables and has a potential application to measuring the similarity between mathematical formulas. We analyze the computational complexity of several variants of this model. In particular, we show that the problem is NP-complete for ordered trees. We also show for unordered trees that the problem of deciding whether or not the distance is 0 is graph isomorphism complete but can be solved in polynomial time if the maximum outdegree of input trees is bounded by a constant. We also present parameterized and exponential-time algorithms for ordered and unordered cases, respectively.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.44/LIPIcs.ISAAC.2022.44.pdf
Tree edit distance
unification
parameterized algorithms
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
45:1
45:18
10.4230/LIPIcs.ISAAC.2022.45
article
On the Cop Number of String Graphs
Das, Sandip
1
Gahlawat, Harmender
2
https://orcid.org/0000-0001-7663-6265
Indian Statistical Institute, Kolkata, India
Ben-Gurion University of the Negev, Beer-Sheva, Israel
Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The cop number of a graph is the minimum number of cops required to capture the robber. We show that the cop number of a string graph is at most 13, improving upon a result of Gavenčiak et al. [Eur. J. of Comb. 72, 45-69 (2018)]. Using similar techniques, we also show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al. [Austr. J. Comb. 76(2), 248-265 (2020)].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.45/LIPIcs.ISAAC.2022.45.pdf
Cop number
string graphs
intersection graphs
planar graphs
pursuit-evasion games
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
46:1
46:16
10.4230/LIPIcs.ISAAC.2022.46
article
On the Parameterized Intractability of Determinant Maximization
Ohsaka, Naoto
1
https://orcid.org/0000-0001-9584-4764
CyberAgent, Inc., Tokyo, Japan
In the Determinant Maximization problem, given an n × n positive semi-definite matrix A in ℚ^{n × n} and an integer k, we are required to find a k × k principal submatrix of A having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to k by Koutis (2006); i.e., a f(k)n^𝒪(1)-time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, we show that Determinant Maximization is solvable in polynomial time on tridiagonal matrices. Thereafter, we demonstrate the W[1]-hardness with respect to the rank r of an input matrix. Our result is stronger than Koutis' result in the sense that any k × k principal submatrix is singular whenever k > r. We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by k within a factor of 2^{-c√k} for some universal constant c > 0. Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov, Ramanujan, Saurab, and Zehavi (2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an ε-additive approximation algorithm that runs in ε^{-r²}⋅r^𝒪(r³)⋅n^𝒪(1) time for the rank r of an input matrix, provided that the diagonal entries are bounded.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.46/LIPIcs.ISAAC.2022.46.pdf
Determinant maximization
Parameterized complexity
Approximability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
47:1
47:15
10.4230/LIPIcs.ISAAC.2022.47
article
One-Face Shortest Disjoint Paths with a Deviation Terminal
Kobayashi, Yusuke
1
https://orcid.org/0000-0001-9478-7307
Terao, Tatsuya
1
https://orcid.org/0000-0002-3530-2194
Research Institute for Mathematical Sciences, Kyoto University, Japan
For an undirected graph G and distinct vertices s₁, t₁, … , s_k, t_k called terminals, the shortest k-disjoint paths problem asks for k pairwise vertex-disjoint paths P₁, … , P_k such that P_i connects s_i and t_i for i = 1, … , k and the sum of their lengths is minimized. This problem is a natural optimization version of the well-known k-disjoint paths problem, and its polynomial solvability is widely open. One of the best results on the shortest k-disjoint paths problem is due to Datta et al. [Datta et al., 2018], who present a polynomial-time algorithm for the case when G is planar and all the terminals are on one face. In this paper, we extend this result by giving a polynomial-time randomized algorithm for the case when all the terminals except one are on some face of G. In our algorithm, we combine the arguments of Datta et al. with some results on the shortest disjoint (A + B)-paths problem shown by Hirai and Namba [Hirai and Namba, 2018]. To this end, we present a non-trivial bijection between k disjoint paths and disjoint (A + B)-paths, which is a key technical contribution of this paper.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.47/LIPIcs.ISAAC.2022.47.pdf
shortest disjoint paths
polynomial time algorithm
planar graph
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
48:1
48:16
10.4230/LIPIcs.ISAAC.2022.48
article
Optimizing Quantum Circuit Parameters via SDP
Lee, Eunou
1
Sunkyunkwan University, Seoul, South Korea
In recent years, parameterized quantum circuits have become a major tool to design quantum algorithms for optimization problems.
The challenge in fully taking advantage of a given family of parameterized circuits lies in finding a good set of parameters in a non-convex landscape that can grow exponentially to the number of parameters.
We introduce a new framework for optimizing parameterized quantum circuits: round SDP solutions to circuit parameters.
Within this framework, we propose an algorithm that produces approximate solutions for a quantum optimization problem called Quantum Max Cut.
The rounding algorithm runs in polynomial time to the number of parameters regardless of the underlying interaction graph.
The resulting 0.562-approximation algorithm for generic instances of Quantum Max Cut improves on the previously known best algorithms by Anshu, Gosset, and Morenz with a ratio 0.531 and by Parekh and Thompson with a ratio 0.533.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.48/LIPIcs.ISAAC.2022.48.pdf
Quantum algorithm
Optimization
Rounding algorithm
Quantum Circuit
Approximation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
49:1
49:16
10.4230/LIPIcs.ISAAC.2022.49
article
Package Delivery Using Drones with Restricted Movement Areas
Erlebach, Thomas
1
https://orcid.org/0000-0002-4470-5868
Luo, Kelin
2
https://orcid.org/0000-0003-2006-0601
Spieksma, Frits C.R.
3
https://orcid.org/0000-0002-2547-3782
Department of Computer Science, Durham University, UK
Institute of Computer Science, Universität Bonn, Germany
Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands
For the problem of delivering a package from a source node to a destination node in a graph using a set of drones, we study the setting where the movements of each drone are restricted to a certain subgraph of the given graph. We consider the objectives of minimizing the delivery time (problem DDT) and of minimizing the total energy consumption (problem DDC). For general graphs, we show a strong inapproximability result and a matching approximation algorithm for DDT as well as NP-hardness and a 2-approximation algorithm for DDC. For the special case of a path, we show that DDT is NP-hard if the drones have different speeds. For trees, we give optimal algorithms under the assumption that all drones have the same speed or the same energy consumption rate. The results for trees extend to arbitrary graphs if the subgraph of each drone is isometric.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.49/LIPIcs.ISAAC.2022.49.pdf
Mobile agents
approximation algorithm
inapproximability
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
50:1
50:16
10.4230/LIPIcs.ISAAC.2022.50
article
Parameterized Approximation Algorithms for TSP
Zhou, Jianqi
1
Li, Peihua
1
Guo, Jiong
1
School of Computer Science and Technology, Shandong University, Qingdao, China
We study the Traveling Salesman problem (TSP), where given a complete undirected graph G = (V,E) with n vertices and an edge cost function c:E↦R_{⩾0}, the goal is to find a minimum-cost cycle visiting every vertex exactly once. It is well-known that unless P = NP, TSP cannot be approximated in polynomial time within a factor of ρ(n) for any computable function ρ, while the metric case of TSP, that the edge cost function satisfies the △-inequality, admits a polynomial-time 1.5-approximation. We investigate TSP on general graphs from the perspective of parameterized approximability. A parameterized ρ-approximation algorithm returns a ρ-approximation solution in f(k)⋅|I|^O(1) time, where f is a computable function and k is a parameter of the input I. We introduce two parameters, which measure the distance of a given TSP-instance from the metric case, and achieve the following two results:
- A 3-approximation algorithm for TSP in O((3k₁)! 8^k₁⋅ n²+n³) time, where k₁ is the number of triangles in which the edge costs violate the △-inequality.
- A 3-approximation algorithm for TSP in O(n^O(k₂)) time and a (6k₂+9)-approximation algorithm for TSP in O(k₂^O(k₂)⋅n³) time, where k₂ is the minimum number of vertices, whose removal results in a metric graph.
To our best knowledge, the above algorithms are the first non-trivial parameterized approximation algorithms for TSP on general graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.50/LIPIcs.ISAAC.2022.50.pdf
FPT-approximation algorithms
the Traveling Salesman problem
the triangle inequality
fixed-parameter tractability
metric graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
51:1
51:16
10.4230/LIPIcs.ISAAC.2022.51
article
Partial and Simultaneous Transitive Orientations via Modular Decompositions
Münch, Miriam
1
https://orcid.org/0000-0002-6997-8774
Rutter, Ignaz
1
https://orcid.org/0000-0002-3794-4406
Stumpf, Peter
1
https://orcid.org/0000-0003-0531-9769
Faculty of Computer Science and Mathematics, Universität Passau, Germany
A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.51/LIPIcs.ISAAC.2022.51.pdf
representation extension
simultaneous representation
comparability graph
permutation graph
circular permutation graph
modular decomposition
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
52:1
52:12
10.4230/LIPIcs.ISAAC.2022.52
article
Polynomial Threshold Functions for Decision Lists
Podolskii, Vladimir
1
2
https://orcid.org/0000-0001-7154-138X
Proskurin, Nikolay V.
3
Courant Institute of Mathematical Sciences, New York University, NY, USA
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russian Federation
HSE University, Moscow, Russian Federation
For S ⊆ {0,1}ⁿ a Boolean function f : S → {-1,1} is a polynomial threshold function (PTF) of degree d and weight W if there is a polynomial p with integer coefficients of degree d and with sum of absolute coefficients W such that f(x) = sign p(x) for all x ∈ S. We study a representation of decision lists as PTFs over Boolean cubes {0,1}ⁿ and over Hamming balls {0,1}ⁿ_{≤ k}.
As our first result, we show that for all d = O((n/(log n))^{1/3}) any decision list over {0,1}ⁿ can be represented by a PTF of degree d and weight 2^O(n/d²). This improves the result by Klivans and Servedio [Adam R. Klivans and Rocco A. Servedio, 2006] by a log² d factor in the exponent of the weight. Our bound is tight for all d = O((n/(log n))^{1/3}) due to the matching lower bound by Beigel [Richard Beigel, 1994].
For decision lists over a Hamming ball {0,1}ⁿ_{≤ k} we show that the upper bound on weight above can be drastically improved to n^O(√k) for d = Θ(√k). We also show that similar improvement is not possible for smaller degrees by proving the lower bound W = 2^Ω(n/d²) for all d = O(√k).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.52/LIPIcs.ISAAC.2022.52.pdf
Threshold function
decision list
Hamming ball
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
53:1
53:16
10.4230/LIPIcs.ISAAC.2022.53
article
Pop & Push: Ordered Tree Iteration in 𝒪(1)-Time
Lapey, Paul
1
Williams, Aaron
1
https://orcid.org/0000-0001-6816-4368
Department of Computer Science, Williams College, Williamstown, MA, USA
The number of ordered trees (also known as plane trees) with n nodes is the (n-1)st Catalan number C_{n-1}. An ordered tree can be stored directly using nodes and pointers, or represented indirectly by a Dyck word. This paper presents a loopless algorithm for generating ordered trees with n nodes using pointer-based representations. In other words, we spend 𝒪(C_{n-1})-time to generate all of the trees, and moreover, the delay between consecutive trees is worst-case 𝒪(1)-time.
To achieve this run-time, each tree must differ from the previous by a constant amount. In other words, the algorithm must create a type of Gray code order. Our algorithm operates on the children of a node like a stack, by popping the first child off of one node’s stack and pushing the result onto another node’s stack. We refer to this pop-push operation as a pull, and consecutive trees in our order differ by one or two pulls. There is a simple two-case successor rule that determines the pulls to apply directly from the current tree. When converted to Dyck words, our rule corresponds to a left-shift, and these shift generate a cool-lex variant of lexicographic order.
Our results represent the first pull Gray code for ordered trees, and the first fully published loopless algorithm for ordered trees using pointer representations. More importantly, our algorithm is incredibly simple: A full implementation in C, including initialization and output, uses only three loops and three if-else blocks. Our work also establishes a simultaneous Gray code for Dyck words, ordered trees, and also binary trees, using cool-lex order.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.53/LIPIcs.ISAAC.2022.53.pdf
combinatorial generation
Gray code
simultaneous Gray code
ordered trees
plane trees
Dyck words
binary trees
Catalan objects
loopless algorithm
cool-lex order
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
54:1
54:14
10.4230/LIPIcs.ISAAC.2022.54
article
Popular Edges with Critical Nodes
Chatterjee, Kushagra
1
Nimbhorkar, Prajakta
2
National University of Singapore, Singapore
Chennai Mathematical Institute, India
In the popular edge problem, the input is a bipartite graph G = (A ∪ B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A∪ B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M' such that the number of vertices that prefer M' to M is more than the number of vertices that prefer M to M'. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018].
We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings.
We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.54/LIPIcs.ISAAC.2022.54.pdf
Matching
Stable Matching
Popular feasible Matching
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
55:1
55:13
10.4230/LIPIcs.ISAAC.2022.55
article
Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average
Kobayashi, Yusuke
1
https://orcid.org/0000-0001-9478-7307
Mahara, Ryoga
1
https://orcid.org/0000-0002-4471-7914
Research Institute for Mathematical Sciences, Kyoto University, Japan
We study the problem of fairly allocating a set of indivisible goods to multiple agents and focus on the proportionality, which is one of the classical fairness notions. Since proportional allocations do not always exist when goods are indivisible, approximate notions of proportionality have been considered in the previous work. Among them, proportionality up to the maximin good (PROPm) has been the best approximate notion of proportionality that can be achieved for all instances. In this paper, we introduce the notion of proportionality up to the least valued good on average (PROPavg), which is a stronger notion than PROPm, and show that a PROPavg allocation always exists. Our results establish PROPavg as a notable non-trivial fairness notion that can be achieved for all instances. Our proof is constructive, and based on a new technique that generalizes the cut-and-choose protocol.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.55/LIPIcs.ISAAC.2022.55.pdf
Discrete Fair Division
Indivisible Goods
Proportionality
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
56:1
56:13
10.4230/LIPIcs.ISAAC.2022.56
article
Pursuit-Evasion in Graphs: Zombies, Lazy Zombies and a Survivor
Bose, Prosenjit
1
De Carufel, Jean-Lou
2
Shermer, Thomas
3
School of Computer Science, Carleton University, Ottawa, Canada
School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa, Canada
School of Computing Science, Simon Fraser University, Burnaby, Canada
We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let z(G) be the smallest number of zombies required to catch the survivor on a graph G with n vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that z(G) = Θ(n). We also show that there exist maximum-degree-3 outerplanar graphs such that z(G) = Ω(n/log(n)).
Let z_L(G) be the smallest number of lazy zombies (zombies that can stay still on their turn) required to catch the survivor on a graph G. We show that lazy zombies are more powerful than normal zombies but less powerful than cops. We prove that z_L(G) ≤ 2 for connected outerplanar graphs and this bound is tight in the worst case. We show that z_L(G) ≤ k for connected graphs with treedepth k. This result implies that z_L(G) is at most (k+1)log n for connected graphs with treewidth k, O(√n) for connected planar graphs, O(√{gn}) for connected graphs with genus g and O(h√{hn}) for connected graphs with any excluded h-vertex minor. Our results on lazy zombies still hold when an adversary chooses the initial positions of the zombies.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.56/LIPIcs.ISAAC.2022.56.pdf
Pursuit-evasion games
Outerplanar
Graphs
Treedepth
Treewidth
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
57:1
57:16
10.4230/LIPIcs.ISAAC.2022.57
article
Range Updates and Range Sum Queries on Multidimensional Points with Monoid Weights
Lu, Shangqi
1
Tao, Yufei
1
Chinese University of Hong Kong, New Territories, Hong Kong
Let P be a set of n points in ℝ^d where each point p ∈ P carries a weight drawn from a commutative monoid (ℳ, +, 0). Given a d-rectangle r_upd (i.e., an orthogonal rectangle in ℝ^d) and a value Δ ∈ ℳ, a range update adds Δ to the weight of every point p ∈ P∩ r_upd; given a d-rectangle r_qry, a range sum query returns the total weight of the points in P ∩ r_qry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of Õ(n) space that handles an update in Õ(T_upd) time and a query in Õ(T_qry) time for arbitrary functions T_upd(n) and T_qry(n) satisfying T_upd ⋅ T_qry = n. The result holds for any fixed dimensionality d ≥ 2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.57/LIPIcs.ISAAC.2022.57.pdf
Range Updates
Range Sum Queries
Data Structures
Lower Bounds
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
58:1
58:16
10.4230/LIPIcs.ISAAC.2022.58
article
Segment Visibility Counting Queries in Polygons
Buchin, Kevin
1
https://orcid.org/0000-0002-3022-7877
Custers, Bram
2
https://orcid.org/0000-0001-9342-319X
van der Hoog, Ivor
3
Löffler, Maarten
4
Popov, Aleksandr
2
https://orcid.org/0000-0002-0158-1746
Roeloffzen, Marcel
2
https://orcid.org/0000-0002-1129-461X
Staals, Frank
4
Department of Computer Science, TU Dortmund, Germany
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Department of Applied Mathematics and Computer Science, DTU, Copenhagen, Denmark
Department of Information and Computing Sciences, Utrecht University, The Netherlands
Let P be a simple polygon with n vertices, and let A be a set of m points or line segments inside P. We develop data structures that can efficiently count the objects from A that are visible to a query point or a query segment. Our main aim is to obtain fast, O(polylog nm), query times, while using as little space as possible.
In case the query is a single point, a simple visibility-polygon-based solution achieves O(log nm) query time using O(nm²) space. In case A also contains only points, we present a smaller, O(n + m^{2+ε} log n)-space, data structure based on a hierarchical decomposition of the polygon.
Building on these results, we tackle the case where the query is a line segment and A contains only points. The main complication here is that the segment may intersect multiple regions of the polygon decomposition, and that a point may see multiple such pieces. Despite these issues, we show how to achieve O(log n log nm) query time using only O(nm^{2+ε} + n²) space. Finally, we show that we can even handle the case where the objects in A are segments with the same bounds.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.58/LIPIcs.ISAAC.2022.58.pdf
Visibility
Data Structure
Polygons
Complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
59:1
59:17
10.4230/LIPIcs.ISAAC.2022.59
article
Shortest Beer Path Queries in Interval Graphs
Das, Rathish
1
He, Meng
2
Kondratovsky, Eitan
3
Munro, J. Ian
3
Naredla, Anurag Murty
3
Wu, Kaiyu
3
https://orcid.org/0000-0001-7562-1336
Department of Computer Science, University of Liverpool, UK
Faculty of Computer Science, Dalhousie University, Halifax, Canada
Cheriton School of Computer Science, University of Waterloo, Canada
Our interest is in paths between pairs of vertices that go through at least one of a subset of the vertices known as beer vertices. Such a path is called a beer path, and the beer distance between two vertices is the length of the shortest beer path.
We show that we can represent unweighted interval graphs using 2n log n + O(n) + O(|B|log n) bits where |B| is the number of beer vertices. This data structure answers beer distance queries in O(log^ε n) time for any constant ε > 0 and shortest beer path queries in O(log^ε n + d) time, where d is the beer distance between the two nodes. We also show that proper interval graphs may be represented using 3n + o(n) bits to support beer distance queries in O(f(n)log n) time for any f(n) ∈ ω(1) and shortest beer path queries in O(d) time. All of these results also have time-space trade-offs.
Lastly we show that the information theoretic lower bound for beer proper interval graphs is very close to the space of our structure, namely log(4+2√3)n - o(n) (or about 2.9 n) bits.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.59/LIPIcs.ISAAC.2022.59.pdf
Beer Path
Interval Graph
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
60:1
60:17
10.4230/LIPIcs.ISAAC.2022.60
article
Simon’s Congruence Pattern Matching
Kim, Sungmin
1
https://orcid.org/0000-0003-3153-0314
Ko, Sang-Ki
2
https://orcid.org/0000-0002-5406-5104
Han, Yo-Sub
1
Department of Computer Science, Yonsei University, Seoul, Republic of Korea
Department of Computer Science & Engineering, Kangwon National University, Chuncheon-si, Republic of Korea
Testing Simon’s congruence asks whether two strings have the same set of subsequences of length no greater than a given integer. In the light of the recent discovery of an optimal linear algorithm for testing Simon’s congruence, we solve the Simon’s congruence pattern matching problem. The problem requires finding all substrings of a text that are congruent to a pattern under the Simon’s congruence. Our algorithm efficiently solves the problem in linear time in the length of the text by reusing results from previous computations with the help of new data structures called X-trees and Y-trees. Moreover, we define and solve variants of the Simon’s congruence pattern matching problem. They require finding the longest and shortest substring of the text as well as the shortest subsequence of the text which is congruent to the pattern under the Simon’s congruence. Two more variants which ask for the longest congruent subsequence of the text and optimizing the pattern matching problem are left as open problems.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.60/LIPIcs.ISAAC.2022.60.pdf
pattern matching
Simon’s congruence
string algorithm
data structure
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
61:1
61:17
10.4230/LIPIcs.ISAAC.2022.61
article
Simple Order-Isomorphic Matching Index with Expected Compact Space
Kim, Sung-Hwan
1
Cho, Hwan-Gue
1
Pusan National University, Busan, South Korea
In this paper, we present a novel indexing method for the order-isomorphic pattern matching problem (also known as order-preserving pattern matching, or consecutive permutation matching), in which two equal-length strings are defined to match when X[i] < X[j] iff Y[i] < Y[j] for 0 ≤ i,j < |X|. We observe an interesting relation between the order-isomorphic matching and the insertion process of a binary search tree, based on which we propose a data structure which not only has a concise structure comprised of only two wavelet trees but also provides a surprisingly simple searching algorithm. In the average case analysis, the proposed method requires 𝒪(R(T)) bits, and it is capable of answering a count query in 𝒪(R(P)) time, and reporting an occurrence in 𝒪(lg |T|) time, where T and P are the text and the pattern string, respectively; for a string X, R(X) is the total time taken for the construction of the binary search tree by successively inserting the keys X[|X|-1],⋯,X[0] at the root, and its expected value is 𝒪(|X|lgσ) where σ is the alphabet size. Furthermore, the proposed method can be viewed as a generalization of some other methods including several heuristics and restricted versions described in previous studies in the literature.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.61/LIPIcs.ISAAC.2022.61.pdf
Compact Data Structure
String Matching
Order-Preserving Matching
Suffix Array
FM-index
Binary Search Tree
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
62:1
62:15
10.4230/LIPIcs.ISAAC.2022.62
article
Space-Efficient Graph Coarsening with Applications to Succinct Planar Encodings
Kammer, Frank
1
https://orcid.org/0000-0002-2662-3471
Meintrup, Johannes
1
https://orcid.org/0000-0003-4001-1153
THM, Technische Hochschule Mittelhessen, Giessen, Germany
We present a novel space-efficient graph coarsening technique for n-vertex planar graphs G, called cloud partition, which partitions the vertices V(G) into disjoint sets C of size O(log n) such that each C induces a connected subgraph of G. Using this partition 𝒫 we construct a so-called structure-maintaining minor F of G via specific contractions within the disjoint sets such that F has O(n/log n) vertices. The combination of (F, 𝒫) is referred to as a cloud decomposition.
For planar graphs we show that a cloud decomposition can be constructed in O(n) time and using O(n) bits. Given a cloud decomposition (F, 𝒫) constructed for a planar graph G we are able to find a balanced separator of G in O(n/log n) time. Contrary to related publications, we do not make use of an embedding of the planar input graph. We generalize our cloud decomposition from planar graphs to H-minor-free graphs for any fixed graph H. This allows us to construct the succinct encoding scheme for H-minor-free graphs due to Blelloch and Farzan (CPM 2010) in O(n) time and O(n) bits improving both runtime and space by a factor of Θ(log n).
As an additional application of our cloud decomposition we show that, for H-minor-free graphs, a tree decomposition of width O(n^{1/2 + ε}) for any ε > 0 can be constructed in O(n) bits and a time linear in the size of the tree decomposition. A similar result by Izumi and Otachi (ICALP 2020) constructs a tree decomposition of width O(k √n log n) for graphs of treewidth k ≤ √n in sublinear space and polynomial time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.62/LIPIcs.ISAAC.2022.62.pdf
planar graph
H-minor-free
space-efficient
separator
tree decomposition
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
63:1
63:14
10.4230/LIPIcs.ISAAC.2022.63
article
Subquadratic Weighted Matroid Intersection Under Rank Oracles
Tu, Ta-Wei
1
https://orcid.org/0000-0002-9706-3790
National Taiwan University, Taipei, Taiwan
Given two matroids ℳ₁ = (V, ℐ₁) and ℳ₂ = (V, ℐ₂) over an n-element integer-weighted ground set V, the weighted matroid intersection problem aims to find a common independent set S^* ∈ ℐ₁ ∩ ℐ₂ maximizing the weight of S^*. In this paper, we present a simple deterministic algorithm for weighted matroid intersection using Õ(nr^{3/4} log{W}) rank queries, where r is the size of the largest intersection of ℳ₁ and ℳ₂ and W is the maximum weight. This improves upon the best previously known Õ(nr log{W}) algorithm given by Lee, Sidford, and Wong [FOCS'15], and is the first subquadratic algorithm for polynomially-bounded weights under the standard independence or rank oracle models. The main contribution of this paper is an efficient algorithm that computes shortest-path trees in weighted exchange graphs.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.63/LIPIcs.ISAAC.2022.63.pdf
Matroids
Weighted Matroid Intersection
Combinatorial Optimization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
64:1
64:18
10.4230/LIPIcs.ISAAC.2022.64
article
Subsequences with Gap Constraints: Complexity Bounds for Matching and Analysis Problems
Day, Joel D.
1
https://orcid.org/0000-0003-0738-9816
Kosche, Maria
2
https://orcid.org/0000-0002-2165-2695
Manea, Florin
3
https://orcid.org/0000-0001-6094-3324
Schmid, Markus L.
4
https://orcid.org/0000-0001-5137-1504
Loughborough University, UK
Computer Science Department, Universität Göttingen, Germany
Computer Science Department and CIDAS, Universität Göttingen, Germany
Humboldt-Universität zu Berlin, Germany
We consider subsequences with gap constraints, i. e., length-k subsequences p that can be embedded into a string w such that the induced gaps (i. e., the factors of w between the positions to which p is mapped to) satisfy given gap constraints gc = (C_1, C_2, …, C_{k-1}); we call p a gc-subsequence of w. In the case where the gap constraints gc are defined by lower and upper length bounds C_i = (L^-_i, L^+_i) ∈ ℕ² and/or regular languages C_i ∈ REG, we prove tight (conditional on the orthogonal vectors (OV) hypothesis) complexity bounds for checking whether a given p is a gc-subsequence of a string w. We also consider the whole set of all gc-subsequences of a string, and investigate the complexity of the universality, equivalence and containment problems for these sets of gc-subsequences.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.64/LIPIcs.ISAAC.2022.64.pdf
String algorithms
subsequences with gap constraints
pattern matching
fine-grained complexity
conditional lower bounds
parameterised complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
65:1
65:17
10.4230/LIPIcs.ISAAC.2022.65
article
Succinct List Indexing in Optimal Time
Holland, William L.
1
School of Computing and Information Systems, The University of Melbourne, Parkville, Australia
An indexed list supports (efficient) access to both the offsets and the items of an arbitrarily ordered set under the effect of insertions and deletions. Existing solutions are engaged in a space-time trade-off. On the one hand, time efficient solutions are composed as a package of data structures: a linked-list, a hash table and a tree-type structure to support indexing. This arrangement observes a memory commitment that is outside the information theoretic lower bound (for ordered sets) by a factor of 12. On the other hand, the memory lower bound can be satisfied, up to an additive lower order term, trivially with an array. However, operations incur time costs proportional to the length of the array.
We revisit the list indexing problem by attempting to balance the competing demands of space and time efficiency. We prepare the first succinct indexed list that supports efficient query and update operations. To implement an ordered set of size n, drawn from the universe {1, …, m}, the solution occupies n(log m + o(log n)) bits (with high probability) and admits all operations optimally in 𝒪(log n/log log n) time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.65/LIPIcs.ISAAC.2022.65.pdf
Succinct Data Structures
Lists
Dynamic Data Structures
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
66:1
66:18
10.4230/LIPIcs.ISAAC.2022.66
article
Super-Cubic Lower Bound for Generalized Karchmer-Wigderson Games
Ignatiev, Artur
1
2
https://orcid.org/0000-0002-1960-5064
Mihajlin, Ivan
3
Smal, Alexander
3
4
https://orcid.org/0000-0002-8241-5503
St.Petersburg State University, Russia
HSE University, St.Petersburg, Russia
St.Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia
Technion, Haifa, Israel
In this paper, we prove a super-cubic lower bound on the size of a communication protocol for generalized Karchmer-Wigderson game for an explicit function f: {0,1}ⁿ → {0,1}^{log n}. Lower bounds for original Karchmer-Wigderson games correspond to De Morgan formula lower bounds, thus the best known size lower bound is cubic. The generalized Karchmer-Wigderson games are similar to the original ones, so we hope that our approach can provide an insight for proving better lower bounds on the original Karchmer-Wigderson games, and hence for proving new lower bounds on De Morgan formula size.
To achieve super-cubic lower bound we adapt several techniques used in formula complexity to communication protocols, prove communication complexity lower bound for a composition of several functions with a multiplexer relation, and use a technique from [Ivan Mihajlin and Alexander Smal, 2021] to extract the "hardest" function from it. As a result, in this setting we are able to show that there is a relatively small set of functions such that at least one of them does not have a small protocol. The resulting lower bound of Ω̃(n^3.156) is significantly better than the bound obtained from the counting argument.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.66/LIPIcs.ISAAC.2022.66.pdf
communication complexity
circuit complexity
Karchmer-Wigderson games
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-12-14
248
67:1
67:18
10.4230/LIPIcs.ISAAC.2022.67
article
The Dispersive Art Gallery Problem
Rieck, Christian
1
https://orcid.org/0000-0003-0846-5163
Scheffer, Christian
2
https://orcid.org/0000-0002-3471-2706
Department of Computer Science, TU Braunschweig, Germany
Faculty of Electrical Engineering and Computer Science, Bochum University of Applied Sciences, Germany
We introduce a new variant of the art gallery problem that comes from safety issues. In this variant we are not interested in guard sets of smallest cardinality, but in guard sets with largest possible distances between these guards. To the best of our knowledge, this variant has not been considered before. We call it the Dispersive Art Gallery Problem. In particular, in the dispersive art gallery problem we are given a polygon 𝒫 and a real number 𝓁, and want to decide whether 𝒫 has a guard set such that every pair of guards in this set is at least a distance of 𝓁 apart.
In this paper, we study the vertex guard variant of this problem for the class of polyominoes. We consider rectangular visibility and distances as geodesics in the L₁-metric. Our results are as follows. We give a (simple) thin polyomino such that every guard set has minimum pairwise distances of at most 3. On the positive side, we describe an algorithm that computes guard sets for simple polyominoes that match this upper bound, i.e., the algorithm constructs worst-case optimal solutions. We also study the computational complexity of computing guard sets that maximize the smallest distance between all pairs of guards within the guard sets. We prove that deciding whether there exists a guard set realizing a minimum pairwise distance for all pairs of guards of at least 5 in a given polyomino is NP-complete.
We were also able to find an optimal dynamic programming approach that computes a guard set that maximizes the minimum pairwise distance between guards in tree-shaped polyominoes, i.e., computes optimal solutions; due to space constraints, details can be found in the full version of our paper [Christian Rieck and Christian Scheffer, 2022]. Because the shapes constructed in the NP-hardness reduction are thin as well (but have holes), this result completes the case for thin polyominoes.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol248-isaac2022/LIPIcs.ISAAC.2022.67/LIPIcs.ISAAC.2022.67.pdf
Art gallery
dispersion
polyominoes
NP-completeness
r-visibility
vertex guards
L₁-metric
worst-case optimal