IA Scholar Query: A polynomial-time algorithm for Outerplanar Diameter Improvement.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 14 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Recognizing weighted and seeded disk graphs
https://scholar.archive.org/work/a5w5be7buzanvk4debowcptl2y
Disk intersection representations realize graphs by mapping vertices bijectively to disks in the plane such that two disks intersect each other if and only if the corresponding vertices are adjacent in the graph. If intersections are restricted to touching points of the boundaries, we call them disk contact representations. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard for both contact and intersection representations. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that in the contact scenario it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights. We also consider a version of the problem in which the disks of a representation are supposed to cover preassigned points, called seeds. We show that both for contact and intersection representations this problem is NP-hard for unit weights even if the given graph is a path. If the disks' radii are not prescribed, the problem remains NP-hard for trees in the contact scenario.Boris Klemz, Martin Nöllenburg, Roman Prutkinwork_a5w5be7buzanvk4debowcptl2yWed, 14 Sep 2022 00:00:00 GMTMetric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters
https://scholar.archive.org/work/ssiae6zazfguhe3imuzii4f2re
For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP ⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, Prafullkumar Tale, Stefan Szeider, Robert Ganian, Alexandra Silvawork_ssiae6zazfguhe3imuzii4f2reMon, 22 Aug 2022 00:00:00 GMTAsymptotically Optimal Vertex Ranking of Planar Graphs
https://scholar.archive.org/work/v54ppwxphbbcjfnrnqjbgpsx2e
A (vertex) ℓ-ranking is a colouring φ:V(G)→ℕ of the vertices of a graph G with integer colours so that for any path u_0,...,u_p of length at most ℓ, φ(u_0)≠φ(u_p) or φ(u_0)<max{φ(u_0),...,φ(u_p)}. We show that, for any fixed integer ℓ≥ 2, every n-vertex planar graph has an ℓ-ranking using O(log n/logloglog n) colours and this is tight even when ℓ=2; for infinitely many values of n, there are n-vertex planar graphs, for which any 2-ranking requires Ω(log n/logloglog n) colours. This result also extends to bounded genus graphs. In developing this proof we obtain optimal bounds on the number of colours needed for ℓ-ranking graphs of treewidth t and graphs of simple treewidth t. These upper bounds are constructive and give O(n)-time algorithms. Additional results that come from our techniques include new sublogarithmic upper bounds on the number of colours needed for ℓ-rankings of apex minor-free graphs and k-planar graphs.Prosenjit Bose, Vida Dujmović, Mehrnoosh Javarsineh, Pat Morinwork_v54ppwxphbbcjfnrnqjbgpsx2eThu, 18 Aug 2022 00:00:00 GMTMosaicSets: Embedding Set Systems into Grid Graphs
https://scholar.archive.org/work/krf4o5qxdzgbvo3upsddt625pe
Visualizing sets of elements and their relations is an important research area in information visualization. In this paper, we present MosaicSets: a novel approach to create Euler-like diagrams from non-spatial set systems such that each element occupies one cell of a regular hexagonal or square grid. The main challenge is to find an assignment of the elements to the grid cells such that each set constitutes a contiguous region. As use case, we consider the research groups of a university faculty as elements, and the departments and joint research projects as sets. We aim at finding a suitable mapping between the research groups and the grid cells such that the department structure forms a base map layout. Our objectives are to optimize both the compactness of the entirety of all cells and of each set by itself. We show that computing the mapping is NP-hard. However, using integer linear programming we can solve real-world instances optimally within a few seconds. Moreover, we propose a relaxation of the contiguity requirement to visualize otherwise non-embeddable set systems. We present and discuss different rendering styles for the set overlays. Based on a case study with real-world data, our evaluation comprises quantitative measures as well as expert interviews.Peter Rottmann, Markus Wallinger, Annika Bonerath, Sven Gedicke, Martin Nöllenburg, Jan-Henrik Haunertwork_krf4o5qxdzgbvo3upsddt625peTue, 16 Aug 2022 00:00:00 GMTThe role of twins in computing planar supports of hypergraphs
https://scholar.archive.org/work/grs5z4wfzvbebk5h6ilrfkrjmi
A support or realization of a hypergraph H is a graph G on the same vertex as H such that for each hyperedge of H it holds that its vertices induce a connected subgraph of G. The NP-hard problem of finding a planar support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins – pairs of vertices that are in precisely the same hyperedges – can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with m hyperedges to have an r-outerplanar support, which depends only on r and m. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing r-outerplanar supports for hypergraphs with m hyperedges if m and r are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters m and r.René van Bevern, Iyad A. Kanj, Christian Komusiewicz, Rolf Niedermeier, Manuel Sorgework_grs5z4wfzvbebk5h6ilrfkrjmiMon, 01 Aug 2022 00:00:00 GMTMetric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters
https://scholar.archive.org/work/qrfznvqlxfglrcmiwwl6xsynki
For a graph G, a subset S ⊆ V(G) is called a resolving set if for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G and a positive integer k, and asks whether there exists a resolving set of size at most k. This problem was introduced in the 1970s and is known to be NP-hard [GT 61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein [CCC 2013] proved that the problem is W[2]-hard when parameterized by the natural parameter k. They also observed that it is FPT when parameterized by the vertex cover number and asked about its complexity under smaller parameters, in particular the feedback vertex set number. We answer this question by proving that Metric Dimension is W[1]-hard when parameterized by the feedback vertex set number. This also improves the result of Bonnet and Purohit [IPEC 2019] which states that the problem is W[1]-hard parameterized by the treewidth. Regarding the parameterization by the vertex cover number, we prove that Metric Dimension does not admit a polynomial kernel under this parameterization unless NP⊆ coNP/poly. We observe that a similar result holds when the parameter is the distance to clique. On the positive side, we show that Metric Dimension is FPT when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.Esther Galby, Liana Khazaliya, Fionn Mc Inerney, Roohani Sharma, Prafullkumar Talework_qrfznvqlxfglrcmiwwl6xsynkiThu, 30 Jun 2022 00:00:00 GMTOn graphs coverable by k shortest paths
https://scholar.archive.org/work/o5q53dylbra5xk33j7d5ryhrsm
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 k2 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.Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todincawork_o5q53dylbra5xk33j7d5ryhrsmThu, 30 Jun 2022 00:00:00 GMTNear-Linear ε-Emulators for Planar Graphs
https://scholar.archive.org/work/e5xgiawv4rcsjcakzrqjnn3iha
We study vertex sparsification for distances, in the setting of planar graphs with distortion: Given a planar graph G (with edge weights) and a subset of k terminal vertices, the goal is to construct an ε-emulator, which is a small planar graph G' that contains the terminals and preserves the distances between the terminals up to factor 1+ε. We construct the first ε-emulators for planar graphs of near-linear size Õ(k/ε^O(1)). In terms of k, this is a dramatic improvement over the previous quadratic upper bound of Cheung, Goranci and Henzinger, and breaks below known quadratic lower bounds for exact emulators (the case when ε=0). Moreover, our emulators can be computed in (near-)linear time, which lead to fast (1+ε)-approximation algorithms for basic optimization problems on planar graphs, including multiple-source shortest paths, minimum (s,t)-cut, graph diameter, and dynamic distace oracle.Hsien-Chih Chang, Robert Krauthgamer, Zihan Tanwork_e5xgiawv4rcsjcakzrqjnn3ihaTue, 21 Jun 2022 00:00:00 GMTTrue Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs
https://scholar.archive.org/work/rn2s4enrtvdf5a727x3wp5ae6a
We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set 𝒟 of n unit disks inducing a unit-disk graph G_𝒟 and a number p ∈ [n], one can partition 𝒟 into p subsets 𝒟₁,... ,𝒟_p such that for every i ∈ [p] and every 𝒟' ⊆ 𝒟_i, the graph obtained from G_𝒟 by contracting all edges between the vertices in 𝒟_i $1𝒟' admits a tree decomposition in which each bag consists of O(p+|𝒟'|) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved very recently by Marx et al. [SODA'22] and Bandyapadhyay et al. [SODA'22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem (CDT) for unit-disk graphs, resolving an open question in the work Panolan et al. [SODA'19]. On the algorithmic side, we obtain a new FPT algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in 2^{O(√k log k)} ⋅ n^{O(1)} time, where k denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA'22] (which works more generally for disk graphs) and is almost optimal, as the problem cannot be solved in 2^{o(√k)} ⋅ n^{O(1)} time assuming the ETH.Sayan Bandyapadhyay, William Lochet, Daniel Lokshtanov, Saket Saurabh, Jie Xue, Xavier Goaoc, Michael Kerberwork_rn2s4enrtvdf5a727x3wp5ae6aWed, 01 Jun 2022 00:00:00 GMTWeak Coloring Numbers of Intersection Graphs
https://scholar.archive.org/work/hpeswll4u5hzfn23raesshnkq4
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for a positive integer k, we seek a vertex ordering such that every vertex can (weakly respectively strongly) reach in k steps only few vertices that precede it in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in structural and algorithmic graph theory. Recently, Dvořák, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in ℝ^d, such as homothets of a compact convex object, or comparable axis-aligned boxes. In this paper, we prove upper and lower bounds for the k-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in k, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).Zdeněk Dvořák, Jakub Pekárek, Torsten Ueckerdt, Yelena Yuditsky, Xavier Goaoc, Michael Kerberwork_hpeswll4u5hzfn23raesshnkq4Wed, 01 Jun 2022 00:00:00 GMTFrom Width-Based Model Checking to Width-Based Automated Theorem Proving
https://scholar.archive.org/work/5oel6ybptfegpeokvzsdv6o4ie
In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we show that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number k as input and correctly determines in time double-exponential in k^O(1) whether the conjecture is valid on all graphs of treewidth at most k. This improves significantly on upper bounds obtained using previously available techniques.Mateus de Oliveira Oliveira, Farhad Vadieework_5oel6ybptfegpeokvzsdv6o4ieSun, 29 May 2022 00:00:00 GMTNarrowing the LOCALx2013CONGEST Gaps in Sparse Networks via Expander Decompositions
https://scholar.archive.org/work/ji4kagikmjaglitwxdty2fhgye
Many combinatorial optimization problems can be approximated within (1 ±ϵ) factors in poly(log n, 1/ϵ) rounds in the LOCAL model via network decompositions [Ghaffari, Kuhn, and Maus, STOC 2018]. These approaches require sending messages of unlimited size, so they do not extend to the CONGEST model, which restricts the message size to be O(log n) bits. In this paper, we develop a generic framework for obtaining poly(log n, 1/ϵ)-round (1±ϵ)-approximation algorithms for many combinatorial optimization problems, including maximum weighted matching, maximum independent set, and correlation clustering, in graphs excluding a fixed minor in the CONGEST model. This class of graphs covers many sparse network classes that have been studied in the literature, including planar graphs, bounded-genus graphs, and bounded-treewidth graphs. Furthermore, we show that our framework can be applied to give an efficient distributed property testing algorithm for an arbitrary minor-closed graph property that is closed under taking disjoint union, significantly generalizing the previous distributed property testing algorithm for planarity in [Levi, Medina, and Ron, PODC 2018 Distributed Computing 2021]. Our framework uses distributed expander decomposition algorithms [Chang and Saranurak, FOCS 2020] to decompose the graph into clusters of high conductance. We show that any graph excluding a fixed minor admits small edge separators. Using this result, we show the existence of a high-degree vertex in each cluster in an expander decomposition, which allows the entire graph topology of the cluster to be routed to a vertex. Similar to the use of network decompositions in the LOCAL model, the vertex will be able to perform any local computation on the subgraph induced by the cluster and broadcast the result over the cluster.Yi-Jun Chang, Hsin-Hao Suwork_ji4kagikmjaglitwxdty2fhgyeTue, 17 May 2022 00:00:00 GMTOriented Diameter of Planar Triangulations
https://scholar.archive.org/work/zjtsaodv4rak5e4zzar7f2ivdq
The diameter of an undirected or a directed graph is defined to be the maximum shortest path distance over all pairs of vertices in the graph. Given an undirected graph G, we examine the problem of assigning directions to each edge of G such that the diameter of the resulting oriented graph is minimized. The minimum diameter over all strongly connected orientations is called the oriented diameter of G. The problem of determining the oriented diameter of a graph is known to be NP-hard, but the time-complexity question is open for planar graphs. In this paper we compute the exact value of the oriented diameter for triangular grid graphs. We then prove an n/3 lower bound and an n/2+O(√(n)) upper bound on the oriented diameter of planar triangulations. It is known that given a planar graph G with bounded treewidth and a fixed positive integer k, one can determine in linear time whether the oriented diameter of G is at most k. In contrast, we consider a weighted version of the oriented diameter problem and show it to be is weakly NP-complete for planar graphs with bounded pathwidth.Debajyoti Mondal, N. Parthiban, Indra Rajasinghwork_zjtsaodv4rak5e4zzar7f2ivdqTue, 08 Mar 2022 00:00:00 GMTFrom Width-Based Model Checking to Width-Based Automated Theorem Proving
https://scholar.archive.org/work/2r2zwse26zfipkxnytp7lzx5wy
In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we show that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number $k$ as input and correctly determines in time double-exponential in $k^{O(1)}$ whether the conjecture is valid on all graphs of treewidth at most $k$. This improves significantly on upper bounds obtained using previously available techniques.Mateus De Oliveira Oliveira, Farhad Vadieework_2r2zwse26zfipkxnytp7lzx5wyMon, 28 Feb 2022 00:00:00 GMTFrom Width-Based Model Checking toWidth-Based Automated Theorem Proving
https://scholar.archive.org/work/wsfqcukjqra3nm2ftaipkkdrda
In the field of parameterized complexity theory, the study of width measures for graphs has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including the width measures treewidth and cliquewidth. As a quantitative application of our framework, we show that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number as input and correctly determines in time double-exponential in k^O(1)whether the conjecture is valid on all graphs of treewidth at most k. This improves significantly on upper bounds obtained using previously available techniques.Anonymouswork_wsfqcukjqra3nm2ftaipkkdrdaMon, 28 Feb 2022 00:00:00 GMTLossy Planarization: A Constant-Factor Approximate Kernelization for Planar Vertex Deletion
https://scholar.archive.org/work/377llva5ajasfplsbob5tng6gi
In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the family F = K_5, K_3,3. Whenever the family F contains at least one planar graph, then F-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [Fomin, Lokshtanov, Misra, and Saurabh, FOCS'12]. The Vertex planarization problem is arguably the simplest setting for which F does not contain a planar graph and the existence of a constant-factor approximation or a polynomial kernelization remains a major open problem. In this work we show that Vertex planarization admits an algorithm which is a combination of both approaches. Namely, we present a polynomial A-approximate kernelization, for some constant A > 1, based on the framework of lossy kernelization [Lokshtanov, Panolan, Ramanujan, and Saurabh, STOC'17]. Simply speaking, when given a graph G and integer k, we show how to compute a graph G' on poly(k) vertices so that any B-approximate solution to G' can be lifted to an (A*B)-approximate solution to G, as long as A*B*OPT(G) <= k. In order to achieve this, we develop a framework for sparsification of planar graphs which approximately preserves all separators and near-separators between subsets of the given terminal set. Our result yields an improvement over the state-of-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time O(n^eps)-approximation algorithm, for any eps > 0, and a quasi-polynomial-time (log n)^O(1) approximation algorithm, both randomized [Kawarabayashi and Sidiropoulos, FOCS'17]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively O(OPT^eps) and (log OPT)^O(1).Bart M. P. Jansen, Michał Włodarczykwork_377llva5ajasfplsbob5tng6giFri, 04 Feb 2022 00:00:00 GMTPath eccentricity of graphs
https://scholar.archive.org/work/jhw2w3prm5a2xczq657nlyub34
Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~$G$ as a parameter. Let pe$(G)$ denote the value of ecc$_G(P)$ for a central path $P$ of $G$. We obtain tight upper bounds for pe$(G)$ in some graph classes. We show that pe$(G) \leq 1$ on biconvex graphs and that pe$(G) \leq 2$ on bipartite convex graphs. Moreover, we design algorithms that find such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe$(G)$ on general graphs and $k$-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that on trees, and bipartite permutation graphs, a longest path is also a central path. Furthermore, for superclasses of these graphs, we exhibit counterexamples for this property.Renzo Gómez, Juan Gutiérrezwork_jhw2w3prm5a2xczq657nlyub34Tue, 01 Feb 2022 00:00:00 GMTDistributed Vertex Cover Reconfiguration
https://scholar.archive.org/work/ztkarhuplbdhbcxwxnkv7vivyy
Reconfiguration schedules, i.e., sequences that gradually transform one solution of a problem to another while always maintaining feasibility, have been extensively studied. Most research has dealt with the decision problem of whether a reconfiguration schedule exists, and the complexity of finding one. A prime example is the reconfiguration of vertex covers. We initiate the study of batched vertex cover reconfiguration, which allows to reconfigure multiple vertices concurrently while requiring that any adversarial reconfiguration order within a batch maintains feasibility. The latter provides robustness, e.g., if the simultaneous reconfiguration of a batch cannot be guaranteed. The quality of a schedule is measured by the number of batches until all nodes are reconfigured, and its cost, i.e., the maximum size of an intermediate vertex cover. To set a baseline for batch reconfiguration, we show that for graphs belonging to one of the classes {{cycles, trees, forests, chordal, cactus, even-hole-free, claw-free}}, there are schedules that use O(ε^{-1}) batches and incur only a 1+ε multiplicative increase in cost over the best sequential schedules. Our main contribution is to compute such batch schedules in a distributed setting O(ε^{-1} {log^*} n) rounds, which we also show to be tight. Further, we show that once we step out of these graph classes we face a very different situation. There are graph classes on which no efficient distributed algorithm can obtain the best (or almost best) existing schedule. Moreover, there are classes of bounded degree graphs which do not admit any reconfiguration schedules without incurring a large multiplicative increase in the cost at all.Keren Censor-Hillel, Yannic Maus, Shahar Romem-Peled, Tigran Tonoyan, Mark Bravermanwork_ztkarhuplbdhbcxwxnkv7vivyyTue, 25 Jan 2022 00:00:00 GMTScattering and Sparse Partitions, and their Applications
https://scholar.archive.org/work/wkzbo5iy3vc2bmafind5mnxmzm
A partition 𝒫 of a weighted graph G is (σ,τ,Δ)-sparse if every cluster has diameter at most Δ, and every ball of radius Δ/σ intersects at most τ clusters. Similarly, 𝒫 is (σ,τ,Δ)-scattering if instead for balls we require that every shortest path of length at most Δ/σ intersects at most τ clusters. Given a graph G that admits a (σ,τ,Δ)-sparse partition for all Δ>0, Jia et al. [STOC05] constructed a solution for the Universal Steiner Tree problem (and also Universal TSP) with stretch O(τσ^2log_τ n). Given a graph G that admits a (σ,τ,Δ)-scattering partition for all Δ>0, we construct a solution for the Steiner Point Removal problem with stretch O(τ^3σ^3). We then construct sparse and scattering partitions for various different graph families, receiving many new results for the Universal Steiner Tree and Steiner Point Removal problems.Arnold Filtserwork_wkzbo5iy3vc2bmafind5mnxmzmThu, 20 Jan 2022 00:00:00 GMT