The arboricity of a graph is the minimum number of forests it can be partitioned into. Previous approximation schemes were nonconstructive, i.e., they only approximated the arboricity as a value without computing a corresponding forest partition, as they operate on the related pseudoforest partitions or the dual problem. We propose an algorithm for converting a partition of @math pseudoforests into a partition of @math forests in @math time, where @math is the inverse Ackermann function, when @math expected time is allowed for pre-computation of a perfect hash function. Without perfect hashing, we obtain @math with a data structure by Brodal and Fagerberg that stores graphs of arboricity @math . For every fixed @math , the latter result implies a constructive @math -approximation algorithm with runtime @math by using Kowalik's approximation scheme for pseudoforest partitions. Our algorithm might help in designing a faster exact arboricity algorithm. We also make several remarks on approximation algorithms for the pseudoarboricity and the equivalent graph orientations with smallest maximum indegree, and correct some mistakes made in the literature.
Barenboim and Elkin @cite_39 propose a constructive distributed algorithm that computes a @math -approximation of @math . @cite_12 describe an algorithm that distinguishes with high constant probability between graphs that are @math -close to and graphs that are @math -far from having arboricity @math , for some constant @math . Several upper bounds of the type @math for the arboricity were given by Chiba and Nishizeki @cite_25 , Gabow and Westermann @cite_9 , @cite_35 and Blumenstock @cite_28 . Let @math denote the bounds, and write @math if @math holds plus an example exists where the bounds differ. One can show @math . We do not know whether the second inequality is strict. The bound @math of is optimal.
@cite_35 @cite_28 @cite_9 @cite_39 @cite_25 @cite_12
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We prove that the thickness and the arboricity of a graph with e edges are at most ⌊e3 + 32⌋ and ⌈e2⌉, respectively, and that the latter bound is best possible. The densest subgraph problem, which asks for a subgraph with the maximum edges-to-vertices ratio d∗, is solvable in polynomial time. We discuss algorithms for this problem and the computation of a graph orientation with the lowest maximum indegree, which is equal to ⌈d∗⌉. This value also equals the pseudoarboricity of the graph. We show that it can be computed in O(|E| √ log log d∗) time, and that better estimates can be given for graph classes where d∗ satisfies certain asymptotic bounds. These runtimes are achieved by accelerating a binary search with an approximation scheme, and a runtime analysis of Dinitz’s algorithm on flow networks where all arcs, except the source and sink arcs, have unit capacity. We experimentally compare implementations of various algorithms for the densest subgraph and pseudoarboricity problems. In flow-based algorithms, Dinitz’s algorithm performs significantly better than push-relabel algorithms on all instances tested. This paper presents improved algorithms for matroid partitioning problems, such as finding a maximum cardinality set of edges of a graph that can be partitioned into k forests. The notion of a clamp in a matroid sum is introduced. Efficient algorithms for problems involving clumps are presented. Applications of these algorithms to problems arising in the study of structural rigidity of graphs, the Shannon switching game and others are given. We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graph-theoretic structure, called Nash-Williams forests-decomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearly-tight lower bounds on the running time of any distributed algorithm for computing a forests-decomposition. In this paper we introduce a new simple strategy into edge-searching of a graph, which is useful to the various subgraph listing problems. Applying the strategy, we obtain the following four algorithms. The first one lists all the triangles in a graph G in @math time, where m is the number of edges of G and @math the arboricity of G. The second finds all the quadrangles in @math time. Since @math is at most three for a planar graph G, both run in linear time for a planar graph. The third lists all the complete subgraphs @math of order l in @math time. The fourth lists all the cliques in @math time per clique. All the algorithms require linear space. We also establish an upper bound on @math for a graph @math , where n is the number of vertices in G. In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are ϵ-close to having arboricity α and graphs that c · ϵ-far from having arboricity 3α, where c is an absolute small constant. The query complexity and running time of the algorithm are1 [EQUATION] where n denotes the number of vertices and m denotes the number of edges. In terms of the dependence on n and m this bound is optimal up to poly-logarithmic factors since [EQUATION] queries are necessary (and the arboricity of a graph is always [EQUATION]. We leave it as an open question whether the dependence on 1 ϵ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forest-decomposition algorithm as well as a tailored procedure of edge sampling.