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.
The approximation scheme of Worou and Galtier @cite_19 computes for @math a @math -approximation of the fractional arboricity @math in time @math . It constructs a subgraph of this density (in the sense of ), but apparently no forest partition is computed.
@cite_19
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In this paper, we develop some algorithmic aspects of the fractional arboricity of a graph in order to study some new approaches for graph clustering. For a given undirected graph G ( V , E ) with m edges and n vertices, the fractional arboricity γ ( G ) measures the maximum edge density of the subgraphs of G ( V , E ) . It is the fractional covering number of the corresponding graphic matroid. The fractional arboricity, for applications in networks reliability or the approximation of the chromatic number of the graphs, has been studied for many years. There are some algorithms in polynomial time to compute the fractional arboricity and its integer part. But, for large graphs such as the graph of the Web or graphs of social networks, the exact algorithms are not fast enough for practical use. That is why we describe a new FPTAS to compute an e -approximation of the fractional arboricity (with e 0 as small as desired). Our algorithm uses the principle of the multiplicative weights update method, needs a memory of size O ( m ) and has a complexity of O ( m log 2 ( m ) log ( m n ) e 2 ) . We also give a 2-approximation of γ ( G ) with computation time O ( m ) , which is a quick preprocessing for the main algorithm. Finally, we present a fast algorithm to extract a subgraph which achieves the value of the approximation of the fractional arboricity. Y. Bartal (1996, 1998) gave a randomized polynomial time algorithm that given any n point metric G, constructs a tree T such that the expected stretch (distortion) of any edge is at most O (log n log log n). His result has found several applications and in particular has resulted in approximation algorithms for many graph optimization problems. However approximation algorithms based on his result are inherently randomized. In this paper we derandomize the use of Bartal's algorithm in the design of approximation algorithms. We give an efficient polynomial time algorithm that given a finite n point metric G, constructs O(n log n) trees and a probability distribution spl mu on them such that the expected stretch of any edge of G in a tree chosen according to spl mu is at most O(log n log log n). Our result establishes that finite metrics can be probabilistically approximated by a small number of tree metrics. We obtain the first deterministic approximation algorithms for buy-at-bulk network design and vehicle routing; in addition we subsume results from our earlier work on derandomization. Our main result is obtained by a novel view of probabilistic approximation of metric spaces as a deterministic optimization problem via linear programming. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. Given a set U with n elements and a family of subsets S 2^U we show how to count the number of k-partitions S_1 ? ? ? S_k = U into subsets S_i S in time 2^ n n^ O(1) . The only assumption on S is that it can be enumerated in time 2^ n n^ O(1) . In effect we get exact algorithms in time 2^ n n^ O(1) for several well-studied partition problems including Domatic Number, Chromatic Number, Bounded Component Spanning Forest, Partition into Hamiltonian Subgraphs, and Bin Packing. If only polynomial space is available, our algorithms run in time 3^ n n^ O(1) if membership in S can be decided in polynomial time. For Chromatic Number, we present a version that runs in time O(2.2461^n ) and polynomial space. For Domatic Number, we present a version that runs in time O(2.8718^n ). Finally, we present a family of polynomial space approximation algorithms that find a number between ( G ) and ( 1 + ) ( G ) in time O(1.2209^n + 2.2461^ e^ - n ).