No abstract

Shimon Even's Graph Algorithms, published in 1979, was a seminal introductory book on algorithms read by everyone engaged in the field. This thoroughly revised second edition, with a foreword by Richard M. Karp and notes by Andrew V. Goldberg, continues the exceptional presentation from the first edition and explains algorithms in a formal but simple language with a direct and intuitive presentation. The book begins by covering basic material, including graphs and shortest paths, trees, depth-first-search and breadth-first search. The main part of the book is devoted to network flows and applications of network flows, and it ends with chapters on planar graphs and testing graph planarity.

This paper deals with approximating feedback sets in directed graphs. We consider two related problems: the weighted feedback vertex set (FVS) problem, and the weighted feedback edge set (FES) problem. In the FVS (resp. FES) problem, one is given a directed graph with weights (each of which is at least one) on the vertices (resp. edges), and is asked to find a subset of vertices (resp. edges) with minimum total weight that intersects every directed cycle in the graph. These problems are among the classical NP-hard problems and have many applications. We also consider a generalization of these problems: SUBSET-FVS and SUBSET-FES, in which the feedback set has to intersect only a subset of the directed cycles in the graph. This subset consists of all the cycles that go through a distinguished input subset of vertices and edges, denoted by X . This generalization is also NP-hard even when |X | = 2. We present approximation algorithms for the SUBSET-FVS and SUBSET-FES problems. The first algorithm we present achieves an approximation factor of O(log 2 |X |). The second algorithm achieves an approximation factor of O(min{log τ * log log τ * , log n log log n)}, where τ * is the value of the optimum fractional solution of the problem at hand, and n is the number of vertices in the graph. We also define a multicut problem in a special type of directed networks which we call circular networks, and show that the SUBSET-FES and SUBSET-FVS problems are equivalent to this multicut problem. Another contribution of our paper is a combinatorial algorithm that computes a (1 + ε) approximation to the fractional optimal feedback vertex set. Computing the approximate solution is much simpler and more efficient than general linear programming methods. All of our algorithms use this approximate solution.

We present a novel divide-and-conquer paradigm for approximating NP-hard graph optimization problems. The paradigm models graph optimization problems that satisfy two properties: First, a divide-and-conquer approach is applicable. Second, a fractional spreading metric is computable in polynomial time. The spreading metric assigns fractional lengths to either edges or vertices of the input graph, such that all subgraphs on which the optimization problem is non-trivial have large diameters. In addition, the spreading metric provides a lower bound, T , on the cost of solving the optimization problem.We present a polynomial time approximation algorithm for problems modelled by our paradigm whose approximation factor is 0 (min{log Tlog log T , log k log log k}), where k denotes the number of "interesting" vertices in the problem instance, and is at most the number of vertices.We present six problems that can be formulated to fit the paradigm.For all these problems our algorithm improves previous results. The problems are: (a) linear arrangement; (b) embedding a graph in a ddimensional mesh; (c) interval graph completion; (d) minimizing storage-time product; (e) (subset) feedback sets in directed graphs and multicuts in circular networks; (f) symmetric multicuts in directed networks. For the first four problems, we improve the best known approximation factor from O(log2 n ) to OQog nlog log n), where n denotes the number of vertices.For the last two problems we improve the approximation factor from o (min{log T log log T , log nlog log n, log2 le) to 0 (min{log T log log T , log k log log k}), where k denotes the number of source-sink pairs.

We present a 1.5-approximation algorithm for the following NP-hard problem: given a connected graph G = (V, E) and an edge set E on V disjoint to E, find a minimum size subset of edges F ⊆ E such that (V, E ∪ F ) is 2-edge-connected. Our result improves and significantly simplifies the approximation algorithm with ratio 1.875 + ε of Nagamochi.