Graph Augmentation Research Articles

An approximation for finding a smallest 2-edge-connected subgraph containing a specified spanning tree

Given a graph G=( V, E) and a tree T=( V, F) with E∩ F=∅ such that G+ T=( V, F∪ E) is 2-edge-connected, we consider the problem of finding a smallest 2-edge-connected spanning subgraph ( V, F∪ E′) of G+ T containing T. The problem, which is known to be NP-hard, admits a 2-approximation algorithm. However, obtaining a factor better than 2 for this problem has been one of the main open problems in the graph augmentation problem. In this paper, we show that the problem is (1.875+ ε)-approximable in O( n 1/2 m+ n 2) time for any constant ε>0, where n=| V| and m=| E∪ F|.

A Unifying Augmentation Algorithm for Two-Edge Connectivity and Biconnectivity

Given an undirected graph G and two vertex subsets H1 and H2, the bi-level augmentation problem is that of adding to G the smallest number of edges such that the resulting graph contains two internally vertex-disjoint paths between every pair of vertices in H1 and two edge-disjoint paths between every pair of vertices in H2. We present an algorithm to solve this problem in linear time. By properly setting H1 and H2, this augmentation algorithm subsumes existing optimal algorithms for several graph augmentation problems.

On Triangulating Planar Graphs under the Four-Connectivity Constraint

Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included.

Comparability graph augmentation for some multiprocessor scheduling problems

A comparability graph is a graph which admits a transitive orientation. In this paper we consider the problem of augmenting a graph to a comparability graph in such a way that the maximum weight of its cliques is minimum. The problem is equivalent to a multiprocessor scheduling problem and to the interval coloring problem; and in the unweighted case also to the chromatic number problem. In the general case, the problem is NP-hard in the strong sense even on some very simple types of perfect graphs. We give complexity and approximation results for two subclasses of perfect graphs, namely for split graphs and stars of cliques, for which the problem still remains intractable but admits efficient estimations.

Approximation Algorithms for Graph Augmentation

The problem of increasing both edge and vertex connectivity of a graph at an optimal cost is studied. Since the general problem is NP-hard, we focus on efficient approximation schemes that come within a constant factor from the optima]. Previous algorithms either do not take edge costs into consideration, or they run slower than our algorithm. Our algorithm takes as input an undirected graph G 0 = ( V, E 0) on n vertices, that is not necessarily connected, and a set Feasible of m weighted edges on V, and outputs a subset Aug of edges which when added to G 0 make it two-connected. The weight of Aug, when G 0 is initially connected, is no more than twice the weight of the least weight subset of edges of Feasible that increases the connectivity to two. The running time of our algorithm is O( m + n log n). As a consequence of our results, we can find an approximation to the least-weight two-connected spanning subgraph of a two-connected weighted graph.

Causal augmentation of bond graphs with algebraic loops

An algorithm for the causal augmentation of bond graphs with algebraic loops is given. Unlike previous algorithms, this involves bond graph—rather than equation— manipulation. A number of illustrative examples are given.

Finding Augmented-Set Bases

The problem of finding a minimum-cost, augmented-set basis is NP-complete. In this paper we show that this problem is not approximable. That is, if ${\text{P}} \ne {\text{NP}}$, then no constants c and d exist so that $A \leqq c{\text{ASB}} + d$, where A is the cost provided by a polynomial-time approximation algorithm and ASB is the optimal cost. We also provide a brief characterization of the cost functions for which this result remains valid. The proof technique used in the augmented-set basis problem is applied directly to other NP-complete problems, such as several graph augmentation and deletion problems, to show that they are also not approximable.

Approximation Algorithms for Several Graph Augmentation Problems

Graph augmentation problems on a weighted graph involve determining a minimum-cost set of edges to add to a graph to satisfy a specified property, such as biconnectivity, bridge-connectivity or strong connectivity. These augmentation problems are shown to be NP-complete in the restricted case of the graph being initially connected. Approximation algorithms with favorable time complexity are presented and shown to have constant worst-case performance ratios.

ボンドグラフにおけるストローク・矢印の適切な割りつけ方

ボンドグラフにおけるストローク・矢印の適切な割りつけ方