Computing the isoperimetric number of a graph

Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 167–172, May–June, 1994     

A depth first search algorithm to generate the family of maximal independent sets of a graph lexicographically

In this paper we present a depth first search algorithm to generate the family of maximal independent sets of an undirected graph lexicographically. Extensive computational experience on more than 1000 randomly generated graphs ranging from 20 to 220 vertices and from 20% to 90% density has shown that the proposed algorithm is (a) two to fifteen times faster than the Bron and Kerbosch algorithm and (b) at least three times faster than the algorithm of Tsukiyamaet al. and becomes increasingly more efficient as both the density and size of the graph increase. A further characteristic of the proposed algorithm is that it employs four one-dimensional arrays of working computer memory only, each of which has length equal to the number of vertices of the graph.     

Computing the hyperbolicity constant of a cubic graph

In this paper we obtain information about the hyperbolicity constant of cubic graphs. They are a very interesting class of graphs with many applications; furthermore, they are also very important in the study of Gromov hyperbolicity, since for any graph G with bounded maximum degree there exists a cubic graph G* such that G is hyperbolic if and only if G* is hyperbolic. We find some characterizations for the cubic graphs which have small hyperbolicity constants, i.e. the graphs which are like trees (in the Gromov sense). Besides, we obtain bounds for the hyperbolicity constant of the complement graph of a cubic graph; our main result of this kind says that for any finite cubic graph G which is not isomorphic either to K₄ or to K_{3, 3}, the inequalities 5k/4≤δ (?)≤3k/2 hold, if k is the length of every edge in G.     

Computing in the RAIN: a reliable array of independent nodes

The RAIN project is a research collaboration between Caltech and NASA-JPL on distributed computing and data-storage systems for future spaceborne missions. The goal of the project is to identify and develop key building blocks for reliable distributed systems built with inexpensive off-the-shelf components. The RAIN platform consists of a heterogeneous cluster of computing and/or storage nodes connected via multiple interfaces to networks configured in fault-tolerant topologies. The RAIN software components run in conjunction with operating system services and standard network protocols. Through software-implemented fault tolerance, the system tolerates multiple node, link, and switch failures, with no single point of failure. The RAIN-technology has been transferred to Rainfinity, a start-up company focusing on creating clustered solutions for improving the performance and availability of Internet data centers. In this paper, we describe the following contributions: 1) fault-tolerant interconnect topologies and communication protocols providing consistent error reporting of link failures, 2) fault management techniques based on group membership, and 3) data storage schemes based on computationally efficient error-control codes. We present several proof-of-concept applications: a highly-available video server, a highly-available Web server, and a distributed checkpointing system. Also, we describe a commercial product, Rainwall, built with the RAIN technology     

Parameterizing cut sets in a graph by the number of their components

For a connected graph G=(V,E), a subset U⊆V is a disconnected cut if U disconnects G and the subgraph G[U] induced by U is disconnected as well. A cut U is a k-cut if G[U] contains exactly k(≥1) components. More specifically, a k-cut U is a (k,?)-cut if V?U induces a subgraph with exactly ?(≥2) components. The Disconnected Cut problem is to test whether a graph has a disconnected cut and is known to be NP-complete. The problems k-Cut and (k,?)-Cut are to test whether a graph has a k-cut or (k,?)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k,?)-Cut is in P for k=1 and any fixed constant ?≥2, while it is NP-complete for any fixed pair k,?≥2. We then prove that k-Cut is in P for k=1 and NP-complete for any fixed k≥2. On the other hand, for every fixed integer g≥0, we present an FPT algorithm that solves (k,?)-Cut on graphs of Euler genus at most g when parameterized by k+?. By modifying this algorithm we can also show that k-Cut is in FPT for this graph class when parameterized by k. Finally, we show that Disconnected Cut is solvable in polynomial time for minor-closed classes of graphs excluding some apex graph.     

Enumeration of all minimum feedback edge sets in a directed graph

The purpose of the present paper is to propose an efficient algorithm to enumerate all the minimum feedback edge sets of a given directed graph. The algorithm has been developed in two distinct phases, i.e. all the directed circuits of the given graph are generated at the first phase and then the minimum feedback edge sets are determined from the information of the directed circuits. The proposed algorithm offers easy programmability on a digital computer, and it seems economic in terms of computation time and required storage space.     

Breaking symmetries in graph search with canonizing sets

There are many complex combinatorial problems which involve searching for an undirected graph satisfying given constraints. Such problems are often highly challenging because of the large number of isomorphic representations of their solutions. This paper introduces effective and compact, complete symmetry breaking constraints for small graph search. Enumerating with these symmetry breaks generates all and only non-isomorphic solutions. For small search problems, with up to 10 vertices, we compute instance independent symmetry breaking constraints. For small search problems with a larger number of vertices we demonstrate the computation of instance dependent constraints which are complete. We illustrate the application of complete symmetry breaking constraints to extend two known sequences from the OEIS related to graph enumeration. We also demonstrate the application of a generalization of our approach to fully-interchangeable matrix search problems.     

Generation of all possible trees of a graph in independent groups

A special graph, the PS graph, is introduced and an algorithm is developed to generate all trees of such graphs. It is proved that for any given graph, $\text{G}$ there exists certain number of PS graphs, obtained from $\text{G}$, such that the collection of all trees of all such PS graph span all trees of $\text{G}$ with no duplication. In addition to a number of properties of PS graphs indicated, the procedure seems very useful for topological analysis and design of networks, or any other types of systems that can be represented by a linear graph.     

Computing the generalized aspect graph for objects with movingparts

Algorithms for computing the aspect graph representation are generalized to include a larger, more realistic domain of objects known as articulated assemblies those objects composed of rigid parts with articulated connections allowed between parts. The generalization suggests two slightly different representations: one that directly summarizes the possible general views of the object and another (hierarchical) form summarizing the possible general configurations and their respective views. Algorithms are outlined for computing both representations. The generalized aspect graphs of assemblies formed using translational connections are examined     

Large independent sets in random regular graphs

We present algorithmic lower bounds on the size _sd$s_d$ of the largest independent sets of vertices in random d$d$-regular graphs, for each fixed d≥3$d\ge 3$. For instance, for d=3$d=3$ we prove that, for graphs on n$n$ vertices, _sd≥0.43475n$s_d\ge 0.43475n$ with probability approaching one as n$n$ tends to infinity.     

On disjoint maximal independent sets in graphs

An old problem in graph theory is to characterize the graphs that admit two disjoint maximal independent sets.     

Finding hidden independent sets in interval graphs

We design efficient competitive algorithms for discovering hidden information using few queries. Specifically, consider a game in a given set of intervals (and their implied interval graph G) in which our goal is to discover an (unknown) independent set X by making the fewest queries of the form “Is point p covered by an interval in X?” Our interest in this problem stems from two applications: experimental gene discovery with PCR technology and the game of Battleship (in a 1-dimensional setting). We provide adaptive algorithms for both the verification scenario (given an independent set, is it X?) and the discovery scenario (find X without any information). Under some assumptions, these algorithms use an asymptotically optimal number of queries in every instance.     

Computing median and antimedian sets in median graphs

The median (antimedian) set of a profile π=(u ₁,…,u _k ) of vertices of a graph G is the set of vertices x that minimize (maximize) the remoteness ∑ _i d(x,u _i ). Two algorithms for median graphs G of complexity O(n idim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes.     

An efficient parallel algorithm for computing a large independent set in a planar graph

Let α(G) denote the independence number of a graphG, that is the maximum number of pairwise independent vertices inG. We present a parallel algorithm that computes in a planar graphG = (V, E), an independent set \(I \subseteq V\) such that ¦I¦≥ α (G)/2. The algorithm runs in timeOlog² n) and requires a linear number of processors. This is achieved by denning a new set of reductions that can be executed “locally” and simultaneously; furthermore, it is shown that a constant fraction of the vertices in the graph are reducible. This is the best known approximation scheme when the number of processors available is linear; parallel implementation of known sequential algorithms requires many more processors.     

An efficient parallel algorithm for computing a large independent set in a planar graph

Let (G) denote the independence number of a graphG, that is the maximum number of pairwise independent vertices inG. We present a parallel algorithm that computes in a planar graphG = (V, E), an independent set such that ¦I¦ (G)/2. The algorithm runs in timeOlog² n) and requires a linear number of processors. This is achieved by denning a new set of reductions that can be executed locally and simultaneously; furthermore, it is shown that a constant fraction of the vertices in the graph are reducible. This is the best known approximation scheme when the number of processors available is linear; parallel implementation of known sequential algorithms requires many more processors.Joseph Naor was supported by Contract ONR N00014-88-K-0166. Most of this work was done while he was a post-doctoral fellow at the Department of Computer Science, University of Southern California, Los Angeles, CA 90089-0782, USA.     

Computing the aspect graph for line drawings of polyhedral objects

An algorithm for computing the aspect graph for polyhedral objects is described. The aspects graph is a representation of three-dimensional objects by a set of two-dimensional views. The set of viewpoints on the Gaussian sphere is partitioned into regions such that in each region the qualitative structure of the line drawing remains the same. At the boundaries between adjacent regions are the accidental viewpoints where the structure for the line drawing changes. It is shown that for polyhedral objects there are two fundamental visual events: (1) the projections of an edge and a vertex coincide; and (2) the projections of three nonadjacent edges intersect at a point. The geometry of the object is reflected in the locus of the accidental viewpoints. The algorithm computes the partition together with a representative view for each region of the partition     

Parallel algorithms for finding a near-maximum independent set of acircle graph

A parallel algorithm for finding a near-maximum independent set in a circle graph is presented. An independent set in a graph is a set of vertices, no two of which are adjacent. A maximum independent set is an independent set whose cardinality is the largest among all independent sets of a graph. The algorithm is modified for predicting the secondary structure in ribonucleic acids (RNA). The proposed system, composed of an n neural network array (where n is the number of edges in the circle graph of the number of possible base pairs), not only generates a near-maximum independent set but also predicts the secondary structure of ribonucleic acids within several hundred iteration steps. The simulator discovered several solutions which are more stable structures, in a sequence of 359 bases from the potato spindle tuber viroid, than previously proposed structures.