Dynamic bottleneck optimization for k-edge and 2-vertex connectivity

We consider the problem of updating efficiently the minimum value b over a weighted graph, so that edges with a cost less than b induce a spanning subgraph satisfying a k-edge or 2-vertex connectivity constraint, when the cost of an edge of the graph is updated. Our results include update algorithms of almost linear time (up to poly-logarithmic factors) in the number of vertices for all considered connectivity constraints (for fixed k), and a worst case construction showing that these algorithms are almost optimal in their class.     

Quasi-Fully Dynamic Algorithms for Two-Connectivity and Cycle Equivalence

We introduce a new class of dynamic graph algorithms called quasi-fully dynamic algorithms , which are much more general than backtracking algorithms and are much simpler than fully dynamic algorithms. These algorithms are especially suitable for applications in which a certain core connected portion of the graph remains fixed, and fully dynamic updates occur on the remaining edges in the graph. We present very simple quasi-fully dynamic algorithms with O(log n) worst-case time per operation for 2-edge connectivity and O(log n) amortized time per operation for cycle equivalence. The former is deterministic while the latter is Monte-Carlo-type randomized. For 2-vertex connectivity, we give a deterministic quasi-fully dynamic algorithm with O(log ³ n) amortized time per operation. The quasi-fully dynamic algorithm we present for cycle equivalence (which has several applications in optimizing compilers) is of special interest since the algorithm is quite simple, and no special-purpose incremental or backtracking algorithm is known for this problem. Received October 26, 1998; revised October 1, 1999, and April 15, 2001.     

Lower Bounds for Fully Dynamic Connectivity Problems in Graphs

We prove lower bounds on the complexity of maintaining fully dynamic k -edge or k -vertex connectivity in plane graphs and in (k-1) -vertex connected graphs. We show an amortized lower bound of (log n / {k (log log n} + log b)) per edge insertion, deletion, or query operation in the cell probe model, where b is the word size of the machine and n is the number of vertices in G . We also show an amortized lower bound of (log n /(log log n + log b)) per operation for fully dynamic planarity testing in embedded graphs. These are the first lower bounds for fully dynamic connectivity problems. Received January 1995; revised February 1997.     

A counterexample for the proof of implication conjecture on independent spanning trees

Khuller and Schieber (1992) in [1] developed a constructive algorithm to prove that the existence of k-vertex independent trees in a k-vertex connected graph implies the existence of k-edge independent trees in a k-edge connected graph. In this paper, we show a counterexample where their algorithm fails.     

Optimal parallel algorithms for multiple updates of minimum spanning trees

Parallel updates of minimum spanning trees (MSTs) have been studied in the past. These updates allowed a single change in the underlying graph, such as a change in the cost of an edge or an insertion of a new vertex. Multiple update problems for MSTs are concerned with handling more than one such change. In the sequential case multiple update problems may be solved using repeated applications of an efficient algorithm for a single update. However, for efficiency reasons, parallel algorithms for multiple update problems must consider all changes to the underlying graph simultaneously. In this paper we describe parallel algorithms for updating an MST whenk new vertices are inserted or deleted in the underlying graph, when the costs ofk edges are changed, or whenk edge insertions and deletions are performed. For multiple vertex insertion update, our algorithm achieves time and processor bounds ofO(log n·logk) and nk/(logn·logk), respectively, on a CREW parallel random access machine. These bounds are optimal for dense graphs. A novel feature of this algorithm is a transformation of the previous MST andk new vertices to a bipartite graph which enables us to obtain the above-mentioned bounds.     

Optimal 1-edge fault-tolerant designs for ladders

A graph G∗ is 1-edge fault-tolerant with respect to a graph G, denoted by 1-EFT(G), if every graph obtained by removing any edge from G∗ contains G. A 1-EFT(G) graph is optimal if it contains the minimum number of edges among all 1-EFT(G) graphs. The kth ladder graph, L_k, is defined to be the cartesian product of the P_k and P₂ where P_n is the n-vertex path graph. In this paper, we present several 1-edge fault-tolerant graphs with respect to ladders. Some of these graphs are proven to be optimal.     

Erratum: An Approximation Algorithm for Minimum-Cost Vertex-Connectivity Problems

Abstract. There is an error in our paper ``An Approximation Algorithm for Minimum-Cost Vertex- Connectivity Problems' (Algorithmica (1997), 18:21—43). In that paper we considered the following problem: given an undirected graph and values r _ij for each pair of vertices i and j , find a minimum-cost set of edges such that there are r _ij vertex-disjoint paths between vertices i and j . We gave approximation algorithms for two special cases of this problem. Our algorithms rely on a primal—dual approach which has led to approximation algorithms for many edge-connectivity problems. The algorithms work in a series of stages; in each stage an augmentation subroutine augments the connectivity of the current solution. The error is in a lemma for the proof of the performance guarantee of the augmentation subroutine. In the case r _ij = k for all i,j , we described a polynomial-time algorithm that claimed to output a solution of cost no more than 2 H (k) times optimal, where H = 1 + 1/2 + · · · + 1/n . This result is erroneous. We describe an example where our primal—dual augmentation subroutine, when augmenting a k -vertex connected graph to a (k+1) -vertex connected graph, gives solutions that are a factor Ω(k) away from the minimum. In the case r _ij ∈ {0,1,2} for all i,j , we gave a polynomial-time algorithm which outputs a solution of cost no more than three times the optimal. In this case we prove that the statement in the lemma that was erroneous for the k -vertex connected case does hold, and that the algorithm performs as claimed.     

Approximating Biconnectivity in Parallel

Consider the following NP-hard problems: Given a graph G , find minimum 2-edge connected and 2-vertex connected subgraphs spanning all vertices of G . Over the past few years, exciting sequential algorithms for approximating such minimum subgraphs have been produced [6],[10]. The approximation factors are improved from 2 to 3/2 for both of the problems. Yet the techniques involved are all based on augmenting depth-first-search trees and no similar progress has been carried to the parallel context. This paper presents NC algorithms to achieve approximation factors of 3/2 + ε for these two problems without computing depth-first-search trees. Received June 21, 1995; revised December 24, 1996, and June 2, 1997.     

The electrical resistance of a graph captures its commute and cover times

View ann-vertex,m-edge undirected graph as an electrical network with unit resistors as edges. We extend known relations between random walks and electrical networks by showing that resistance in this network is intimately connected with thelengths of random walks on the graph. For example, thecommute time between two verticess andt (the expected length of a random walk froms tot and back) is precisely characterized by the effective resistanceR _st betweens andt: commute time=2mR _st . As a corollary, thecover time (the expected length of a random walk visiting all vertices) is characterized by the maximum resistanceR in the graph to within a factor of logn:mR<-cover time<-O(mRlogn). For many graphs, the bounds on cover time obtained in this manner are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix. In particular, we improve known bounds on cover times for high-degree graphs and expanders, and give new proofs of known results for multi-dimensional meshes. Moreover, resistance seems to provide an intuitively appealing and tractable approach to these problems.     

Solving the Euclidean bottleneck matching problem byk-relative neighborhood graphs

Given a set of pointsV in the plane, the Euclidean bottleneck matching problem is to match each point with some other point such that the longest Euclidean distance between matched points, resulting from this matching, is minimized. To solve this problem, we definek-relative neighborhood graphs, (kRNG) which are derived from Toussaint's relative neighborhood graphs (RNG). Two points are calledk-relative neighbors if and only if there are less thank points ofV which are closer to both of the two points than the two points are to each other. AkRNG is an undirected graph (V,E _r ^k ) whereE _r ^k is the set of pairs of points ofV which arek-relative neighbors. We prove that there exists an optimal solution of the Euclidean bottleneck matching problem which is a subset ofE _r ¹⁷ . We also prove that ¦E _r ^k ¦ < 18kn wheren is the number of points in setV. Our algorithm would construct a 17RNG first. This takesO(n ²) time. We then use Gabow and Tarjan's bottleneck maximum cardinality matching algorithm for general graphs whose time-complexity isO((n logn)^0.5 m), wherem is the number of edges in the graph, to solve the bottleneck maximum cardinality matching problem in the 17RNG. This takesO(n ^1.5 log^0.5 n) time. The total time-complexity of our algorithm for the Euclidean bottleneck matching problem isO(n ² +n ^1.5 log^0.5 n).This research was partially supported by a grant from the National Science Council of the Republic of China under Grant NSC-78-0408-E-007-05.     

The Dense k -Subgraph Problem

This paper considers the problem of computing the dense k -vertex subgraph of a given graph, namely, the subgraph with the most edges. An approximation algorithm is developed for the problem, with approximation ratio O(n ^δ ) , for some δ < 1/3 . Received April 29, 1997; revised June 26, 1998, and April 13, 1999.     

The spanning connectivity of folded hypercubes

A k-container of a graph G is a set of k internally disjoint paths between u and v. A k-container of G is a k∗-container if it contains all vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices, and a bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u and v from different partite sets of G for a given k. A k-connected graph (respectively, bipartite graph) G is f-edge fault-tolerant spanning connected (respectively, laceable) if G − F is w∗-connected for any w with 1 ? w ? k − f and for any set F of f faulty edges in G. This paper shows that the folded hypercube FQ_n is f-edge fault-tolerant spanning laceable if n(?3) is odd and f ? n − 1, and f-edge fault-tolerant spanning connected if n (?2) is even and f ? n − 2.     

Property Testing in Bounded Degree Graphs

Abstract. We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre-determined property from the case when it is ``far' from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure the distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown bounded-degree graph is connected, k -connected (for k>1 ), cycle-free and Eulerian. Our algorithms work in time polynomial in 1/ɛ , always accept the graph when it has the tested property, and reject with high probability if the graph is ɛ -far from having the property. For example, the 2-connectivity algorithm rejects (with high probability) any N -vertex d -degree graph for which more than ɛ dN edges need to be added in order to make the graph 2-edge-connected. In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.     

A constant approximation algorithm for the densest k-subgraph problem on chordal graphs

The densest k-subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The DkS problem is NP-hard even for special graph classes including bipartite, planar, comparability and chordal graphs, while no constant approximation algorithm is known for any of these classes. In this paper we present a 3-approximation algorithm for the class of chordal graphs. The analysis of our algorithm is based on a graph theoretic lemma of independent interest.     

Maintaining the Classes of 4-Edge-Connectivity in a Graph On-Line

Two vertices of an undirected graph are called k -edge-connected if there exist k edge-disjoint paths between them (equivalently, they cannot be disconnected by the removal of less than k edges from the graph). Equivalence classes of this relation are called classes of k -edge-connectivity or k -edge-connected components. This paper describes graph structures relevant to classes of 4 -edge-connectivity and traces their transformations as new edges are inserted into the graph. Data structures and an algorithm to maintain these classes incrementally are given. Starting with the empty graph, any sequence of q Same-4-Class? queries and n Insert-Vertex and m Insert-Edge updates can be performed in O(q + m + n log n) total time. Each individual query requires O(1) time in the worst-case. In addition, an algorithm for maintaining the classes of k -edge-connectivity (k arbitrary) in a (k-1) -edge-connected graph is presented. Its complexity is O(q + m + n) , with O(M +k ² n log (n/k)) preprocessing, where M is the number of edges initially in the graph and n is the number of its vertices. Received July 5, 1995; revised October 21, 1996.     

Forests,frames, and games: Algorithms for matroid sums and applications

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 intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.     

Forests, frames, and games: Algorithms for matroid sums and applications

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 intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.This is a revised and expanded version of a paper appearing in theProceedings of the 20th Annual ACM Symposium on Theory of Computing, 1988. This research was supported in part by National Science Foundation Grants MCS-8302648 and DCR-851191.     

Finding Maximum Induced Matchings in Subclasses of Claw-Free and <Emphasis Type="Italic">P</Emphasis><Subscript>5</Subscript>-Free Graphs,and in Graphs with Matching and Induced Matching of Equal Maximum Size

In a graph G a matching is a set of edges in which no two edges have a common endpoint. An induced matching is a matching in which no two edges are linked by an edge of G. The maximum induced matching (abbreviated MIM) problem is to find the maximum size of an induced matching for a given graph G. This problem is known to be NP-hard even on bipartite graphs or on planar graphs. We present a polynomial time algorithm which given a graph G either finds a maximum induced matching in G, or claims that the size of a maximum induced matching in G is strictly less than the size of a maximum matching in G. We show that the MIM problem is NP-hard on line-graphs, claw-free graphs, chair-free graphs, Hamiltonian graphs and r-regular graphs for r \geq 5. On the other hand, we present polynomial time algorithms for the MIM problem on (P ₅,D _m )-free graphs, on (bull, chair)-free graphs and on line-graphs of Hamiltonian graphs.     

Component connectivity of generalized Petersen graphs

Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λ_r (G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this paper we establish bounds of the r component edge connectivity of an important family of interconnection network models, the generalized Petersen graphs GP(n, k) in which n and k are relatively prime integers.     

A Randomized Linear-Work EREW PRAM Algorithm to Find a Minimum Spanning Forest

We present a randomized EREW PRAM algorithm to find a minimum spanning forest in a weighted undirected graph. On an n -vertex graph the algorithm runs in o(( log n) ^{1+ ɛ} ) expected time for any ɛ >0 and performs linear expected work. This is the first linear-work, polylog-time algorithm on the EREW PRAM for this problem. This also gives parallel algorithms that perform expected linear work on two general-purpose models of parallel computation—the QSM and the BSP.