Space-time tradeoffs in negative cycle detection - An empirical analysis of the Stressing Algorithm

This paper discusses space-time tradeoffs associated with algorithms for the problem of detecting negative cost cycles in networks (NCCD). NCCD is one of the more ubiquitous problems in computer science and operations research, with applications ranging from program verification and real-time scheduling to image segmentation and shortest path identification. Typical algorithmic analysis, whether theoretical or empirical, focuses almost exclusively on the running time of the algorithm. However, there exist applications in which space is just as important a parameter as time is. This is especially true when the problem instances are very large, as is the case in program verification. Consequently, an algorithm that minimizes running time while ignoring the space overhead may be impractical. In this paper, we analyze a number of the more common algorithms for NCCD from the perspectives of both time and space, with a view towards providing a space-time tradeoff for the practitioner. All the algorithms discussed in this paper (with the exception of Network Simplex) run in O(m·n) time on a network with m arcs and n vertices; however, their space requirements range from O(1) (Stressing Algorithm) to Ω(n) (all the Bellman-Ford and Network Simplex variants). Our empirical results demonstrate that in the cases where space is paramount, the Stressing Algorithm is a useful alternative to the Bellman-Ford variants.     

An Incremental Algorithm for the Maximum Flow Problem

An incremental algorithm may yield an enormous computational time saving to solve a network flow problem. It updates the solution to an instance of a problem for a unit change in the input. In this paper we have proposed an efficient incremental implementation of maximum flow problem after inserting an edge in the network G. The algorithm has the time complexity of O((n)² m), where n is the number of affected vertices and m is the number of edges in the network. We have also discussed the incremental algorithm for deletion of an edge in the network G.     

Cycles through a prescribed vertex set in N-connected graphs

A well-known result of Dirac (Math. Nachr. 22 (1960) 61) says that given n vertices in an n-connected G, G has a cycle through all of them. In this paper, we generalize Dirac's result as follows:Given at most vertices in an n-connected graph G when n3 and , then G has a cycle through exactly n vertices of them.This improves the previous known bound given by Kaneko and Saito (J. Graph Theory 15(6) (1991) 655).     

New scaling algorithms for the assignment and minimum mean cycle problems

In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recentauction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes,m arcs, and integer arc costs bounded byC, the algorithm runs in O( m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under thesimilarity assumption, i.e.,C = O(n ^k ) for somek. We next consider the minimum mean cycle problem. Themean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. Theminimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O( m lognC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.     

Generalizing Pancyclic and <Emphasis Type="Italic">k</Emphasis>-Ordered Graphs

Given positive integers kmn, a graph G of order n is (k,m)-pancyclic if for any set of k vertices of G and any integer r with mrn, there is a cycle of length r containing the k vertices. Minimum degree conditions and minimum sum of degree conditions of nonadjacent vertices that imply a graph is (k,m)-pancylic are proved. If the additional property that the k vertices must appear on the cycle in a specified order is required, then the graph is said to be (k,m)-pancyclic ordered. Minimum degree conditions and minimum sum of degree conditions for nonadjacent vertices that imply a graph is (k,m)-pancylic ordered are also proved. Examples showing that these constraints are best possible are provided.Acknowledgments. The authors would like to thank the referees for their careful reading of the paper and their useful suggestions.Final version received: January 26, 2004     

A Note on the Robust 1-Center Problem on Trees

We consider the robust 1-center problem on trees with uncertainty in vertex weights and edge lengths. The weights of the vertices and the lengths of the edges can take any value in prespecified intervals with unknown distribution. We show that this problem can be solved in O(n ³ logn) time thus improving on Averbakh and Berman's algorithm with time complexity O(n ⁶). For the case when the vertices of the tree have weights equal to 1 we show that the robust 1-center problem can be solved in O(nlogn) time, again improving on Averbakh and Berman's time complexity of O(n ² logn).     

Constructing disjoint paths on expander graphs

In a typical parallel or distributed computation model processors are connected by a spars interconnection network. To establish open-line communication between pairs of processors that wish to communicate interactively, a set of disjoint paths has to be constructed on the network. Since communication needs vary in time, paths have to be dynamically constructed and destroyed.We study the complexity of constructing disjoint paths between given pairs of vertices on expander interconnection graphs. These graphs have been shown before to possess desirable properties for other communication tasks.We present a sufficient condition for the existence ofKn ^Q edge-disjoint paths connecting any set ofK pairs of vertices on an expander graph, wheren is the number of vertices and<1 is some constant. We then show that the computational problem of constructing these paths lies in the classes Deterministic-P and Random-P C.Furthermore, we show that the set of paths can be constructed in probabilistic polylog time in the parallel-distributed model of computation, in which then participating processors reside in the nodes of the communication graph and all communication is done through edges of the graph. Thus, the disjoint paths are constructed in the very computation model that uses them.Finally, we show how to apply variants of our parallel algorithms to find sets ofvertex-disjoint paths when certain conditions are satisfied.Supported in part by a Weizmann fellowship and by contract ONR N00014-85-C-0731.     

Cycles in weighted graphs

A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n–1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n–1). This generalizes, to weighted graphs, a classical result of Erds and Gallai [4].     

A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles

The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n ^2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n ^2K+3) algorithm for testing membership inG This work was supported by NSF grant ENG75-00568 to Cornell University. Some of the results of this paper have appeared in Hsu's unpublished Ph.D. dissertation [9].     

Minimal Degree and (k, m)-Pancyclic Ordered Graphs

Given positive integers k m n, a graph G of order n is (k, m)-pancyclic ordered if for any set of k vertices of G and any integer r with m r n, there is a cycle of length r encountering the k vertices in a specified order. Minimum degree conditions that imply a graph of sufficiently large order n is (k, m)-pancylic ordered are proved. Examples showing that these constraints are best possible are also provided.     

On the Nullity of Bipartite Graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. We obtain some lower bounds for the nullity of graphs and we then find the nullity of bipartite graphs with no cycle of length a multiple of 4 as a subgraph. Among bipartite graphs on n vertices, the star has the greatest nullity (equal to n − 2). We generalize this by showing that among bipartite graphs with n vertices, e edges and maximum degree Δ which do not have any cycle of length a multiple of 4 as a subgraph, the greatest nullity is . G. R. Omidi: This research was in part supported by a grant from IPM (No.87050016).     

Graphs S(n, k) and a Variant of the Tower of Hanoi Problem

For any n 1 and any k 1, a graph S(n, k) is introduced. Vertices of S(n, k) are n-tuples over {1, 2,. . . k} and two n-tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S(n, 3) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S(n, k). Together with a formula for the distance, this result is used to compute the distance between two vertices in O(n) time. It is also shown that for k 3, the graphs S(n, k) are Hamiltonian.     

A lower bound for the optimal crossing-free Hamiltonian cycle problem

Consider a drawing in the plane ofK _n , the complete graph onn vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing ofK _n . If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let (n) represent the maximum number of cfhc's of any drawing ofK _n , and (n) the maximum number of cfhc's of any rectilinear drawing ofK _n . The problem of determining (n) and (n), and determining which drawings have this many cfhc's, is known as the optimal cfhc problem. We present a brief survey of recent work on this problem, and then, employing a recursive counting argument based on computer enumeration, we establish a substantially improved lower bound for (n) and (n). In particular, it is shown that (n) is at leastk × 3.2684 ⁿ . We conjecture that both (n) and (n) are at mostc × 4.5 ⁿ .This research, part of which was conducted at Queen's University, was supported by an N.S.E.R.C. postgraduate scholarship.     

Tight bounds for christofides' traveling salesman heuristic

Christofides [1] proposes a heuristic for the traveling salesman problem that runs in polynomial time. He shows that when the graphG = (V, E) is complete and the distance matrix defines a function onV × V that is metric, then the length of the Hamiltonian cycle produced by the heuristic is always smaller than 3/2 times the length of an optimal Hamiltonian cycle. The purpose of this note is to refine Christofides' worst-case analysis by providing a tight bound for everyn 3, wheren is the number of vertices of the graph. We also show that these bounds are still tight when the metric is restricted to rectilinear distances, or to Euclidean distances for alln 6.This work was supported, in part, by NSF Grant ENG 75-00568 to Cornell University. This work was done when the authors were affiliated with the Center for Operations Research and Econometrics, University of Louvain, Belgium.     

Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph

The existence problem for a Hamiltonian cycle decomposition of (the so called cocktail party graph) with a dihedral automorphism group acting sharply transitively on the vertices is completely solved. Such Hamiltonian cycle decompositions exist for all even n while, for n odd, they exist if and only if the following conditions hold: (i) n is not a prime power; (ii) there is a suitable ? such that (mod 2^?) for all prime factors p of n and the number of the prime factors (counted with their respective multiplicities) such that (mod ) is even. Thus in particular one has a dihedral Hamiltonian cycle decomposition of the cocktail party graph on 8k vertices for all k, while it is known that no such cyclic Hamiltonian cycle decomposition exists. Finally, this paper touches on a recently introduced symmetry requirement by proving that there exists a dihedral and symmetric Hamiltonian cycle system of if and only if (mod 4).     

A Construction of Almost Steiner Systems

Let n, k, and t be integers satisfying . A Steiner system with parameters t, k, and n is a k‐uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for . In this note we prove that for every and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k‐uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.     

On The Number Of Sets Of Cycle Lengths

A set S of integers is called a cycle set on {1, 2, . . .,n} if there exists a graph G on n vertices such that the set of lengths of cycles in G is S. Erds conjectured that the number of cycle sets on {1, 2, . . .,n} is o(2 ⁿ ). In this paper, we verify this conjecture by proving that there exists an absolute constant c 0.1 such that the number of cycle sets on {1, 2, . . .,n} is .     

Broadcasting from multiple originators

We begin an investigation of broadcasting from multiple originators, a variant of broadcasting in which any k vertices may be the originators of a message in a network of n vertices. The requirement is that the message be distributed to all n vertices in minimum time. A minimumk-originator broadcast graph is a graph on n vertices with the fewest edges such that any subset of k vertices can broadcast in minimum time. B_k(n) is the number of edges in such a graph. In this paper, we present asymptotic upper and lower bounds on B_k(n). We also present an exact result for the case when . We also give an upper bound on the number of edges in a relaxed version of this problem in which one additional time unit is allowed for the broadcast.     

A Network Improvement Problem Under Different Norms

In this paper, we first consider a network improvement problem, called vertex-to-vertices distance reduction problem. The problem is how to use a minimum cost to reduce lengths of the edges in a network so that the distances from a given vertex to all other vertices are within a given upper bound. We use l , l ₁ and l ₂ norms to measure the total modification cost respectively. Under l norm, we present a strongly polynomial algorithm to solve the problem, and under l ₁ or weighted l ₂ norm, we show that achieving an approximation ratio O(log(|V|)) is NP-hard. We also extend the results to the vertex-to-points distance reduction problem, which is to reduce the lengths of edges most economically so that the distances from a given vertex to all points on the edges of the network are within a given upper bound.