204999323
Acyclic Directed Hypercubes may have Exponential Diameter
Abstract The diameter of a directed graph is the maximum of the lengths of the shortest paths between all pairs of vertices (if such paths exist). We prove that for every n , the edges of the n -dimensional hypercube can be acyclically oriented such that the diameter of the cube is F n + 1 , the ( n + 1)st Fibonacci number.
26737462
The Parallel Complexity of Tree Embedding Problems
Abstract The question of the sequential complexity of various tree embedding problems arises in the recent work of Robertson and Seymour on graph miners; here we consider the parallel complexity of such problems. In particular, we present two CREW PRAM algorithms: an O (log 3 n ) time randomized algorithm for determining whether there is a topological embedding of one rooted tree in another and an O (log 2 n log log n ) time deterministic algorithm for determining whether or not a rooted tree with a degree constraint is a minor of a general rooted tree. These algorithms are two examples of a general technique that can be used for solving other problems on trees.
42435460
Embeddings of k-connected graphs of pathwidth k
We present O(n^3) embedding algorithms (subgraph isomorphism and its generalizations) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the first polynomial-time algorithm for minor containment and the first O(n^c) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees, as well as an O(n^2) algorithm for subgraph isomorphism. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more efficient solutions. Finally, we describe n^k^+^O^(^1^) algorithms for solving these problems on arbitrary graphs of pathwidth at most k.
38885821
Embeddings of k-Connected Graphs of Pathwidth k
We present O(n3) embedding algorithms (generalizing subgraph isomorphism) for classes of graphs of bounded pathwidth, where n is the number of vertices in the graph. These include the first polynomialtime algorithm for minor containment and the first O(nc) algorithm (c a constant independent of k) for topological embedding of graphs from subclasses of partial k-trees. Of independent interest are structural properties of k-connected graphs of bounded pathwidth on which our algorithms are based. We also describe special cases which reduce to various generalizations of string matching, permitting more efficient solutions.
28349058
A Constructive Proof that Trees Are Well-Quasi-Ordered Under Minors
The recent proof of Robertson and Seymour that graphs are well-quasi-ordered under minors immediately implies that a number of intesting problems have polynomial time algorithms. However, partially because of their non-constructive nature, these proofs do not yield any information about the algorithms. Here we present a constructive proof that trees are well-quasi-ordered under minors. This extends the results of Murty and Russell [MR90] who give a constructive proof of Higman's Lemma. Our proof is based on transforming finite sequences of trees to ordinals. We begin by describing a transformation which carries trees to finite strings of numbers and give an ordering on these strings which preserves the minor ordering on the underlying trees. We show that in our well-quasi-ordering argument, these strings are actually over a finite alphabet. This allows us to conclude the result. We require the well-ordering of the ordinal e0 for our proof.
2367227
Parallel Algorithms For Vehicle Routing Problems
Vehicle routing problems involve the navigation of one or more vehicles through a network of locations. Locations have associated handling times as well as time windows during which they are active. The arcs connecting locations have time costs associated with them. In this paper, we consider two different problems in single vehicle routing. The first is to find least time cost routes between all pairs of nodes in a network for navigating vehicles; we call this the all pairs routing problem. We show that there is an O(log3 n) time parallel algorithm using a polynomial number of processors for this problem on a CREW PRAM. We next consider the problem in which a vehicle services all locations in a network. Here, locations can be passed through at any time but only serviced during their time window. The general problem is -complete under even fairly stringent restrictions but polynomial algorithms have been developed for some special cases. In particular, when the network is a line, there is no time cost in servicing a location, and all time windows are unbounded at either their lower or upper end, O(n2) algorithms have been developed. We show that under the same conditions, we can reduce this problem to the all pairs routing problem and therefore obtain an O(log3 n) time parallel algorithm on a CREW PRAM.
10298588
Using Statically Computed Invariants Inside the Predicate Abstraction and Refinement Loop
Predicate abstraction is a powerful technique for extracting finite-state models from often complex source code. This paper reports on the usage of statically computed invariants inside the predicate abstraction and refinement loop. The main idea is to selectively strengthen (conjoin) the concrete transition relation at a given program location by efficiently computed invariants that hold at that program location. We experimentally demonstrate the usefulness of transition relation strengthening in the predicate abstraction and refinement loop. We use invariants of the form ±x ±y ≤c where c is a constant and x,y are program variables. These invariants can be discovered efficiently at each program location using the octagon abstract domain. We observe that the abstract models produced by predicate abstraction of strengthened transition relation are more precise leading to fewer spurious counterexamples, thus, decreasing the total number of abstraction refinement iterations. Furthermore, the length of relevant fragments of spurious traces needing refinement shortens. This leads to an addition of fewer predicates for refinement. We found a consistent reduction in the total number of predicates, maximum number of predicates tracked at a given program location, and the overall verification time.
14215965
On the Covering Steiner Problem
The Covering Steiner problem is a common generalization of the k-MST and Group Steiner problems. An instance of the Covering Steiner problem consists of an undirected graph with edge-costs, and some subsets of vertices called groups, with each group being equipped with a non-negative integer value (called its requirement); the problem is to find a minimum-cost tree which spans at least the required number of vertices from every group. When all requirements are equal to 1, this is the Group Steiner problem.
2125509
Improved results for directed multicut
We give a simple algorithm for the Minimum Directed Multicut problem, and show that it gives an <i>O</i>(√<i>n</i>)-approximation. This improves on the previous approximation guarantee of <i>O</i>(√<i>n</i> log <i>k</i>) of Cheriyan, Karloff and Rabani [1], which was obtained by a more sophisticated algorithm.
9848793
Stochastic Steiner Trees Without a Root
This paper considers the Steiner tree problem in the model of two-stage stochastic optimization with recourse. This model, the focus of much recent research [11, 16, 8, 18], tries to capture the fact that many infrastructure planning problems have to be solved in the presence of uncertainty, and that we have make decisions knowing merely market forecasts (and not the precise set of demands); by the time the actual demands arrive, the costs may be higher due to inflation. 
 
In the context of the Stochastic Steiner Tree problem on a graph G = (V,E), the model can be paraphrased thus: on Monday, we are given a probability distribution π on subsets of vertices, and can build some subset EM of edges. On Tuesday, a set of terminals D materializes (drawn from the same distribution π). We now have to buy edges ET so that the set EM ∪ ET forms a Steiner tree on D. The goal is to minimize the expected cost of the solution. 
 
We give the first constant-factor approximation algorithm for this problem. To the best of our knowledge, this is the first O(1)-approximation for the stochastic version of a non sub-additive problem. In fact, algorithms for the unrooted stochastic Steiner tree problem we consider are powerful enough to solve the Multicommodity Rent-or-Buy problem, itself a topic of recent interest [3, 7, 15].
575078
An Approach for Detecting Self-propagating Email Using Anomaly Detection
This paper develops a new approach for detecting self-propagating email viruses based on statistical anomaly detection. Our approach assumes that a key objective of an email virus attack is to eventually overwhelm mail servers and clients with a large volume of email traffic. Based on this assumption, the approach is designed to detect increases in traffic volume over what was observed during the training period. This paper describes our approach and the results of our simulation-based experiments in assessing the effectiveness of the approach in an intranet setting. Within the simulation setting, our results establish that the approach is effective in detecting attacks all of the time, with very few false alarms. In addition, attacks could be detected sufficiently early so that clean up efforts need to target only a fraction of the email clients in an intranet.