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.
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.
1019520
Vertex Partitioning Problems On Partial k-Trees
We describe a general approach to obtain polynomial-time algorithms over partial k-trees for graph problems in which the vertex set is to be partitioned in some way. We encode these problems with formulae of the Extended Monadic Second-order (or EMS) logic. Such a formula can be translated into a polynomial-time algorithm automatically. We focus on the problem of partitioning a partial k-tree into induced subgraphs isomorphic to a fixed pattern graph; a distinct algorithm is derived for each pattern graph and each value of k. We use a “pumping lemma” to show that (for some pattern graphs) this problem cannot be encoded in the “ordinary” Monadic Second-order logic—from which a linear-time algorithm over partial k-trees would be obtained. Hence, an EMS formula is in some sense the strongest possible. As a further application of our general approach, we derive a polynomial-time algorithm to determine the maximum number of co-dominating sets into which the vertices of a partial k-tree can be partitioned. (A co-dominating set of a graph is a dominating set of its complement graph).
17407466
Point-Based Approaches to Qualitative Temporal Reasoning
We are interested in the problem of qualitative, point-based temporal reasoning. That is, given a set of assertions concerning the relative positions of points in time, we are interested in compiling the assertions into a representation that supports efficient reasoning, determining consistency, computing the strongest entailed relation between any two points and updating the set of assertions. We begin by considering a general set of operations and their corresponding algorithms, applicable to general point-based temporal domains. We also consider special-purpose reasoners, which may be expected to perform well in an application domain with a restricted structure. Lastly we consider reasoners which incorporate restricted structures into a more general reasoning framework. Finally, we present instances of each such approach and give a comparison of their performance on random data sets.
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.
2981830
Characterizing the Complexity of Subgraph Isomorphism for Graphs of Bounded Path-Width
We show that the complexity of the subgraph isomorphism problem on graphs of bounded path-width is inherently dependent on the connectivity of the source and target graphs. In particular, for the problem of determining whether a source graph G of path-width k is a subgraph of a target graph H of path-width k, we present an O(n3) algorithm for G and H both k-connected, for n the sum of the sizes of the graphs, and NP-completeness results for connectivity less than k. In previous polynomial-time algorithms, the degree of the polynomial in the running time was a function of k. In contrast, we show that when neither G nor H is k-connected, the problem becomes NP-complete. The same result also holds if one of the graphs has at least k vertices of unbounded degree. Since bounded path-width graphs are also bounded tree-width graphs, our hardness results immediately extend to this larger class. A further NP-completeness result applies to the situation in which both graphs have tree-width k, but only the target graph is k-connected. This provides a complete characterization of the subgraph isomorphism problem on bounded tree-width graphs, thus answering an open question of Matousek and Thomas.
10440176
Task assignment with imprecise information for real-time operation in a supply chain
Abstract Real-time operation is often required in a supply chain, where it is desirable to re-optimize the schedules each time a new order is submitted. An organization in a supply chain may not be able to perform all the tasks it has received completely due to lack of resources. It has to delegate those subtasks which it cannot perform to its suppliers in the supply chain. In a practical situation, the suppliers may be parts of more than one supply chain. Hence the organization does not have a complete control over the schedules of its suppliers. There should be coordination within and across organizations for producing the schedules. Usually, multiple objectives (or criteria) like quality, cost, and duration are associated with a task. Preference ratings, assigned by the end-user or the organization itself, may be associated with these criteria. The preference ratings may be specified in an imprecise manner by using ordinal values or linguistic terms. The assignment of the tasks to the suppliers, which is a multiobjective decision making problem, has to be done quickly for real-time operation. In this paper, a supply chain is first modeled using a multiagent framework and an architecture for the agents located at every organization in the chain is presented. We then propose a real-time scheduler that can schedule new orders with soft real-time deadlines. An algorithm for the task assignment using the fuzzy set approach for the real-time operation in a supply chain is then proposed. Comparison of the algorithm versus other commonly used heuristic methods for task assignment is done in a simulated environment. The experimental results show significant performance gains for a large range of task arrival rates. However, the performance deteriorates for very high arrival rates. In order to overcome this limitation, we present a hybrid algorithm that augments the fuzzy technique and does not suffer from performance degradation even for high arrival rates.
6555974
Lazy constraints and SAT heuristics for proof-based abstraction
SAT proof analysis techniques have been used recently with BMC in order to generate proof-based abstractions that preserve the correctness of the property up to a certain depth. In this paper, we propose various techniques for handling BMC constraints that are targeted at reducing the size of the abstract model while preserving bounded correctness. We propose lazy constraints, where the idea is to delay propagation of certain implied values without modifying a standard SAT solver. We propose different methods for automatic use of lazy constraints in SAT-based BMC. Their use can be regarded as a heuristic, which typically leads to significant reduction in the size of the abstract model, frequently accompanied by an overall performance improvement. We also use SAT proof analysis to loosen the user-specified environmental constraints, in order to generate smaller abstract models for conservative verification. We briefly describe other SAT-based heuristics for reducing the size of the abstract model. We have implemented our techniques in a prototype verification platform, and demonstrate their effectiveness on several large industry designs.
854648
Approximation algorithms for low-distortion embeddings into low-dimensional spaces
We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. Among other results, we give an O(√n)-approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. We give an improved Õ(n1/3) approximation for the case of metrics generated by unweighted trees. This is the first result of this type.
32901959
Efficient embeddings of ternary trees into hypercubes
In this paper we present efficient graph embeddings for complete k-ary trees into boolean hypercubes. In particular, we describe an efficient embedding of a complete ternary tree (k = 3) of height h into a hypercube, which achieves dilation 3 and expansion Θ(1.0104...h). The previously best-known result is dilation 2 and expansion Θ(1.333...h). Our embedding achieves exponentially better expansion at the cost of an increase of 1 in the dilation. We also describe efficient embeddings of k-ary trees into hypercubes when k = 2p * 3q for some p, q > 0 such that the embeddings achieve small constant dilation.
8374849
Quorum placement in networks to minimize access delays
A quorum system is a family of sets (themselves called quorums), each pair of which intersect. In many distributed algorithms, the basic unit accessed by a client is a quorum of nodes. Such algorithms are used for applications such as mutual exclusion, data replication, and dissemination of information. However, accessing spread-out quorums causes access delays that we would like to minimize. Furthermore, every member of the quorum incurs processing load to handle quorum accesses by clients.In this paper we study the problem of placing quorums in a physical network so as to minimize the delay that clients incur by accessing quorums, and while respecting each physical node's capacity (in terms of the load of client requests it can handle). We provide approximation algorithms for this problem for two natural measures of delay (the max-delay and total-delay). All our algorithms ensure that each node's load is within a constant factor of its capacity, and minimize delay to within a constant factor of the optimal delay for all capacity-respecting solutions. We also provide better approximations for several well-known quorum systems.