Richard M. Karp

Reducibility Among Combinatorial Problems

Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.

An

The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to

n^{5/2}

(m + n)\sqrt n

Algorithm for Maximum Matchings in Bipartite Graphs

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LogP: towards a realistic model of parallel computation

A vast body of theoretical research has focused either on overly simplistic models of parallel computation, notably the PRAM, or overly specific models that have few representatives in the real world. Both kinds of models encourage exploitation of formal loopholes, rather than rewarding development of techniques that yield performance across a range of current and future parallel machines. This paper offers a new parallel machine model, called LogP, that reflects the critical technology trends underlying parallel computers. it is intended to serve as a basis for developing fast, portable parallel algorithms and to offer guidelines to machine designers. Such a model must strike a balance between detail and simplicity in order to reveal important bottlenecks without making analysis of interesting problems intractable. The model is based on four parameters that specify abstractly the computing bandwidth, the communication bandwidth, the communication delay, and the efficiency of coupling communication and computation. Portable parallel algorithms typically adapt to the machine configuration, in terms of these parameters. The utility of the model is demonstrated through examples that are implemented on the CM-5.

The traveling-salesman problem and minimum spanning trees: Part II

The relationship between the symmetric traveling-salesman problem and the minimum spanning tree problem yields a sharp lower bound on the cost of an optimum tour. An efficient iterative method for approximating this bound closely from below is presented. A branch-and-bound procedure based upon these considerations has easily produced proven optimum solutions to all traveling-salesman problems presented to it, ranging in size up to sixty-four cities. The bounds used are so sharp that the search trees are minuscule compared to those normally encountered in combinatorial problems of this type.

Efficient randomized pattern-matching algorithms

We present randomized algorithms to solve the following string-matching problem and some of its generalizations: Given a string X of length n (the pattern) and a string Y (the text), find the first occurrence of X as a consecutive block within Y. The algorithms represent strings of length n by much shorter strings called fingerprints, and achieve their efficiency by manipulating fingerprints instead of longer strings. The algorithms require a constant number of storage locations, and essentially run in real time. They are conceptually simple and easy to implement. The method readily generalizes to higher-dimensional patternmatching problems.

Parallel program schemata

This paper introduces a model called the parallel program schema for the representation and study of programs containing parallel sequencing. The model is related to Ianovs program schema, but extends it, both by modelling memory structure in more detail and by admitting parallel computation. The emphasis is on decision procedures, both for traditional properties, such as equivalence, and for new properties particular to parallel computation, such as determinacy and boundedness.

Topologically-aware overlay construction and server selection

A number of large-scale distributed Internet applications could potentially benefit from some level of knowledge about the relative proximity between its participating host nodes. For example, the performance of large overlay networks could be improved if the application-level connectivity between the nodes in these networks is congruent with the underlying IP-level topology. Similarly, in the case of replicated Web content, client nodes could use topological information in selecting one of multiple available servers. For such applications, one need not find the optimal solution in order to achieve significant practical benefits. Thus, these applications, and presumably others like them, do not require exact topological information and can instead use sufficiently informative hints about the relative positions of Internet hosts. In this paper, we present a binning scheme whereby nodes partition themselves into bins such that nodes that fall within a given bin are relatively close to one another in terms of network latency. Our binning strategy is simple (requiring minimal support from any measurement infrastructure), scalable (requiring no form of global knowledge, each node only needs knowledge of a small number of well-known landmark nodes) and completely distributed (requiring no communication or cooperation between the nodes being binned). We apply this binning strategy to the two applications mentioned above: overlay network construction and server selection. We test our binning strategy and its application using simulation and Internet measurement traces. Our results indicate that the performance of these applications can be significantly improved by even the rather coarse-grained knowledge of topology offered by our binning scheme.

Application-Level Multicast Using Content-Addressable Networks

Most currently proposed solutions to application-level multicast organise the group members into an application-level mesh over which a Distance-Vector routing protocol, or a similar algorithm, is used to construct source-rooted distribution trees. The use of a global routing protocol limits the scalability of these systems. Other proposed solutions that scale to larger numbers of receivers do so by restricting the multicast service model to be single-sourced. In this paper, we propose an application-level multicast scheme capable of scaling to large group sizes without restricting the service model to a single source. Our scheme builds on recent work on Content-Addressable Networks (CANs). Extending the CAN framework to support multicast comes at trivial additional cost and, because of the structured nature of CAN topologies, obviates the need for a multicast routingalg orithm. Given the deployment of a distributed infrastructure such as a CAN, we believe our CAN-based multicast scheme offers the dual advantages of simplicity and scalability.