Introduction: New approaches to linear programming

This issue ofAlgorithmica present papers on various aspects of nonlinear methods for solving linear programming problems, inspired by the work of Karmarkar. This introduction describes some of these aspects and briefly mentions other recent developments in the field. A bibliography of recent articles is included.     

Algorithms and complexity analysis for some flow problems

Several network-flow problems with additional constraints are considered. They are all special cases of the linear-programming problem and are shown to be -complete. It is shown that the existence of a strongly polynomial-time algorithm for any of these problems implies the existence of such an algorithm for the general linear-programming problem. On the positive side, strongly polynomial algorithms for some parametric flow problems are given, when the number of parameters is fixed. These algorithms are applicable to constrained flow problems when the number of additional constraints is fixed.Work on the paper was done while at Stanford University and IBM Almaden Research Center. This research was partially supported by NSF PYI Grant CCR-8858097.     

Extending NC and RNC algorithms

A technique is presented by which NC and RNC algorithms for some problems can be extended into NC and RNC algorithms, respectively, that solve more general parametric problems. The technique is demonstrated on explicit bounded degree circuits. Applications include parametric extensions of the shortest-path and spanning-tree problems and, in particular, the minimum-ratio-cycle problem, showing all these problems are in NC.     

A logic for reasoning about probabilities

We consider a language for reasoning about probability which allows us to make statements such as “the probability of E₁ is less than ” and “the probability of E₁ is at least twice the probability of E₂,” where E₁ and E₂ are arbitrary events. We consider the case where all events are measurable (i.e., represent measurable sets) and the more general case, which is also of interest in practice, where they may not be measurable. The measurable case is essentially a formalization of (the propositional fragment of) Nilsson's probabilistic logic. As we show elsewhere, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions. In both cases, we provide a complete axiomatization and show that the problem of deciding satisfiability is NP-complete, no worse than that of propositional logic. As a tool for proving our complete axiomatizations, we give a complete axiomatization for reasoning about Boolean combinations of linear inequalities, which is of independent interest. This proof and others make crucial use of results from the theory of linear programming. We then extend the language to allow reasoning about conditional probability and show that the resulting logic is decidable and completely axiomatizable, by making use of the theory of real closed fields.     

Is binary encoding appropriate for the problem-language relationship?

It is proved that there exist encoding schemes which are arbitrarily as efficient as the binary encoding (in terms of compactness and arithmetic operations), with respect to which Khachiyan's algorithm for Linear Programming is exponential. This constitutes an objection to the standard translation of problems into languages via the binary encoding.     

Cyclic ordering is NP-complete

The cyclic ordering problem is to recognize whether a collection of cyclically ordered triples of elements of a set T is derived from an arrangement of all the elements of T on a circle. This problem is shown to be NP-complete.     

Consistent selectivity estimation via maximum entropy

Cost-based query optimizers need to estimate the selectivity of conjunctive predicates when comparing alternative query execution plans. To this end, advanced optimizers use multivariate statistics to improve information about the joint distribution of attribute values in a table. The joint distribution for all columns is almost always too large to store completely, and the resulting use of partial distribution information raises the possibility that multiple, non-equivalent selectivity estimates may be available for a given predicate. Current optimizers use cumbersome ad hoc methods to ensure that selectivities are estimated in a consistent manner. These methods ignore valuable information and tend to bias the optimizer toward query plans for which the least information is available, often yielding poor results. In this paper we present a novel method for consistent selectivity estimation based on the principle of maximum entropy (ME). Our method exploits all available information and avoids the bias problem. In the absence of detailed knowledge, the ME approach reduces to standard uniformity and independence assumptions. Experiments with our prototype implementation in DB2 UDB show that use of the ME approach can improve the optimizer’s cardinality estimates by orders of magnitude, resulting in better plan quality and significantly reduced query execution times. For almost all queries, these improvements are obtained while adding only tens of milliseconds to the overall time required for query optimization.     

Extending NC and RNC algorithms

A technique is presented by which NC and RNC algorithms for some problems can be extended into NC and RNC algorithms, respectively, that solve more general parametric problems. The technique is demonstrated on explicit bounded degree circuits. Applications include parametric extensions of the shortest-path and spanning-tree problems and, in particular, the minimum-ratio-cycle problem, showing all these problems are in NC.     

Outperforming LRU with an adaptive replacement cache algorithm

The self-tuning, low-overhead, scan-resistant adaptive replacement cache algorithm outperforms the least-recently-used algorithm by dynamically responding to changing access patterns and continually balancing between workload recency and frequency features. Caching, a fundamental metaphor in modern computing, finds wide application in storage systems, databases, Web servers, middleware, processors, file systems, disk drives, redundant array of independent disks controllers, operating systems, and other applications such as data compression and list updating. In a two-level memory hierarchy, a cache performs faster than auxiliary storage, but it is more expensive. Cost concerns thus usually limit cache size to a fraction of the auxiliary memory's size.     

Introduction: New approaches to linear programming

This issue ofAlgorithmica present papers on various aspects of nonlinear methods for solving linear programming problems, inspired by the work of Karmarkar. This introduction describes some of these aspects and briefly mentions other recent developments in the field. A bibliography of recent articles is included.