Computability and complexity of ray tracing

The ray-tracing problem is, given an optical system and the position and direction of an initial light ray, to decide if the light ray reaches some given final position. For many years ray tracing has been used for designing and analyzing optical systems. Ray tracing is now used extensively in computer graphics to render scenes with complex curved objects under global illumination. We show that ray-tracing problems in some three-dimensional simple optical systems (purely geometrical optics) are undecidable. These systems may consist of either reflective objects that are represented by rational quadratic equations, or refractive objects that are represented by rational linear equations. Some problems in more restricted models are shown to be PSPACE-hard or sometimes in PSPACE. A preliminary version of this paper appeared as “The Computability and Complexity of Optical Beam Tracing” in theProceedings of the 31st Annual Symposium on Foundations of Computer Science, October 1990, Vol. I, pp. 106–114. The research of J. H. Reif and A. Yoshida was supported in part by Air Force Contract No. AFOSR-87-0386, DARPA/ARO Contract No. DAAL03-88-K-0195, DARPA/ISTO Contract No. N00014-88-K-0458, and NASA subcontract 550-63 of primecontract NASS-30428. J.D. Tygar's research was supported in part by the Defense Advanced Research Projects Agency under Contract No. F33615-87-C-1449 and by a National Science Foundation Presidential Young Investigator Award under Contract No. CCR-8858087.     

Packet routing and job-shop scheduling inO(congestion+dilation) steps

In this paper, we prove that there exists a schedule for routing any set of packets with edge-simple paths, on any network, inO(c+d) steps, wherec is the congestion of the paths in the network, andd is the length of the longest path. The result has applications to packet routing in parallel machines, network emulations, and job-shop scheduling.This research was conducted while the authors were at MIT. Support was provided by the Defense Advanced Research Projects Agency under Contract N00014-87-K-825, the Office of Naval Research under Contract N00014-86-K-0593, the Air Force under Contract OSR-86-0076, and the Army under Contract DAAL-03-86-K-0171. Tom Leighton is supported by an NSF Presidential Young Investigator Award with matching funds provided by IBM.     

Stopping rules,permutation invariance and sufficiency principle

In the context of sequential (point as well as interval) estimation, a general formulation of permutation-invariant stopping rules is considered. These stopping rules lead to savings in the ASN at the cost of some elevation of the associated risk—a phenomenon which may be attributed to the violation of the sufficiency principle. For the (point and interval) sequential estimation of the mean of a normal distribution, it is shown that such permutation-invariant stopping rules may lead to a substantial saving in the ASN with only a small increase in the associated risk.Work partially supported by (i) Office of Naval Research, Contract Number N00014-85-K-0548, and (ii) Office of Naval Research, Contract Number N00014-83-K-0387.     

Mean stochastic comparison of diffusions

Summary Stochastic bounds are derived for one dimensional diffusions (and somewhat more general random processes) by dominating one process pathwise by a convex combination of other processes. The method permits comparison of diffusions with different diffusion coefficients. One interpretation of the bounds is that an optimal control is identified for certain diffusions with controlled drift and diffusion coefficients, when the reward function is convex. An example is given to show how the bounds and the Liapunov function technique can be applied to yield bounds for multidimensional diffusions.This work was supported by the Office of Naval Research under Contract N00014-82-K-0359 and the U.S. Army Research Office under Contract DAAG29-82-K-0091 (administered through the University of California at Berkeley).     

Markov's Rule revisited

I am grateful to Boris Kushner for informing me about [Nov 43], the impetus for this note, and to Anne Troelstra for comments on an early sketch. Research partially supported by ONR grant N00014-84-K-0415 and by DARPA grant F33615-81-K-1539     

Edge-clique graphs

The edge-clique graphK(G) of a graphG is that graph whose vertices correspond to the edges ofG and where two vertices ofK(G) are adjacent whenever the corresponding edges ofG belong to a common clique. It is shown that every edge-clique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G). An algorithm is provided for determining whether a graph is an edge-clique graph. A new graph called the STP graph is introduced and a relationship involving this graph, the edge-clique graph, and the line graph is presented. The STP graphs are also characterized.Research supported in part by Office of Naval Research Contract N00014-88-K-0018.Research supported in part by Office of Naval Research Contract N00014-88-K-0163.     

Simplified Voronoi diagrams

We are interested in Voronoi diagrams as a tool in robot path planning, where the search for a path in anr-dimensional space may be simplified to a search on an (r–1)-dimensional Voronoi diagram. We define a Voronoi diagramV based on a measure of distance which is not a true metric. This formulation has lower algebraic complexity than the usual definition, which is a considerable advantage in motion-planning problems with many degrees of freedom. In its simplest form, the measure of distance between a point and a polytope is the maximum of the distances of the point from the half-spaces which pass through faces of the polytope. More generally, the measure is defined in configuration spaces which represent rotation. The Voronoi diagram defined using this distance measure is no longer a strong deformation retract of free space, but it has the following useful property: any path through free space which starts and ends on the diagram can be continuously deformed so that it lies entirely on the diagram. Thus it is still complete for motion planning, but it has lower algebraic complexity than a diagram based on the Euclidean metric.This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the Laboratory's Artificial Intelligence research is provided in part by the Office of Naval Research under Office of Naval Research Contract N00014-81-K-0494 and in part by the Advanced Research Projects Agency under Office of Naval Research Contracts N00014-85-K-0124 and N00014-82-K-0334. John Canny was supported by an IBM fellowship. Bruce Donald was funded in part by a NASA fellowship administered by the Jet Propulsion Laboratory.     

On the number of nondismantlable posets with the fixed point property

We give a lower bound for the number of orders on a finite set that are not dismantlable and have the fixed point property. To do so we also derive an upper bound for the number of orders on a finite set that are dismantlable.The author was partially funded by the Office of Naval Research under Contract N00014-88-K-0455.     

On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method

Interest in linear programming has been intensified recently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a “projected Newton barrier” method. This method is shown to be equivalent to Karmarkar’s projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several non-trivial test problems, and the implications for future developments in linear programming are discussed. The research of the Stanford authors was supported by the U.S. Department of Energy Contract DE-AA03-76SF00326, PA No. DE-AS03-76ER72018; National Science Foundation Grants DCR-8413211 and ECS-8312142; the Office of Naval Research Contract N00014-85-K-0343; and the U.S. Army Research Office Contract DAAG29-84-K-0156. The research of J.A. Tomlin was supported by Ketron, Inc. and the Office of Naval Research Contract N00014-85-C-0338.     

Bounds on the chromatic polynomial and on the number of acyclic orientations of a graph

An upper bound is given on the number of acyclic orientations of a graph, in terms of the spectrum of its Laplacian. It is shown that this improves upon the previously known bound, which depended on the degree sequence of the graph. Estimates on the new bound are provided.A lower bound on the number of acyclic orientations of a graph is given, with the help of the probabilistic method. This argument can take advantage of structural properties of the graph: it is shown how to obtain stronger bounds for small-degree graphs of girth at least five, than are possible for arbitrary graphs. A simpler proof of the known lower bound for arbitrary graphs is also obtained.Both the upper and lower bounds are shown to extend to the general problem of bounding the chromatic polynomial from above and below along the negative real axis.Partially supported by the NSF under grant CCR-9404113. Most of this research was done while the author was at the Massachusetts Institute of Technology, and was supported by the Defense Advanced Research Projects Agency under Contracts N00014-92-J-1799 and N00014-91-J-1698, the Air Force under Contract F49620-92-J-0125, and the Army under Contract DAAL-03-86-K-0171.Supported by an ONR graduate fellowship, grants NSF 8912586 CCR and AFOSR 89-0271, and an NSF postdoctoral fellowship.     

On stable laws for estimating functions and derived estimators

Summary Stable laws forM-estimators, maximum likelihood and other estimators and obtained through parallel results for the estimating functions and relative compactness of some related estimating functional processes. Work supported by the Office of Naval Research, Contract No. N00014-83-K-0387.     

Consistency of infinitesimal perturbation analysis estimators with rates

We study a class of infinitesimal perturbation analysis (IPA) algorithms for queueing systems with load-dependent service and/or arrival rates. Such IPA algorithms were originally motivated by applications to large queueing systems in conjunction with aggregation algorithms. We prove strong consistency of these estimators through a type of birth and death queue. This work was supported in part by the NSF under Grants Nos. ECS85-15449 and CDR-8803012, by ONR under Contracts Nos. N00014-89-J-0075 and N00014-90-K-1093, and by the US Army under Contract No. DAAL-03-83-K-0171. This paper was written while the author was with the Division of Applied Sciences at Harvard University.     

Effect of model uncertainty on some optimal routing problems

We study the effect of model uncertainties on optimal routing in a system of parallel queues. The uncertainty arises in modeling the service time distribution for the customers (jobs, packets) to be served. For a Poisson arrival process and Bernoulli routing, the optimal mean system delay generally depends on the variance of this distribution. However, as the input traffic load approaches the system capacity, the optimal routing assignment and corresponding mean system delay are shown to converge to a variance-invariant point. The implications of these results are examined in the context of gradient-based routing algorithms. An example of a model-independent algorithm using on-line gradient estimation is also included and its performance compared with that of model-based algorithms.This work was supported in part by the National Science Foundation under Grant ECS-88-01912, by the Office of Naval Research under Contract N00014-87-K-0304, and by NASA under Contract NAG 2-595.     

Nonlocal regularization of L. C. Young's tacking problem

L. C. Young's tacking problem is a prototype of a nonconvex variational problem for which minimizing sequences for the energy do not attain a minimum. The minimizer of the energy is usually described as a Young-measure or generalized curve. In many studies, the tacking problem is regularized by adding a higher-order viscosity term to the energy. This regularized energy has classical minimizers. In this paper we regularize instead with a spatially nonlocal term. This weakly regularized problem still has measure-valued minimizers, but as the nonlocal term becomes stronger, the measure-valued solutions organize, coalesce, and eventually turn into classical solutions. The information on the measure-valued solutions is obtained by studying equivalent variational problems involving moments of the measures.The research of D. Brandon has been partially supported by the Office of Naval Research under Grant Number N00014-88-K-0417 and by DARPA Grant F4920-87-C-0116, and that of R. C. Rogers has been partially supported by the Office of Naval Research under Grant Number N00014-88-K-0417 and by the National Science Foundation under Grant Number DMS-8801412.     

A relaxed version of Karmarkar's method

A relaxed version of Karmarkar's method is developed. This method is proved to have the same polynomial time complexity as Karmarkar's method and its efficient implementation using inexact projections is discussed. Computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable.This research was supported in part by NSF Grants CDR 84-21402 and DMS-85-12277 and ONR Contract N00014-87-K-0214.     

Exploiting special structure in Karmarkar's linear programming algorithm

We propose methods to take advantage of specially-structured constraints in a variant of Karmarkar's projective algorithm for standard form linear programming problems. We can use these constraints to generate improved bounds on the optimal value of the problem and also to compute the necessary projections more efficiently, while maintaining the theoretical bound on the algorithm's performance. It is shown how various upper-bounding constraints can be handled implicitly in this way. Unfortunately, the situation for network constraints appears less favorable.Research supported in part by National Science Foundation Grant ECS-8602534, ONR Contract N00014-87-K-0212 and the US Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Linear/nonlinear forms and the normal law: Characterization by high order correlations

Summary The problem of characterizing the normal law associated with linear forms and processes, as well as with quadratic forms, is considered. The classical condition of constancy of regression is replaced by a distinct condition of high-order uncorrelatedness. The work of E. Masry was supported by the Office of Naval Research under Contract N00014-84-K-0042.     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

Use of dynamic trees in a network simplex algorithm for the maximum flow problem

Goldfarb and Hao (1990) have proposed a pivot rule for the primal network simplex algorithm that will solve a maximum flow problem on ann-vertex,m-arc network in at mostnm pivots and O(n ² m) time. In this paper we describe how to extend the dynamic tree data structure of Sleator and Tarjan (1983, 1985) to reduce the running time of this algorithm to O(nm logn). This bound is less than a logarithmic factor larger than those of the fastest known algorithms for the problem. Our extension of dynamic trees is interesting in its own right and may well have additional applications.Research partially supported by a Presidential Young Investigator Award from the National Science Foundation, Grant No. CCR-8858097, an IBM Faculty Development Award, and AT&T Bell Laboratories.Research partially supported by the Office of Naval Research, Contract No. N00014-87-K-0467.Research partially supported by the National Science Foundation, Grant No. DCR-8605961, and the Office of Naval Research, Contract No. N00014-87-K-0467.     

A generalization of Dirac’s theorem for K(1,3)-free graphs

It is known that if a 2-connected graphG of sufficiently large ordern satisfies the property that the union of the neighborhoods of each pair of vertices has cardinality at leastn/2, thenG is hamiltonian. In this paper, we obtain a similar generalization of Dirac’s Theorem forK(1,3)-free graphs. In particular, we show that ifG is a 2-connectedK(1,3)-free graph of ordern with the cardinality of the union of the neighborhoods of each pair of vertices at least (n+1)/3, thenG is hamiltonian. We also investigate several other related properties inK(1,3)-free graphs such as traceability, hamiltonian-connectedness, and pancyclicity. Partially Supported by O. N. R. Contract Number N00014-88-K-0070. Partially Supported by O. N. R. Contract Number N00014-85-K-0694.