Venkatesh Radhakrishnan

The Complexity of Planar Counting Problems

We prove the #P-hardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover. We also prove the NP-completeness of the Ambiguous Satisfiability} problems [J. B. Saxe, Two Papers on Graph Embedding Problems, Tech. Report CMU-CS-80-102, Dept. of Computer Science, Carnegie Mellon Univ., Pittsburgh, PA, 1980] and the DP-completeness (with respect to random polynomial reducibility) of the unique satisfiability problems [L. G. Valiant and V. V. Vazirani, NP is as easy as detecting unique solutions, in Proc. 17th ACM Symp. on Theory of Computing, 1985, pp. 458--463] associated with several of the above problems, when restricted to planar instances. Previously, very few #P}-hardness results, no {\sf NP}-hardness results, and no DP-completeness results were known for counting problems, ambiguous satisfiability problems, and unique satisfiability problems, respectively, when restricted to planar instances. Assuming {\sf P \neq

A Unified Approach to Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs

NP}, one corollary of the above results is that there are no

Compact location problems

\epsilon

Approximation Algorithms for PSPACE-Hard Hierarchically and Periodically Specified Problems

-approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers.

Hierarchically specified unit disk graphs

We present for the first time NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner [CCJ90, DS84, MHR92, Te91]. Our NC-approximation schemes exhibit the same time versus performance trade-off as those of Baker [Ba83]. We also define the concept of -precision unit disk graphs and show that for such graphs our NC approximation schemes have a better time versus performance trade-off. Moreover, compared to unit disk graphs, we show that for -precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer [BOW83]. Our approximation schemes for hierarchically specified unit disk graphs along with our results in [MHSR94] are the first approximation schemes in the literature for natural PSPACE-hard optimization problems.

Approximation Schemes Using L-Reductions

We consider the problem of placing a specified number (p) of facilities on the nodes of a network so as to minimize some measure of the distances between facilities. This formulation models a number of problems arising in facility location, statistical clustering, pattern recognition, and also a processor allocation problem in multiprocessor systems.

Compact Location Problems

We study the efficient approximability of basic graph and logic problems in the literature when instances are specified hierarchically as in [T. Lengauer, J. Assoc. Comput. Mach., 36(1989), pp. 474--509] or are specified by one-dimensional finite narrow periodic specifications as in [E. Wanke, Paths and cycles in finite periodic graphs, in Lecture Notes in Comp. Sci. 711, Springer-Verlag, New York, 1993, pp. 751--760]. We show that, for most of the problems

Efficient Algorithms for Solving Systems of Linear Equations and Path Problems

\Pi

Parallel Approximation Schemes for a Class of Planar and Near Planar Combinatorial Optimization Problems

considered when specified using k-level-restricted hierarchical specifications or k-narrow periodic specifications, the following hold. Let

Approximation schemes for PSPACE-complete problems for succinct specifications (preliminary version)

\rho