Madhav V. Marathe

Simple heuristics for unit disk graphs

Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NP-hard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an on-line coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.

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. nAssuming {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

Approximation Schemes Using L-Reductions

epsilon

Compact Location 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.

Efficient approximation algorithms for domatic partition and on-line coloring of circular arc 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 for PSPACE-complete problems for succinct specifications (preliminary version)

We present parallel approximation schemes for a large class of problems in logic and graph theory, when these problems are restricted to planar graphs and δ-near-planar graphs. Our results are based on the positive use of L-reductions, and the decomposition of a given graph into subgraphs for which the given problem can be solved optimally in polynomial time. The problems considered include MAX 3SAT, MAX SAT(S), maximum independent set, and minimum dominating set. Our NC-approximation schemes exhibit the same time versus performance tradeoff as those of Baker [Ba83]. For δ-near-planar graphs, this is the first time that approximation schemes of any kind have been obtained for these problems.

Hierarchically specified unit disk graphs

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.

Theory of Periodically Specified Problems: Complexity and Approximability

Exploits the close relationship between circular arc graphs and interval graphs to design efficient approximation algorithms for NP-hard optimization problems on circular arc graphs. The problems considered are maximum domatic partition and online minimum vertex coloring. We present a heuristic for the domatic partition problem with a performance ratio of 4. For online coloring, we consider two different online models. In the first model, arcs are presented in the increasing order of their left endpoints. For this model, our heuristic guarantees a solution which is within a factor of 2 of the optimal (off-line) value; and we show that no online coloring algorithm can achieve a performance guarantee of less than 3/2. In the second online model, arcs are presented in an arbitrary order; and it is known that no online coloring algorithm can achieve a performance guarantee of less than 3. For this model, we present a heuristic which provides a performance guarantee of 4.<<ETX>>

Computational Complexity of Analyzing the Dynamic Reliability of Interdependent Infrastructures

We study the existence of polynomial time approximation algorithms and approximation schemes for a number of basic problems in the literature including the various natural problems in [Ba83, Ka91, Ka92, PY9 1], when instances are specified using the succinct specification languages of [Le89, Wa93, Ga82]. We introduce the concept of k-level-restricted specifications, and show that this syntactic restriction can be applied to each of the succinct specification languages of [Le89, Wa93, Ga82]. Although many basic problems are PSPACE-hard even for level-restricted specifications of planar instances, we show that . Many of these basic problems II have efficient approximation algorithms with performance guarantees which are asymptotically equal to the best known performance guarantee for H for flat specification. This result holds, for both sequential and parallel capproximations and approximation schemes. As a corollary of the above result, we get that for every fixed k >1, all planar graph problems shown to have a PTAS–in Baker [Ba83] when instances are specified non-succinctly, have a PTAS when graphs are specified by k-leveI-restricted succinct specifications. Thus our results answer an open question by Condon et. al. in their recent paper [CF+93a]. 1Email addresses: {madhav, hunt ,res}c?!cs.albany. edu QSuPPOrted bY NSF Grants CCR 89-CISS1