Sambuddha Roy

Decision trees for entity identification: approximation algorithms and hardness results

We consider the problem of constructing decision trees for entity identification from a given relational table. The input is a table containing information about a set of entities over a fixed set of attributes and a probability distribution over the set of entities that specifies the likelihood of the occurrence of each entity. The goal is to construct a decision tree that identifies each entity unambiguously by testing the attribute values such that the average number of tests is minimized. This classical problem finds such diverse applications as efficient fault detection, species identification in biology, and efficient diagnosis in the field of medicine. Prior work mainly deals with the special case where the input table is binary and the probability distribution over the set of entities is uniform. We study the general problem involving arbitrary input tables and arbitrary probability distributions over the set of entities. We consider a natural greedy algorithm and prove an approximation guarantee of O(rK • log N), where N is the number of entities and K is the maximum number of distinct values of an attribute. The value rK is a suitably defined Ramsey number, which is at most log K. We show that it is NP-hard to approximate the problem within a factor of Ω(log N), even for binary tables (i.e. K=2). Thus, for the case of binary tables, our approximation algorithm is optimal up to constant factors (since r2=2). In addition, our analysis indicates a possible way of resolving a Ramsey-theoretic conjecture by Erdos.

The directed planar reachability problem

We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete. We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case. We also consider a previously-studied subclass of planar graphs known as grid graphs. We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs. A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “layered grid graph reachability problem”. We show that this problem lies in the complexity class UL.

Grid graph reachability problems

We study the complexity of reachability problems on various classes of grid graphs. Reachability on certain classes of grid graphs gives natural examples of problems that are hard for NC1 under AC0 reductions but are not known to be hard far L; they thus give insight into the structure of L. In addition to explicating the structure of L, another of our goals is to expand the class of digraphs for which connectivity can be solved in logspace, by building on the work of Jakoby et al. (2001), who showed that reachability in series-parallel digraphs is solvable in L. We show that reachability for single-source multiple sink planar dags is solvable in L

Time-space tradeoffs in the counting hierarchy

Extends the lower-bound techniques of L. Fortnow (2000) to the unbounded-error probabilistic model. A key step in the argument is a generalization of V.A. Nepomnjas/spl caron/c/spl caron/ii/spl breve/s (1970) theorem from the Boolean setting to the arithmetic setting. This generalization is made possible due to the recent discovery of logspace-uniform TC/sup 0/ circuits for iterated multiplication (A. Chiu et al., 2000). As an example of the sort of lower bounds that we obtain, we show that MAJ-MAJSAT is not contained in PrTiSp(n/sup 1+o(1)/, n/sup /spl epsiv//) for any /spl epsiv/<1. We also extend one of Fortnows lower bounds, from showing that S~A~T~ does not have uniform NC/sup 1/ circuits of size n/sup 1+o(1)/, to a similar result for SAC/sup 1/ circuits.

Oblivious symmetric alternation

We introduce a new class

Approximating Decision Trees with Multiway Branches

\rm {O}^p_2

Extracting dense communities from telecom call graphs

as a subclass of the symmetric alternation class

Derandomization and distinguishing complexity

\rm {S}^p_2

Minimum Cost Resource Allocation for Meeting Job Requirements

. An

Improved Algorithms for Resource Allocation under Varying Capacity

\rm {O}^p_2