Anand Srivastav

Finding Dense Subgraphs with Semidefinite Programming

In this paper we consider the problem of computing the heaviest k-vertex induced subgraph of a given graph with nonnegative edge weights. This problem is known to be NP-hard, but its approximation complexity is not known. For the general problem only an approximation ratio of O(n0.3885) has been proved (Kortsarz and Peleg (1993)). In the last years several authors analyzed the case k=Ω(n). In this case Asahiro et al. (1996) showed a constant factor approximation, and for dense graphs Arora et al. (1995) obtained even a polynomial-time approximation scheme. We give a new approximation algorithm for arbitrary graphs and k=n/c for c > 1 based on semidefinite programming and randomized rounding which achieves for some c the presently best (randomized) approximation factors.

Algorithmic Chernoff-Hoeffding inequalities in integer programming

Raghavans paper on derandomized approximation algorithms for 0–1 packing integer programs raised two challenging problems [11]: 1. Are there more examples of NP-hard combinatorial optimization problems for which derandomization yields constant factor approximations in polynomial-time ? 2. The pessimistic estimator technique shows an O(mn)-time implementation of the conditional probability method on the RAM model of computation in case of m large deviation events associated to m unweighted sums of n indepependent Bernoulli trials. Is there a fast algorithm also in case of rational weighted sums of Bernoulli trials ?

Finding optimal volume subintervals with k points and calculating the star discrepancy are NP-hard problems

The well-known star discrepancy is a common measure for the uniformity of point distributions. It is used, e.g., in multivariate integration, pseudo random number generation, experimental design, statistics, or computer graphics. We study here the complexity of calculating the star discrepancy of point sets in the d-dimensional unit cube and show that this is an NP-hard problem. To establish this complexity result, we first prove NP-hardness of the following related problems in computational geometry: Given n points in the d-dimensional unit cube, find a subinterval of minimum or maximum volume that contains k of the n points. Our results for the complexity of the subinterval problems settle a conjecture of E. Thiemard [E. Thiemard, Optimal volume subintervals with k points and star discrepancy via integer programming, Math. Meth. Oper. Res. 54 (2001) 21-45].

Multicolour Discrepancies

In this article we introduce combinatorial multicolour discrepancies and generalize several classical results from

Improved Approximation Algorithms for Maximum Graph Partitioning Problems

2

Bounds and constructions for the star-discrepancy via δ-covers

-colour discrepancy theory to

Tight approximations for resource constrained scheduling and bin packing

c

Bipartite Graph Matchings in the Semi-streaming Model

colours (

Quantum-Inspired Evolutionary Algorithm for difficult knapsack problems

c \geq 2

Approximation Algorithms for Pick-and-Place Robots

). We give a recursive method that constructs