The complexity, approximation and algorithmic issues of several clustering problems are studied. These non-traditional clustering problems arise from recent studies in microarray data analysis. We prove the following results. (1) Two variants of the Order-Preserving Submatrix problem are NP-hard. There are polynomial-time algorithms for the Order-Preserving Submatrix problem when the condition ...

Lattice problems are known to be hard to approximate to within sub-polynomial factors. For larger approximation factors, such as √ n, lattice problems are known to be in complexity classes such as NP∩ coNP and are hence unlikely to be NP-hard. Here we survey known results in this area. We also discuss some related zero-knowledge protocols for lattice problems.

We discuss what we consider to be the ten most vexing open questions in the area of polynomial time approximation algorithms for NP-hard deterministic machine scheduling problems. We summarize what is known on these problems, we discuss related results, and we provide pointers to the literature. Prologue In the early days of scheduling, the work of the theoretical research branch mainly consist...

We survey the conceptual framework and several applications of the iterative compression technique introduced in 2004 by Reed, Smith, and Vetta. This technique has proven very useful for achieving a number of recent breakthroughs in the development of fixed-parameter algorithms for NP-hard minimization problems. There is a clear potential for further applications as well as a further developmen...

In this paper we consider a graphical realization of dynamic programming. The concept is discussed on the partition and knapsack problems. In contrast to dynamic programming, the new algorithm can also treat problems with non-integer data without necessary transformations of the corresponding problem. We compare the proposed method with existing algorithms for these problems on small-size insta...

We develop a generic framework for deriving linear-size problem kernels for NP-hard problems on planar graphs. We demonstrate the usefulness of our framework in several concrete case studies, giving new kernelization results for Connected Vertex Cover, Minimum Edge Dominating Set,Maximum Triangle Packing, and Efficient Dominating Set on planar graphs. On the route to these results, we present e...

An α-approximation algorithm is an algorithm guaranteed to output a solution that is within an α ratio of the optimal solution. We are interested in the following question: Given an NP-hard optimization problem, what is the best approximation guarantee that any polynomial time algorithm could achieve? We mostly focus on studying the approximability of two classes of NP-hard problems: Constraint...