Abstract

This paper provides an overview of the field of parameterized parallel complexity by surveying previous work in addition to presenting a few new observations and exploring potential new directions. In particular, we present a general view of how known FPT techniques, such as bounded search trees, color coding, kernelization, and iterative compression, can be modified to produce fixed-parameter parallel algorithms.

Highlights

  • To how computational problems are classified according to their tractability or whether efficient algorithms exist for them, problems are classified according to whether efficient parallel algorithms exist to solve them

  • A problem is fixed-parameter parallel tractable (FPPT) if it can be solved in O( f (k ) ∗ logc1 (n))

  • Iterative compression algorithms are mainly applied to some minimization problems when it is possible to construct a feasible solution whose size is slightly larger than the target solution size k, which is the parameter in this case

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Summary

Introduction

To how computational problems are classified according to their tractability or whether efficient algorithms exist for them, problems are classified according to whether efficient parallel algorithms exist to solve them. A parallel problem is in NC if it can be solved in O(logc (n)) time using O(nc2 ) processors, where c1 and c2 are constants [4] To convert this notion into a parameterized definition that we can use (or relate to) later, and following the same assumptions of [1], we consider a parameterized problem to be in NC if it is in NC for any given value of the parameter. The more restrictive class of fixed-parameter parallel (FPP) problems was defined by Cesati and Ianni in [1] where a strictly poly-logarithmic factor in the running time is required, by replacing h(k ) with a constant It was observed in [6] that the function g(k ). In particular we overview the methods used for obtaining a parameterized parallel algorithm from a known sequential fixed-parameter algorithm

Preliminaries
Parameterized Parallel Complexity Classes
PNC and NC
PNC and FPT
FPP and PNC
FPP and FPPT
Parameterized Parallel Intractability
Key Results and Methods
Implementation Aspects
Input Representation
Input Queries
Input Modification
Avoiding Excessive Use of Successive Operations
Bounded Search Trees
Kernelization
Color Coding
Iterative Compression
Parallel Algorithms Using FPT Methods
Problems not Fixed-Parameter Parallel Tractable
Conclusions
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