Confronting intractability via parameters

Abstract

One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practicioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques both practical and theoretical, and a fine-grained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show both the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts of Downey and Fellows [125], Flum and Grohe [155], Niedermeier [240], and the upcoming undergraduate text Downey and Fellows [127].

Parameterized Approximation via Fidelity Preserving Transformations

“…Instead of expressing the running time as a function of the input size n only, the running time is expressed as a function of n and k, where k is a well-defined parameter of the input instance. We say that a problem (with a particular parameter k) is fixed-parameter tractable (FPT) if it can be solved in time f (k) · p(n), where f is an arbitrary function depending only on k. Thus we relax polynomial time by committing the exponential explosion to the parameter k. For further background on parameterized complexity we refer the reader to the textbooks [14,19,25], and the recent surveys in [15,17].…”

A Multi-Parameter Complexity Analysis of Cost-Optimal and Net-Benefit Planning

“…Choosing a smaller d value in the proof would require a larger k value; if ω is an optimal plan, then c(ω) ≥ 2n d + m d so we must set k ≥ 2n+m d . In a more extensive analysis, one could attempt to derive sharper lower bounds, or even so-called XP optimal bounds (Downey and Thilikos 2011).…”

Parameterized algorithmics for computational social choice: Nine research challenges

“…Parameterized intractability. As a central tool for classifying problems, Parameterized Complexity Theory provides the W-hierarchy consisting of the following classes and interrelations [37,49]:…”

“…Complexity Theory provides the W-hierarchy consisting of the following classes and interrelations [37,49]:…”