Chapter 2 Parameterized Algorithms Fedor Fomin Daniel Lokshtanov

Chapter 2: Parameterized Algorithms Fedor Fomin, Daniel Lokshtanov, Saket Saurabh, Meirav Zehavi 1

Basic idea When standard worst case analysis (expressing running time as a function of size of input instance) is not sufficiently informative: • Identify parameters of interest (like we did with properties of the working set function for online paging, or the margin for the Perceptron algorithm). • With respect to these parameters, perform worst case analysis. For a good choice of parameter, the corresponding worst case analysis becomes more informative. 2

Example – minimum vertex cover Select the smallest number of vertices that cover all edges. NP-hard. 3

Example – minimum vertex cover Select the smallest number of vertices that cover all edges. NP-hard. 4

A parameter for vertex cover • 5

A simple algorithm for min vertex cover • 6

A faster algorithm • 7

Fixed parameter tractability • 8

Some other graph problems 9

Maximum independent set 10

Minimum vertex coloring (chromatic number) 11

Examples of classifications • 12

Randomized FPT algorithms • 13

Set splitting • 14

Randomized algorithm for set splitting • 15

Longest path 16

Longest path 1 2 3 6 7 4 5 17

Longest path and Hamiltonicity • 18

Further comments on Hamilton path • 19

Longest path is fixed parameter tractable • 20

Derandomization [Naor, Schulman and Srinivasan 1995] • 22

Kernelization • 23

Significance of kernelization • 24

Conditional negative results • 26

The subset construction • 28