Families with infants: speeding up algorithms for NP-hard problems using FFT

Assume that a group of people is going to an excursion and our task is to seat them into buses with several constraints each saying that a pair of people does not want to see each other in the same bus. This is a well-known coloring problem and it can be solved in $O^*(2n)$ time by the inclusion-exclusion principle as shown by Bj\"{o}rklund, Husfeldt, and Koivisto in 2009.Another approach to solve this problem in $O^*(2n)$ time is to use the fast Fourier transform. A graph is $k$-colorable if and only if the $k$-th power of a polynomial containing a monomial $\prod{i=1}ⁿ xi^{[i \in I]}$ for each independent set $I \subseteq [n]$ of the graph, contains the monomial $x1x2... x_n$. Assume now that we have additional constraints: the group of people contains several infants and these infants should be accompanied by their relatives in a bus. We show that if the number of infants is linear then the problem can be solved in $O^{*((2-\varepsilon)n)$} time. We use this approach to improve known bounds for several NP-hard problems (the traveling salesman problem, the graph coloring problem, the problem of counting perfect matchings) on graphs of bounded average degree, as well as to simplify the proofs of several known results.