Parameterised counting complexity theory

A little more than two decades ago, Flum and Grohe (STOC 02), and McCartin (MFCS 02) introduced the structural foundations of parameterised counting complexity theory with the goal of applying and generalising the extensive toolkit of parameterised algorithmics to the world of counting.

Counting problems are known to be infamously hard with respect to classical complexity theory, much harder than $\mathrm{NP}$-complete problems under standard assumptions, as shown by Toda (STOC 91). This holds true even for counting problems that admit a tractable decision version, a fact established in Valiant’s seminal work on the complexity of counting perfect matchings (SICOMP 79). Naturally, the central question in parameterised counting complexity theory asks: Can this intractability be alleviated with a multivariate complexity analysis?

We have observed that many tools from the “swiss army knife” of parameterised decision algorithms, such as win–win approaches based on bidimensionality, colour-coding, and, to some extent, kernelisation, often fail in the realm of counting problems (especially for exact counting). Circumventing the inapplicability of well-established algorithmic tools, we have witnessed the development of a flurry of novel techniques and theories tailored to parameterised counting problems, with origins in commutative combinatorial algebra, topology and deep graph theory dating back to early works of Lovász.

In this survey, we will revisit some of the most important frameworks and results discovered and established in the field over the years. Particular focus will be put on the framework of Graph Motif Parameters due to Curticapean, Dell and Marx (STOC 17), one of, if not the most exciting development in parameterised counting since its inception.

We will assume familiarity with basic concepts of parameterised algorithms and complexity theory, but, aside from that, we aim to present the introduction to the world of parameterised counting in a self-contained way.