Parameterized Complexity of Domination Problems Using Restricted Modular Partitions

TL;DR

This paper explores the parameterized complexity of domination problems using restricted modular partitions, revealing conditions under which these problems are fixed-parameter tractable or W[1]-hard based on the structure of solution tables.

Contribution

It introduces the concept of $ ext{G}$-modular cardinality, generalizes module-based graph parameters, and analyzes the complexity of domination problems within this framework.

Findings

Problems are W[1]-hard when solution tables are large.

Problems are FPT when succinct solution tables exist.

New results relate structural parameters like clique-width and modular-width to problem complexity.

Abstract

For a graph class $\mathcal{G}$, we define the $\mathcal{G}$-modular cardinality of a graph $G$ as the minimum size of a vertex partition of $G$ into modules that each induces a graph in $\mathcal{G}$. This generalizes other module-based graph parameters such as neighborhood diversity and iterated type partition. Moreover, if $\mathcal{G}$ has bounded modular-width, the W[1]-hardness of a problem in $\mathcal{G}$-modular cardinality implies hardness on modular-width, clique-width, and other related parameters. On the other hand, fixed-parameter tractable (FPT) algorithms in $\mathcal{G}$-modular cardinality may provide new ideas for algorithms using such parameters. Several FPT algorithms based on modular partitions compute a solution table in each module, then combine each table into a global solution. This works well when each table has a succinct representation, but as we argue, when no such representation exists, the problem is typically W[1]-hard. We illustrate these ideas on the generic $(\alpha, \beta)$-domination problem, which asks for a set of vertices that contains at least a fraction $\alpha$ of the adjacent vertices of each unchosen vertex, plus some (possibly negative) amount $\beta$. This generalizes known domination problems such as Bounded Degree Deletion, $k$-Domination, and $\alpha$-Domination. We show that for graph classes $\mathcal{G}$ that require arbitrarily large solution tables, these problems are W[1]-hard in the $\mathcal{G}$-modular cardinality, whereas they are fixed-parameter tractable when they admit succinct solution tables. This leads to several new positive and negative results for many domination problems parameterized by known and novel structural graph parameters such as clique-width, modular-width, and $cluster$-modular cardinality.