Summary: A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting \(k\) edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this paper, we study the \(\mathcal{F}\)-Contraction problem, where \(\mathcal{F}\) is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph \(G\) and an integer \(k\), \(\mathcal{F}\)-Contraction asks whether there exists \(X\subseteq E(G)\) such that \(G/X\in\mathcal{F}\) and \(|X|\le k\). Here, \(G/X\) is the graph obtained from \(G\) by contracting edges in \(X\). We obtain the following results for the \(\mathcal{F}\)-Contraction problem.

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Clique Contraction is known to be FPT. However, unless \(\text{NP}\subseteq\text{coNP/poly}\), it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme (PSAKS). That is, it admits a \((1+\varepsilon)\)-approximate kernel with \(\mathcal{O}(k^{f(\varepsilon)})\) vertices for every \(\varepsilon>0\).

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Split Contraction is known to be W[1]-Hard. We deconstruct this intractability result in two ways. Firstly, we give a \((2+\varepsilon)\)-approximate polynomial kernel for Split Contraction (which also implies a factor \((2+\varepsilon)\)-FPT-approximation algorithm for Split Contraction). Furthermore, we show that, assuming Gap-ETH, there is no \((\frac{5}{4}-\delta)\)-FPT-approximation algorithm for Split Contraction. Here, \(\varepsilon,\delta>0\) are fixed constants.

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Chordal Contraction is known to be W[2]-Hard. We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming \(\text{FPT}\ne\text{W[1]}\), there is no \(F(k)\)-FPT-approximation algorithm for Chordal Contraction. Here, \(F(k)\) is an arbitrary function depending on \(k\) alone.

We say that an algorithm is an \(h(k)\)-FPT-approximation algorithm for the \(\mathcal{F}\)-Contraction problem, if it runs in FPT time, and on any input \((G,k)\) such that there exists \(X\subseteq E(G)\) satisfying \(G/X\in\mathcal{F}\) and \(|X|\le k\), it outputs an edge set \(Y\) of size at most \(h(k)\cdot k\) for which \(G/Y\) is in \(\mathcal{F}\). We find it extremely interesting that three closely related problems have different behavior with respect to FPT-approximation.
For the entire collection see [Zbl 1445.68009].