Blocker size via matching minors

TL;DR

This paper extends bounds on the number of maximal independent sets from graphs to more general structures called clutters, using minors instead of induced subgraphs, and provides polynomial algorithms for related problems.

Contribution

It generalizes graph results to clutters, establishing bounds on independent sets via minors and introducing polynomial algorithms for certain computational problems.

Findings

Bound on blocker size for minor-free clutters

Polynomial algorithms for Set Cover and k-SAT in restricted cases

Extension of Alekseev's bounds to clutters

Abstract

Finding the maximum number of maximal independent sets in an $n$-vertex graph $G$, $i(G)$, from a restricted class is an extensively studied problem. Let $kK_2$ denote the matching of size $k$, that is a graph with $2k$ vertices and $k$ disjoint edges. A graph with an induced copy of $kK_2$ contains at least $2^k$ maximal independent sets. The other direction was established in a series of papers finally yielding $i(G) \le (n/k)^{2k}$ for a graph $G$ without an induced $(k+1)K_2$. Alekseev proved that $i(G)$ is at most the number of induced matchings of $G$. This work generalises the aforementioned results to clutters. The right substructures in this setting are minors rather than induced subgraphs. Maximal independent sets of a clutter $\mathcal{H}$ are in one-to-one correspondence to the sets of its blocker, $b(\mathcal{H})$, hence $i(\mathcal{H}) = |b(\mathcal{H})|$. We show that \[ |b(\mathcal{H})| \le \sum_{m=0}^{k \cdot f(r)}{|\mathcal{H}| \choose m} {r \choose 2}^m \] for a $(k+1)K_2$-minor-free clutter $\mathcal{H}$ where $f(r) = (2r-3)2^{r-2}$ and $r$ is the maximum size of a set in $\mathcal{H}$. A key step in the proofs is, similarly to Alekseev's result, showing that $i(\mathcal{H})$ is bounded by the number of a substructure called semi-matching, and then proving a dependence between the number of semi-matchings and the number of minor matchings. Note that similarly to graphs, a clutter containing a $kK_2$ minor has at least $2^k$ maximal independent sets. From a computational perspective, a polynomial number of independent sets is particularly interesting. Our results lead to polynomial algorithms for restricted instances of many problems including Set Cover and k-SAT.