The Fine-Grained Complexity of Approximately Counting Proper Connected Colorings (Extended Abstract)

Abstract

A k-proper connected 2-coloring for a graph is an edge bipartition which ensures the existence of at least k vertex disjoint simple alternating paths (i.e., paths where no two adjacent edges belong to the same partition) between all pairs of vertices. In this work, for every $k \in \mathbb {N}_{>0}$, we show that exactly counting such colorings is $\#P$-hard under many-one counting reductions, as well as $\#P$-complete under many-one counting reductions for $k=1$. Furthermore, for every $k \in \mathbb {N}_{>0}$, we rule out the existence of a $2^{o\left( \frac{n}{k^2}\right) }$ time algorithm for finding a k-proper connected 2-coloring of an order n graph under the ETH, or for exactly counting such colorings assuming the moderated Counting Exponential Time Hypothesis (#ETH) of (Dell et al.; ACM Trans. Algorithms 10(4); 2014). Finally, assuming the Exponential Time Hypothesis (ETH), and as a consequence of a recent result of (Dell & Lapinskas; ACM Trans. Comput. Theory 13(2); 2021), for every $k \in \mathbb {N}_{>0}$ and every $\epsilon > 0$, we are able to rule out the existence of a $2^{o\left( \frac{n}{k^2}\right) }/\epsilon ^2$ time algorithm for approximating the number of k-proper connected 2-colorings of an order n graph within a multiplicative factor of $1 + \epsilon$.

This work was supported by JSPS Kakenhi grants {20K21827, 20H05967, 21H04871}, and JST CREST Grant JPMJCR1402JST.