Upper Dominating Set: Tight Algorithms for Pathwidth and Sub-exponential Approximation

An upper dominating set is a minimal in graph. In the Upper Dominating Set problem, goal to find an of maximum size. We study complexity parameterized algorithms for Set, as well its sub-exponential approximation. First, we prove that, under ETH, k -Upper cannot be solved time \(O(n^{o(k)})\) (improving on \(O(n^{o(\sqrt{k})})\)), and same show assumption that any constant ratio r \(\varepsilon > 0\), there no r-approximation algorithm running \(O(n^{k^{1-\varepsilon }})\). Then, settle problem’s by pathwidth giving \(O^*(6^{pw})\) current best \(O^*(7^{pw})\)), lower bound showing our can get SETH. Furthermore, obtain simple approximation this problem: produces \(n^{O(n/r)}\), desired \(r 1\) output \(n^{(n/r)^{1-\varepsilon }}\). Hence, completely characterize approximability problem time.