Parameterized Algorithms for Minimum Sum Vertex Cover

Minimum sum vertex cover of an $n$-vertex graph $G$ is a bijection $\phi : V(G) \to [n]$ that minimizes the cost $\sum_{\{u,v\} \in E(G)} \min \{\phi(u), \phi(v) \}$. Finding a minimum sum vertex cover of a graph (the MSVC problem) is NP-hard. MSVC is studied well in the realm of approximation algorithms. The best-known approximation factor in polynomial time for the problem is $16/9$ [Bansal, Batra, Farhadi, and Tetali, SODA 2021]. Recently, Stankovic [APPROX/RANDOM 2022] proved that achieving an approximation ratio better than $1.014$ for MSVC is NP-hard, assuming the Unique Games Conjecture. We study the MSVC problem from the perspective of parameterized algorithms. The parameters we consider are the size of a minimum vertex cover and the size of a minimum clique modulator of the input graph. We obtain the following results. 1. MSVC can be solved in $2^{2^{O(k)}} n^{O(1)}$ time, where $k$ is the size of a minimum vertex cover. 2. MSVC can be solved in $f(k)\cdot n^{O(1)}$ time for some computable function $f$, where $k$ is the size of a minimum clique modulator.