Parameterized Algorithms for Minimum Sum Vertex Cover

Minimum sum vertex cover of an n-vertex graph G is a bijection φ : V(G) → [n] that minimizes the cost ∑{u,v}ϵE(G) min{φ(u), φ(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.

- MSVC can be solved in 2²O(k) nO(1) time, where k is the size of a minimum vertex cover.

- MSVC can be solved in f(k) · nnO(1) time for some computable function f, where k is the size of a minimum clique modulator.