Parameterized algorithms for minimum sum vertex cover - Aute, Shubhada; Panolan, Fahad (Theoretical computer science, 2025)
A minimum sum vertex cover of an n-vertex graph G is a bijection ϕ:V(G)→[n] that minimizes the cost ∑{u,v}∈E(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)⋅nO(1) time for some computable function f, where k is the size of a minimum clique modulator.