Parameterized Algorithms for Minimum Sum Vertex Cover

TL;DR

This paper studies Minimum Sum Vertex Cover (MSVC) through parameterized complexity, focusing on two natural structural parameters: the size $k$ of a minimum vertex cover and the size $k$ of a minimum clique modulator. For the vertex-cover parameterization, it exploits a partition of the independent set into at most $2^k$ equivalence classes by identical neighborhoods to show each class is consecutive in an optimal ordering and reduces the problem to an Integer Quadratic Program, achieving a running time of $[k! (k+1)^{2^k} + (1.2738)^k] n^{O(1)}$. For clique-modulator parameterization, it demonstrates that after fixing a modulator order, equivalence classes in the clique can be organized block-wise and again reduces to an IQP with at most $2^{k+1}(k+1)+2k$ variables, solvable in $f(k)n^{O(1)}$ time. Together, these results establish fixed-parameter tractability for MSVC under both parameters with complementary approaches and provide a unifying IQP-based framework for vertex-ordering problems.

Abstract

Minimum sum vertex cover of an $n$-vertex graph $G$ is a bijection $φ: V(G) \to [n]$ that minimizes the cost $\sum_{\{u,v\} \in E(G)} \min \{φ(u), φ(v) \}$. 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. 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.

Figures (5)

• Figure 1: Counter example
• Figure 2: $k+1$ blocks in an ordering of $V(G)$
• Figure 3: Vertices of equivalence class $A$ distributed in different blocks
• Figure 4: Cases of Lemma 4
• Figure 5: Graph $G_3$

Theorems & Definitions (14)

• definition thmcounterdefinition
• definition thmcounterdefinition
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
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• proposition thmcounterproposition: Lokshtanov lokshtanov2015parameterized
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• lemma thmcounterlemma