Faster Exponential-Time Approximation Algorithms Using Approximate Monotone Local Search

TL;DR

The potential of approximate monotone local search is demonstrated by deriving new and faster exponential approximation algorithms for Vertex Cover, $3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut.

Abstract

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J. ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size $n$ which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized $α$-approximation algorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solution size, can be used to derive an $α$-approximation randomized algorithm that runs in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in (1,1+\frac{c-1}α)$ such that $\mathcal{D}(\frac{1}α\|\frac{d-1}{c-1})=\frac{\ln c}α$ and $\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for $α=1$, and is strictly better when $α>1$, for any $c > 1$. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, $3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n \cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximation running in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011].

Figures (2)

• Figure 1: Results for Vertex Cover and $3$-Hitting Set. A dot at $(\alpha,d)$ means that the respective algorithm outputs an $\alpha$-approximation in time $\mathcal{O}^*(d^n)$.
• Figure 2: Results for Vertex Cover and $3$-Hitting Set. A value $d$ at the column of an approximation ratio $\alpha$ means that the respective algorithm outputs an $\alpha$-approximation in time $\mathcal{O}^*(d^n)$.

Theorems & Definitions (52)

• Lemma 1.1: $\star$
• Theorem 2.1: Approximate Monotone Local Search
• Theorem 2.2: Derandomization Approximate Monotone Local Search
• Lemma 3.1: Correctness
• proof
• Lemma 3.2: Running time
• proof
• Lemma 3.3: $\star$
• Lemma 3.4
• proof