Approximately covering vertices by order-$5$ or longer paths

TL;DR

A new approximation algorithm is presented which achieves a ratio of $2.511$ and runs in $O(|V|^{2.5} |E|^2)$ time and is based on maximum matching, maximum path-cycle cover, and recursion.

Abstract

This paper studies $MPC^{5+}_v$, which is to cover as many vertices as possible in a given graph $G=(V,E)$ by vertex-disjoint $5^+$-paths (i.e., paths each with at least five vertices). $MPC^{5+}_v$ is NP-hard and admits an existing local-search-based approximation algorithm which achieves a ratio of $\frac {19}7\approx 2.714$ and runs in $O(|V|^6)$ time. In this paper, we present a new approximation algorithm for $MPC^{5+}_v$ which achieves a ratio of $2.511$ and runs in $O(|V|^{2.5} |E|^2)$ time. Unlike the previous algorithm, the new algorithm is based on maximum matching, maximum path-cycle cover, and recursion.

Figures (7)

• Figure 1: A typical case for a single repetition of Step 1.1 that involves four $2$-paths. The filled (respectively, blank) vertices are in (respectively, not in) $V(M)$. The thick (respectively, thin) edges are in the matching $M$ (respectively, in $H$ but not in $M$) and the yellow (respectively, green) edges are in $\mathcal{M}$ (respectively, not in $E(H)$). In the left-hand-side graph, we have $q_4(H) = 0$ and the left green edge and the $5$-path form an augmenting pair; modifying with this augmenting pair results in the right-hand-side graph for which $q_4(H) = 1$. The yellow edge emerges and will be added to $H$ afterwards.
• Figure 2: An illustration of modifying $K$ into $\widetilde{K}$. The filled (respectively, blank) vertices are in (respectively, not in) $V(M)$. The thick (thin, respectively) edges are in the matching $M$ (not in $M \cup C$ but in $H$, respectively) and the green (yellow, respectively) edges are in $C$ (in $\mathcal{M}$, see Step 1.2, respectively).
• Figure 3: All possible structures of a critical component of $H + C$. In these structures, the yellow vertices are $2$-anchors and the filled (white, respectively) vertices are in (not in, respectively) $V(M)$.
• Figure 4: All possible structures of a responsible components of $H + C$. The filled (white, respectively) vertices are in (not in, respectively) $V(M)$. Moreover, the yellow vertices are $2$-anchors while the red vertices are responsible $1$-anchors.
• Figure 5: Illustration of two representative possible cases in Operation \ref{['op01']}, in which $v_2$ is a $0$-anchor and a $1$-anchor, respectively, and the green edge is the edge $\{v_1, v_2\}$.

Theorems & Definitions (43)

• Lemma 1
• Definition 1
• Definition 2
• Definition 3
• Lemma 2
• Lemma 3
• Lemma 4
• Definition 4
• Lemma 5
• Definition 5