Breaking the Barrier $2^k$ for Subset Feedback Vertex Set in Chordal Graphs

TL;DR

This work addresses SFVS-C in chordal graphs by breaking the $2^{k}$-barrier and achieving an $\\(O^{*}(1.820^{k})\\)$-time algorithm. It combines reduction rules, branching analyses, and a structural DM decomposition to exploit the split/chordal graph geometry, including a novel DM Reduction and a divide-and-conquer strategy that reduces SFVS-C to SFVS-S subproblems. The approach hinges on a tailored measure $\mu(\mathcal{I}) = k - \frac{2}{3}|A|$ for Good Instances and a careful analysis of branching factors, yielding substantial improvements over prior $2^{k}$-based bounds. The results advance exact algorithms for SFVS-C and its variants, with implications for related problems like PCMIS and other $3$-Hitting Set reformulations.

Abstract

The Subset Feedback Vertex Set problem (SFVS), to delete $k$ vertices from a given graph such that any vertex in a vertex subset (called a terminal set) is not in a cycle in the remaining graph, generalizes the famous Feedback Vertex Set problem and Multiway Cut problem. SFVS remains NP-hard even in split and chordal graphs, and SFVS in Chordal Graphs (SFVS-C) can be considered as an implicit 3-Hitting Set problem. However, it is not easy to solve SFVS-C faster than 3-Hitting Set. In 2019, Philip, Rajan, Saurabh, and Tale (Algorithmica 2019) proved that SFVS-C can be solved in $\mathcal{O}^{*}(2^{k})$ time, slightly improving the best result $\mathcal{O}^{*}(2.076^{k})$ for 3-Hitting Set. In this paper, we break the "$2^{k}$-barrier" for SFVS-C by giving an $\mathcal{O}^{*}(1.820^{k})$-time algorithm. Our algorithm uses reduction and branching rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer method.

Figures (1)

• Figure 1: A bipartite graph $F$ with bipartition $V(F) = A \cup B$, where $A = \left\{\, u_{i} \,\right\}_{i = 1}^{7}$ and $B = \left\{\, v_{i} \,\right\}_{i = 1}^{7}$. The thick edges form a maximum matching of $F$. The Dulmage-Mendelsohn decomposition of $F$ is $(C, H, R)$ with $C = \left\{\, u_{6}, u_{7}, v_{5}, v_{6}, v_{7} \,\right\}$, $H = \left\{\, u_{4}, u_{5}, v_{4} \,\right\}$, and $R = \left\{\, u_{1}, u_{2}, u_{3}, v_{1}, v_{2}, v_{3} \,\right\}$. If $F$ is an auxiliary subgraph of an instance of SFVS-S, then $\hat{A} = \left\{\, u_{1}, u_{2}, u_{3}, u_{6}, u_{7} \,\right\}$ (denoted by blue vertices) and $\hat{B} = \left\{\, v_{1}, v_{2}, v_{3}, v_{4} \,\right\}$ (denoted by green vertices).

Theorems & Definitions (8)

• Lemma 1: algorithmicaPhilipRST19
• Definition 2: Dulmage-Mendelsohn Decomposition elsevierLocaszP86jcssChenK03
• Lemma 3
• Lemma 4
• Lemma 5
• Definition 6: Good Instances
• Lemma 7
• Definition 8: The Measure of Good Instances