Polyhedral Aspects of Feedback Vertex Set and Pseudoforest Deletion Set

TL;DR

This work advances polyhedral understanding of two related vertex-deletion problems, the Feedback Vertex Set ($FVS$) and Pseudoforest Deletion Set ($PFDS$). It builds polynomial-sized ILPs whose LP relaxations have integrality gaps of at most 2, linking weak-density and orientation-based formulations with a cycle-cover perspective to bound the gap and enable rounding approaches. The authors prove strong extreme-point properties for the weak-density and orientation polyhedra (e.g., a $1/3$-level coordinate bound under certain conditions), derive a $2$-approximation framework via primal-dual analyses, and establish a polynomial-time separation oracle for the $2$-pseudotree cover constraints. They also connect their PFDS formulations to Charikar’s densest-subgraph LP and show that their PFDS results extend to the $FVS$ setting, yielding practical LP-based tools and deeper polyhedral insights that bridge previous gaps in the literature. Overall, the paper provides concrete, scalable LP relaxations with provable tight integrality gaps and multiple robust formulations for $FVS$ and PFDS, contributing to approximation and algorithm design in graph deletion problems.

Abstract

We consider the feedback vertex set problem in undirected graphs (FVS). The input to FVS is an undirected graph $G=(V,E)$ with non-negative vertex costs. The goal is to find a minimum cost subset of vertices $S \subseteq V$ such that $G-S$ is acyclic. FVS is a well-known NP-hard problem and does not admit a $(2-ε)$-approximation for any fixed $ε> 0$ assuming the Unique Games Conjecture. There are combinatorial $2$-approximation algorithms and also primal-dual based $2$-approximations. Despite the existence of these algorithms for several decades, there is no known polynomial-time solvable LP relaxation for FVS with a provable integrality gap of at most $2$. More recent work (Chekuri and Madan, SODA '16) developed a polynomial-sized LP relaxation for a more general problem, namely Subset FVS, and showed that its integrality gap is at most $13$ for Subset FVS, and hence also for FVS. Motivated by this gap in our knowledge, we undertake a polyhedral study of FVS and related problems. In this work, we formulate new integer linear programs (ILPs) for FVS whose LP-relaxation can be solved in polynomial time, and whose integrality gap is at most $2$. The new insights in this process also enable us to prove that the formulation in (Chekuri and Madan, SODA '16) has an integrality gap of at most $2$ for FVS. Our results for FVS are inspired by new formulations and polyhedral results for the closely-related pseudoforest deletion set problem (PFDS). Our formulations for PFDS are in turn inspired by a connection to the densest subgraph problem. We also conjecture an extreme point property for a LP-relaxation for FVS, and give evidence for the conjecture via a corresponding result for PFDS.

Figures (6)

• Figure 1: An example showing that the integrality gap of (\ref{['PFDS-LP:Orient-and-2PT-cover']}) tends to $2$ as $n$ tends to infinity. Consider the graph $G=(V, E)$ as shown above (where $K_n$ denotes the complete graph on $n$ vertices) with cost of every vertex $a\in V-\{u, v, w\}$ being $1$ and costs of vertices $u, v, w$ being infinite. The optimum value of (\ref{['PFDS-IP:Orient-and-2PT-cover']}) is $n-1$. The optimum value of (\ref{['PFDS-LP:Orient-and-2PT-cover']}) is at most $n/2$: the solution $x_a = 1/2$ for every vertex $a\in V-\{u, v, w\}$, $x_u = x_v = x_w = 0$, $y_{e, a}=0$ for all edges $e=ab$ where $a, b\in V-\{u, v, w\}$, $y_{uv, v}=y_{vw, w}=y_{wu,u}=1$, $y_{uv, u} = y_{vw, v} = y_{wu, w} = 0$, and $y_{ua,a}=1/2$, $y_{ua, u} = 0$ for every $a\in V-\{u, v,w\}$ is feasible for (\ref{['PFDS-LP:Orient-and-2PT-cover']}) and has cost $n/2$.
• Figure 2: The unique extreme point optimum for (\ref{['FVS-LP:weak-density-cycle-cover']}) along the all-ones objective direction on the complete graph $K_4$ is as shown. Consequently, the integrality gap of (\ref{['FVS-LP:weak-density-cycle-cover']}) is at least $3$.
• Figure 3: WD-CC Primal and Dual LPs for FVS
• Figure 4: (SFVS-IP: CM)
• Figure 5: (FVS-LP: CM)

Theorems & Definitions (83)

• Conjecture 1
• Theorem 1
• Theorem 2
• Corollary 2.1
• Theorem 3
• Corollary 3.1
• Theorem 4
• Remark 1
• Theorem 5
• Theorem 6