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 least cost subset of vertices $S \subseteq V$ such that $G-S$ is acyclic. FVS is a well-known NP-hard problem with no $(2-\epsilon)$-approximation assuming the Unique Games Conjecture and it admits a $2$-approximation via combinatorial local-ratio methods (Bafna, Berman and Fujito, Algorithms and Computations '95; Becker and Geiger, Artificial Intelligence '96) which can also be interpreted as LP-based primal-dual algorithms (Chudak, Goemans, Hochbaum and Williamson, Operations Research Letters '98). 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.

翻译：本文考虑无向图中的反馈顶点集问题（FVS）。FVS的输入为带非负顶点代价的无向图$G=(V,E)$，目标是找到最小代价顶点子集$S\subseteq V$，使得$G-S$无环。FVS是著名的NP困难问题，在唯一博弈猜想下不存在$(2-\epsilon)$-近似算法，但可通过组合局部比率方法（Bafna, Berman and Fujito, Algorithms and Computations '95; Becker and Geiger, Artificial Intelligence '96）获得2-近似解，该方法亦可解释为基于线性规划的原始对偶算法（Chudak, Goemans, Hochbaum and Williamson, Operations Research Letters '98）。尽管这些算法已存在数十年，目前仍未有已知的多项式时间可解线性规划松弛能保证FVS的整数性间隙不超过2。近期研究（Chekuri and Madan SODA '16）为更一般的子集反馈顶点集问题（Subset FVS）设计了多项式规模线性规划松弛，并证明其整数性间隙对Subset FVS至多为13（从而对FVS亦然）。受此知识空白的驱动，我们对FVS及相关问题开展多面体研究。本文为FVS构建了新的整数线性规划（ILP），其线性规划松弛可在多项式时间内求解，且整数性间隙不超过2。在此过程中产生的新见解还使我们能够证明（Chekuri and Madan, SODA '16）中的公式对FVS的整数性间隙至多为2。本文对FVS的研究灵感来自紧密相关的伪森林删除集问题（PFDS）的新公式及多面体结果，而PFDS的公式又源于与最密集子图问题的联系。我们还提出了FVS线性规划松弛的一个极值点猜想，并通过PFDS的对应结果为该猜想提供证据。