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 $(2-\epsilon)$-approximation for any fixed $\epsilon > 0$ 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.

翻译：我们研究无向图中的反馈顶点集问题。FVS的输入是一个带非负顶点成本的无向图$G=(V,E)$。目标是找到一个最小成本的顶点子集$S \subseteq V$，使得$G-S$是无环的。FVS是一个著名的NP难问题，并且在唯一游戏猜想成立的前提下，对于任意固定的$\epsilon > 0$，不存在$(2-\epsilon)$-近似算法。目前存在组合的2-近似算法以及基于原始对偶的2-近似算法。尽管这些算法已存在数十年，但尚未发现一个多项式时间可解的线性规划松弛，其可证明的整数间隙至多为2。近期研究（Chekuri and Madan, SODA '16）针对一个更一般的问题——子集反馈顶点集，提出了一个多项式规模的线性规划松弛，并证明其整数间隙对于子集反馈顶点集至多为13，因此对于FVS也至多为13。受此认知差距的启发，我们对FVS及相关问题展开了多面体研究。在本工作中，我们为FVS构建了新的整数线性规划模型，其线性规划松弛可在多项式时间内求解，且整数间隙至多为2。这一过程中的新见解也使我们能够证明（Chekuri and Madan, SODA '16）中的模型对于FVS的整数间隙至多为2。我们在FVS上取得的成果受到了与FVS密切相关的伪森林删除集问题的新模型和多面体结果的启发。而我们对PFDS的建模又受到了其与最密子图问题之间联系的启发。我们还针对FVS的一个线性规划松弛提出了极值点性质的猜想，并通过PFDS的相应结果为该猜想提供了证据。