解决线性系统不可行性的参数化方法
Resolving Infeasibility of Linear Systems: A Parameterized Approach
摘要：
判定大型线性方程组和不等式组的可行性是最基本的算法任务之一。然而，由于数据不准确或建模误差，在实际应用中，人们经常面临的线性系统是不可行的。已经提出了广泛的理论和实践方法用于线性系统的后不可行性分析。这通常相当于检测小尺寸$k$的可行性阻塞，小尺寸$k$是一组等式和不等式，从大尺寸$m$的大系统中去除或扰动这些等式和不等式产生可行系统。这激发了一个参数化的方法对后不可行性分析，我们的目标是找到一个可行性阻滞剂的大小最$k$在固定参数的时间$f(k) \cdot m^{O(1)}$。我们建立参数化的棘手性（$W[1]$ -和$NP$ -硬度）的结果已经在非常有限的设置为不同的选择的参数的最大大小的删除集，正/负的右手边，treewide，pathwidth和treedepth。此外，我们排除了多项式压缩MinFB参数化的删除集的大小和负的右手边的数量。此外，我们开发的固定参数的算法参数化这些参数的各种组合时，系统的每一行对应的差异约束。我们的算法捕获的情况下，定向反馈弧集，一个基本的参数化问题，其固定参数的易处理性显示的陈等。（STOC 2008）。
英文摘要：
Deciding feasibility of large systems of linear equations and inequalities is one of the most fundamental algorithmic tasks. However, due to data inaccuracies or modeling errors, in practical applications one often faces linear systems that are infeasible. Extensive theoretical and practical methods have been proposed for post-infeasibility analysis of linear systems. This generally amounts to detecting a feasibility blocker of small size $k$, which is a set of equations and inequalities whose removal or perturbation from the large system of size $m$ yields a feasible system. This motivates a parameterized approach towards post-infeasibility analysis, where we aim to find a feasibility blocker of size at most $k$ in fixed-parameter time $f(k) \cdot m^{O(1)}$. We establish parameterized intractability ($W[1]$- and $NP$-hardness) results already in very restricted settings for different choices of the parameters maximum size of a deletion set, number of positive/negative right-hand sides, treewidth, pathwidth and treedepth. Additionally, we rule out a polynomial compression for MinFB parameterized by the size of a deletion set and the number of negative right-hand sides. Furthermore, we develop fixed-parameter algorithms parameterized by various combinations of these parameters when every row of the system corresponds to a difference constraint. Our algorithms capture the case of Directed Feedback Arc Set, a fundamental parameterized problem whose fixed-parameter tractability was shown by Chen et al. (STOC 2008).
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