本文主要研究一类整数规划的一对新的近似算法。首先，对于覆盖整数规划{mincx：Ax >= b，0 <= x <= d}，其中A每行至多有k个非零值，我们给出了一个k-近似算法. (We假设A，b，c，d都是非负的。对于任意k >= 2且eps>0，如果P！= NP这个比率不能提高到k-1-eps，并且在唯一博弈猜想下这个比率不能提高到k-eps。一个关键的想法是用其他具有更好的舍入特性但具有相同的非负积分解的约束来替换单个约束;另一个关键因素是背包覆盖不等式。其次，对于填充整数规划{max cx：Ax <= b，0 <= x <= d}，其中A每列至多有k个非零值，我们给出了一个（2k^2+2）-近似算法.我们的方法建立在迭代LP松弛框架。此外，我们得到了改进的近似的第二个问题时，k=2，并为这两个问题时，每一个A_{ij}是小的相比，b_i。最后，我们证明了一个17/16-不可逼近覆盖整数规划，每列最多有两个非零。
英文摘要：
The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx: Ax >= b, 0 <= x <= d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k >= 2 and eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the unique games conjecture this ratio cannot be improved to k-eps. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx: Ax <= b, 0 <= x <= d} where A has at most k nonzeroes per column, we give a (2k^2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A_{ij} is small compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.
Chinese University of Hong Kong（香港中文大学）；Microsoft Research New England（微软新英格兰研究院）；UC Berkeley（加州大学伯克利分校）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
Approximability of Electrical Distribution Network Reconfiguration for General Graphs
Forschungszentrum Jülich（于利希研究中心）；Hasso Plattner Institute, University of Potsdam（波茨坦大学哈索·普拉特纳研究所）；RWTH Aachen University（亚琛工业大学）；TU Clausthal（克劳塔尔工业大学）
