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Improving the integrality gap for multiway cut
Mathematical Programming ( IF 2.7 ) Pub Date : 2020-03-06 , DOI: 10.1007/s10107-020-01485-2
Kristóf Bérczi , Karthekeyan Chandrasekaran , Tamás Király , Vivek Madan

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of k terminal nodes, and the goal is to partition the node set of the graph into k non-empty parts each containing exactly one terminal, so that the total weight of the edges crossing the partition is minimized. The multiway cut problem for $$k\ge 3$$ k ≥ 3 is APX-hard. For arbitrary k , the best-known approximation factor is 1.2965 due to Sharma and Vondrák (Proceedings of the forty-sixth annual ACM symposium on theory of computing, STOC, 2014) while the best known inapproximability result due to Angelidakis et al. (Integer programming and combinatorial optimization, IPCO, 2017) rules out efficient algorithms to achieve an approximation factor less than 1.2 under the unique games conjecture (UGC). In this work, we improve the lower bound to $$1.20016$$ 1.20016 under UGC by constructing an integrality gap instance for the CKR relaxation. The CKR relaxation embeds the graph into a simplex and it is known that its integrality gap translates to inapproximability under UGC. A technical challenge in improving the integrality gap has been the lack of geometric tools to understand higher-dimensional simplices. Our instance is a non-trivial 3-dimensional instance that overcomes this technical challenge. We analyze the gap of the instance by viewing it as a convex combination of 2-dimensional instances and a uniform 3-dimensional instance. We believe that this technique could be exploited further to construct instances with larger integrality gap. One of the ingredients of our proof technique is a generalization of a result on Sperner admissible labelings due to Mirzakhani and Vondrák (Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms, SODA, 2015) that might be of independent combinatorial interest.

中文翻译:

改善多路切割的完整性差距

在多路切割问题中,我们给出了一个具有非负边权重的无向图和 k 个终端节点的集合,目标是将图的节点集划分为 k 个非空部分,每个部分只包含一个终端,从而使穿过分区的边的总重量最小化。$$k\ge 3$$ k ≥ 3 的多路切割问题是 APX 难的。对于任意 k ,由于 Sharma 和 Vondrák(第 46 届年度 ACM 计算理论研讨会论文集,STOC,2014 年),最著名的近似因子是 1.2965,而最著名的不可逼近性结果是由 Angelidakis 等人提出的。(整数规划和组合优化,IPCO,2017)排除了在独特游戏猜想(UGC)下实现小于 1.2 的近似因子的有效算法。在这项工作中,我们通过构建 CKR 松弛的完整性差距实例,将 UGC 下的下限提高到 $1.20016$$1.20016。CKR 松弛将图嵌入到单纯形中,并且已知其完整性差距在 UGC 下转化为不可近似性。改善完整性差距的技术挑战是缺乏理解高维单纯形的几何工具。我们的实例是一个非平凡的 3 维实例,它克服了这一技术挑战。我们通过将实例视为 2 维实例和均匀 3 维实例的凸组合来分析实例的间隙。我们相信可以进一步利用这种技术来构建具有更大完整性差距的实例。
更新日期:2020-03-06
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