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LP Relaxation and Tree Packing for Minimum $k$-Cut
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-06-16 , DOI: 10.1137/19m1299359 Chandra Chekuri , Kent Quanrud , Chao Xu
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-06-16 , DOI: 10.1137/19m1299359 Chandra Chekuri , Kent Quanrud , Chao Xu
SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1334-1353, January 2020.
Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46--76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum $k$-cut problem via greedy recursive tree packings [M. Thorup, Minimum $k$-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for cͅut proposed by Naor and Rabani [Tree packing and approximating $k$-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the $k$-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of $n$. We also improve the bound on the number of $\alpha$-approximate $k$-cuts. Second, we give a simple proof that the integrality gap of the LP is $2(1-1/n)$. Third, we show that an optimum solution to the LP relaxation, for all values of $k$, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.
中文翻译：
最少$ k $的LP放松和树木包装
SIAM离散数学杂志，第34卷，第2期，第1334-1353页，2020年1月。
Karger使用生成树填充[DR Karger，J. ACM，47（2000），第46--76页]导出了针对全局最小割问题的近似线性时间随机算法以及近似数的界最低限度的削减。这与他著名的随机收缩算法[DR Karger，图优化问题中的随机采样，博士，论文，斯坦福大学，加利福尼亚州斯坦福，1995，DR Karger和C. Stein，J。ACM，43（1996），第601--640页]。Thorup通过贪婪的递归树包装为最小的$ k $ -cut问题开发了一种快速确定性算法[M. Thorup，通过确定性贪心树打包进行的最低$ k $方式削减，在第40届ACM计算理论年度研讨会论文集中，ACM，2008年，第159--166页]。在本文中，我们将回顾Naor和Rabani提出的LP松弛的性质[在第十二届ACM-SIAM离散算法年度学术会议论文集中，“树包装和近似$ k $削减”。103，SIAM，费城，2001年，第26--27页]，并在[C. Chekuri，S。Guha和J. Naor，SIAM J. Discrete Math。，第20卷，2006年，第261--271页]。我们证明了LP的对偶会产生一个树包装，当与LP的完整性缺口的上限结合时，可以轻松透明地将Karger的mincut分析扩展到$ k $ -cut问题。除了简化算法及其分析之外，这还使我们可以将Thorup算法的运行时间缩短$ n $倍。我们还改善了$ \ alpha $-近似$ k $削减数量的界限。第二，我们给出一个简单的证明，LP的完整性差距为$ 2（1-1 / n）$。第三，我们表明，对于$ k $的所有值，LP松弛的最佳解决方案完全由输入图的分区的主序列确定。这使我们可以将LP松弛与Barahona的Lagrangean松弛方法联系起来[Oper。Res。Lett。，26（2000），pp.99--105]和Ravi和Sinha [European J.Oper。Res。186（2008），第77--90页]；它还表明，Thorup考虑的理想化递归树包装为LP提供了最佳的双重解决方案。Res。Lett。，26（2000），pp.99--105]和Ravi和Sinha [European J.Oper。Res。186（2008），第77--90页]；它还表明，Thorup考虑的理想化递归树包装为LP提供了最佳的双重解决方案。Res。Lett。，26（2000），pp.99--105]和Ravi和Sinha [European J.Oper。Res。186（2008），第77--90页]；它还表明，Thorup考虑的理想化递归树包装为LP提供了最佳的双重解决方案。
更新日期：2020-06-30
Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46--76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum $k$-cut problem via greedy recursive tree packings [M. Thorup, Minimum $k$-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for cͅut proposed by Naor and Rabani [Tree packing and approximating $k$-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the $k$-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of $n$. We also improve the bound on the number of $\alpha$-approximate $k$-cuts. Second, we give a simple proof that the integrality gap of the LP is $2(1-1/n)$. Third, we show that an optimum solution to the LP relaxation, for all values of $k$, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.
中文翻译：
最少$ k $的LP放松和树木包装
SIAM离散数学杂志，第34卷，第2期，第1334-1353页，2020年1月。
Karger使用生成树填充[DR Karger，J. ACM，47（2000），第46--76页]导出了针对全局最小割问题的近似线性时间随机算法以及近似数的界最低限度的削减。这与他著名的随机收缩算法[DR Karger，图优化问题中的随机采样，博士，论文，斯坦福大学，加利福尼亚州斯坦福，1995，DR Karger和C. Stein，J。ACM，43（1996），第601--640页]。Thorup通过贪婪的递归树包装为最小的$ k $ -cut问题开发了一种快速确定性算法[M. Thorup，通过确定性贪心树打包进行的最低$ k $方式削减，在第40届ACM计算理论年度研讨会论文集中，ACM，2008年，第159--166页]。在本文中，我们将回顾Naor和Rabani提出的LP松弛的性质[在第十二届ACM-SIAM离散算法年度学术会议论文集中，“树包装和近似$ k $削减”。103，SIAM，费城，2001年，第26--27页]，并在[C. Chekuri，S。Guha和J. Naor，SIAM J. Discrete Math。，第20卷，2006年，第261--271页]。我们证明了LP的对偶会产生一个树包装，当与LP的完整性缺口的上限结合时，可以轻松透明地将Karger的mincut分析扩展到$ k $ -cut问题。除了简化算法及其分析之外，这还使我们可以将Thorup算法的运行时间缩短$ n $倍。我们还改善了$ \ alpha $-近似$ k $削减数量的界限。第二，我们给出一个简单的证明，LP的完整性差距为$ 2（1-1 / n）$。第三，我们表明，对于$ k $的所有值，LP松弛的最佳解决方案完全由输入图的分区的主序列确定。这使我们可以将LP松弛与Barahona的Lagrangean松弛方法联系起来[Oper。Res。Lett。，26（2000），pp.99--105]和Ravi和Sinha [European J.Oper。Res。186（2008），第77--90页]；它还表明，Thorup考虑的理想化递归树包装为LP提供了最佳的双重解决方案。Res。Lett。，26（2000），pp.99--105]和Ravi和Sinha [European J.Oper。Res。186（2008），第77--90页]；它还表明，Thorup考虑的理想化递归树包装为LP提供了最佳的双重解决方案。Res。Lett。，26（2000），pp.99--105]和Ravi和Sinha [European J.Oper。Res。186（2008），第77--90页]；它还表明，Thorup考虑的理想化递归树包装为LP提供了最佳的双重解决方案。
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