Joint Economic Lot-sizing Problems (JELPs) have received a great deal of attention in recent years because they are the bedrock of pushing forward the frontiers of knowledge in supply chain (SC). J...

Joint Economic Lot-sizing in Multi-product Multi-level Integrated Supply Chains: Generalized Benders Decomposition

Survey: Hierarchical facility location problem: Models, classifications, techniques, and applications

ACM-SIAM Symposium on Discrete Algorithms

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.

Are stable instances easy

Multi-level facility location problems

We consider the facility location problem with submodular penalties (FLPSP) and the facility location problem with linear penalties (FLPLP), two extensions of the classical facility location problem (FLP). First, we introduce a general algorithmic framework for a class of covering problems with submodular penalties, extending the recent result of Geunes et al. (Math Program 130:85---106, 2011) with linear penalties. This framework leverages existing approximation results for the original covering problems to obtain corresponding results for their counterparts with submodular penalties. Specifically, any LP-based $$\alpha $$?-approximation for the original covering problem can be leveraged to obtain an $$\left( 1-e^{-1/\alpha }\right) ^{-1}$$1-e-1/?-1-approximation algorithm for the counterpart with submodular penalties. Consequently, any LP-based approximation algorithm for the classical FLP (as a covering problem) can yield, via this framework, an approximation algorithm for the counterpart with submodular penalties. Second, by exploiting some special properties of submodular/linear penalty function, we present an LP rounding algorithm which has the currently best $$2$$2-approximation ratio over the previously best $$2.375$$2.375 by Li et al. (Theoret Comput Sci 476:109---117, 2013) for the FLPSP and another LP-rounding algorithm which has the currently best $$1.5148$$1.5148-approximation ratio over the previously best $$1.853$$1.853 by Xu and Xu (J Comb Optim 17:424---436, 2008) for the FLPLP, respectively.

http://www.optimization-online.org/DB_FILE/2012/02/3343.pdf

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

We consider the facility location problem with submodular penalties (FLPSP), introduced by Hayrapetyan et al. (Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 933–942, 2005), who presented a 2.50-approximation algorithm that is non-combinatorial because this algorithm has to solve the LP-relaxation of an integer program with exponential number of variables. The only known polynomial algorithm for this exponential LP is via the ellipsoid algorithm as the corresponding separation problem for its dual program can be solved in polynomial time. By exploring the properties of the submodular function, we offer a primal-dual 3-approximation combinatorial algorithm for this problem.

A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties

The k-level facility location game

In order to cluster the textual data with high dimension in modern data analysis, the spherical k-means clustering is presented. It aims to partition the given points with unit length into k sets so as to minimize the within-cluster sum of cosine dissimilarity. In this paper, we mainly study seeding algorithms for spherical k-means clustering, for its special case (with separable sets), as well as for its generalized problem ($$\alpha $$-spherical k-means clustering). About the spherical k-means clustering with separable sets, an approximate algorithm with a constant factor is presented. Moreover, it can be generalized to the $$\alpha $$-spherical separable k-means clustering. By slickly constructing a useful function, we also show that the famous seeding algorithms such as k-means++ and k-means|| for k-means problem can be applied directly to solve the $$\alpha $$-spherical k-means clustering. Except for theoretical analysis, the numerical experiment is also included.

Dachuan Xu

Papers

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties

The k-level facility location game

The seeding algorithms for spherical k -means clustering

A cost-sharing method for an economic lot-sizing game