Discrete Optimization Talks (DOTs)

Abstract: Strong branching is a well-known variable selection rule that is known experimentally to produce significantly smaller branch-and-bound trees in comparison to all other known variable selection rules. In this paper, we attempt an analysis of the performance of the strong-branching rule both from a theoretical and a computational perspective. On the positive side, we identify vertex cover as a class of instances where this rule provably works well. In particular, for vertex cover we present: 1) an upper bound on the size of the branch-and-bound tree using strong-branching as a function of the additive integrality gap; 2) show how the Nemhauser-Trotter property of persistency, which can be used as a pre-solve technique for vertex cover, is being recursively and consistently used throughout the strong-branching tree; 3) and finally provide an example of a vertex cover instance where not using strong-branching leads to a tree that has at least exponentially more nodes than the branch-and-bound tree based on strong-branching. On the negative side for strong-branching, we identify another class of instances where the strong-branching tree is exponentially larger than another branch-and-bound tree for solving these instances. On the computational side, we conduct experiments on various types of instances, like the lot-sizing problem and its variants, packing integer programs (IP), covering IPs, chance constrained IPs, vertex cover, etc., to understand how much larger is the size of the strong-branching based branch-and-bound tree in comparison to the optimal branch-and-bound tree. The main take-away from these experiments (on small instances) is that for all these instances the size of the strong-branching tree is within a factor of two of the size of the optimal branch-and-bound tree. 

Abstract: Modern traceability technologies promise to improve supply chain management by simplifying recalls, increasing visibility, or verifying sustainable supplier practices. Initiatives leading the implementation of traceability technologies must choose the least-costly set of firms — or seed set — to target for early adoption. Choosing this seed set is challenging because firms are part of supply chains interlinked in complex networks, yielding an inherent supply chain effect: benefits obtained from traceability are conditional on technology adoption by a subset of firms in a product’s supply chain. We prove that the problem of selecting the least- costly seed set in a supply chain network is hard to solve and even approximate within a polylogarithmic factor. Nevertheless, we provide a novel linear programming-based algorithm to identify the least-costly seed set. The algorithm is fixed-parameter tractable in the supply chain network’s treewidth, which we show to be low in real-world supply chain networks. The algorithm also enables us to derive easily-computable bounds on the cost of selecting an optimal seed set. Finally, we leverage our algorithms to conduct large-scale numerical experiments that provide insights into how the supply chain network structure influences diffusion. These insights can help managers optimize their technology diffusion strategy.