8.997 Topics in Combinatorial Optimization

8.997 Topics in Combinatorial Optimization 1.1 Example: Circulation

1 Special cases of submodular flows We saw last time that orientation of a 2k-edge connected graph into a k-arc connected digraph and the Lucchesi and Younger Theorem were special cases of submodular flows. Other familiar problems can also be phrased as submodular flows. Let C = 2 V \ {∅, V } and let f be identically zero. Then for any U ∈ C, x(δ in (U)) − x(δ out (U) ≤ 0 (1) x(δ in (V \ U)) − ...

8.997 Topics in Combinatorial Optimization 18 Orientations, Directed Cuts and Submodular Flows 18.1 Graph Orientations

Proof: ⇐: Fix a strongly-connected orientation D. For any non-empty U ⊂ V , we may choose u ∈ U and v ∈ V \U . Since D is strongly connected, there is a directed u-v path and a directed v-u δ in path. Thus |δ out (U)| ≥ 1 and | D (U)| ≥ 1, implying |δG(U)| ≥ 2. D ⇒: Since G is 2-edge-connected, it has an ear decomposition. We proceed by induction on the number of ears. If G is a cycle then we m...

Topics in random graphs, combinatorial optimization, and statistical inference

Optimization of profit and customer satisfaction in combinatorial production and purchase model by genetic algorithm

Optimization of inventory costs is the most important goal in industries. But in many models, the constraints are considered simple and relaxed. Some actual constraints are to consider the combinatorial production and purchase models in multi-products environment. The purpose of this article is to improve the efficiency of inventory management and find the economic order quantity and economic p...

Winner Determination in Combinatorial Auctions using Hybrid Ant Colony Optimization and Multi-Neighborhood Local Search

A combinatorial auction is an auction where the bidders have the choice to bid on bundles of items. The WDP in combinatorial auctions is the problem of finding winning bids that maximize the auctioneer’s revenue under the constraint that each item can be allocated to at most one bidder. The WDP is known as an NP-hard problem with practical applications like electronic commerce, production manag...