8.438 Advanced Combinatorial Optimization
نویسنده
چکیده
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 path. Thus |δ D(U)| ≥ 1 and |δ − D(U)| ≥ 1, implying |δG(U)| ≥ 2. ⇒: 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 may orient the edges to form a directed cycle D, which is obviously strongly connected. Otherwise, G consists of an ear P and subgraph G′ with a strongly connected orientation D′. The ear is an undirected path with endpoints x, y ∈ V (G′) (possibly x = y). We orient P so that it is a directed path from x to y and add this to D′, thereby obtaining an orientation D of G.
منابع مشابه
8.438 Advanced Combinatorial Optimization 2 Iwata and Orlin's Algorithm
Given a finite set V with n elements, a function f : 2 → Z is submodular if for all X,Y ⊆ V , f(X ∪ Y ) + f(X ∩ Y ) ≤ f(X) + f(Y ). Submodular functions frequently arise in combinatorial optimization. For example, the cut function in a weighted undirected graph and the rank function of a matroid are both submodular. Submodular function minimization is the problem of finding the global minimum o...
متن کاملOptimization 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...
متن کاملWinner 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...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009