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Unformatted text preview: Chapter 15 Network Flow IV - Applications II By Sariel Har-Peled , October 19, 2007 Version: 0.11 15.1 Airline Scheduling Problem 15.1.1 Given information about flights that an airline needs to provide, generate a prof- itable schedule. The input is a detailed information about legs of flight that the airline need to serve. We denote this set of flights by F . We would like to find the minimum number of airplanes needed to carry out this schedule. For an example of possible input, see Figure 15.1 (i). 1: Boston (depart 6 A.M.) - Washington DC (arrive 7 A.M,). 2: Urbana (depart 7 A.M.) - Champaign (arrive 8 A.M.) 3: Washington (depart 8 A.M.) - Los Angeles (arrive 11 A.M.) 4: Urbana (depart 11 A.M.) - San Francisco (arrive 2 P.M.) 5: San Francisco (depart 2:15 P.M.) - Seattle (arrive 3:15 P.M.) 6: Las Vegas (depart 5 P.M.) - Seattle (arrive 6 P.M.). 1 2 3 4 5 6 (i) (ii) Figure 15.1: (i) a set F of flights that have to be served, and (ii) the corresponding graph G representing these flights. We can use the same airplane for two segments i and j if the destination of i is the origin of the segment j and there is enough time in between the two flights for required maintenance. Alternatively, the airplane can fly from dest( i ) to origin( j ) (assuming that the time constraints are satisfied). This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. 1 Example 15.1.2 As a concrete example, consider the flights: 1. Boston (depart 6 A.M.) - Washington D.C. (arrive 7 A.M,). 2. Washington (depart 8 A.M.) - Los Angeles (arrive 11 A.M.) 3. Las Vegas (depart 5 P.M.) - Seattle (arrive 6 P.M.) This schedule can be served by a single airplane by adding the leg Los-Angeles (depart 12 noon)- Las Vegas (1 P,M.) to this schedule. 15.1.1 Modeling the problem The idea is to model the feasibility constraints by a graph. Specifically, G is going to be a directed graph over the flight legs. For i and j , two given flight legs, the edge ( i j ) will be present in the graph G if the same airplane can serve both i and j ; namely, the same airplane can perform leg i and afterwards serves the leg j . Thus, the graph G is acyclic. Indeed, since we can have an edge ( i j ) only if the flight j comes after the flight i (in time), it follows that we can not have cycles. We need to decide if all the required legs can be served using only k airplanes? 15.1.2 Solution The idea is to perform a reduction of this problem to the computation of circulation. Specifically, we construct a graph H , as follows: u 1 v 1 u 2 v 2 u 3 v 3 u 4 v 4 u 5 v 5 u 6 v 6 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1- k s k t k Figure 15.2: The resulting graph H for the instance of airline scheduling from Fig- ure 15.1....
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