Combinatorial Optimization Chapter 1. Problems and Algorithms  1.1 Two Problems (representative problems)  The Traveling Salesman Problem 47 drilling platforms off the coast of Nigeria. Need to visit platforms by a helicopter to regulate the rates of flows. Starting from the onshore base, what is the cheapest (shortest) route to visit each platform once and return to the base? Euclidean traveling salesman problem. Input: a set V of points in the Euclidean plane. Objective: Find a simple circuit passing through the points for which the sum of the lengths of the edges is minimized. Enumeration: (n-1)!/2 ways (For 23 platforms, takes 178 centuries if can check one possibility in one nanosec.) No efficient algorithm is known. (NP-hard)

 Heuristics? nearest neighbor, nearest insertion, farthest insertion, cheapest insertion, 2-opt and 3-opt, Lin-Kernighan (meta heuristic), … Fig 1.1 shows nearest neighbor solution on 1173 node problem: cost 67,822 (opt sol is 56892). Empirical tests showed nearest neighbor give solutions about 1.26 times optimal. Lin-Kernighan:  TSP site: (largest problem solved optimally: 85,900 cities, 2006)) Combinatorial Optimization

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4  There exist graphs for which Hamiltonian cycle doesn’t exist. Petersen graph

Combinatorial Optimization

6  The Matching Problem Drawing a logic circuit using a plotter, assume graph is connected. Total drawing time = pen-down + pen-up minimize pen-up time odd nodes even nodes

Combinatorial Optimization  Seven bridges in Königsberg: Euler’s problem

Combinatorial Optimization

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Combinatorial Optimization

Combinatorial Optimization

Combinatorial Optimization

Combinatorial Optimization  Example:  Note the difference between the circuit drawing and the chinese postman problem.  Other variations: directed graph (easy), mixed graph (NP-hard)  More on this problem later when we study T-join.

Combinatorial Optimization

Combinatorial Optimization