BranchAnBound分支定界.pptVIP

Data Compression and Multimedia Communication Lab. Branch and BoundSearching Strategies Branch-and-bound strategy 2 mechanisms: A mechanism to generate branches A mechanism to generate a bound so that many braches can be terminated. Branch-and-bound strategy It is efficient in the average case because many branches can be terminated very early. Although it is usually very efficient, a very large tree may be generated in the worst case. Recall the NPC problem. (Many NPC problem can be solved by BB efficiently in the average case; however, the worst case time complexity is still exponential.) Personnel Assignment Problem A linearly ordered set of persons P={P1, P2, …, Pn} where P1P2…Pn A partially ordered set of jobs J={J1, J2, …, Jn} Suppose that Pi and Pj are assigned to jobs f(Pi) and f(Pj) respectively. If f(Pi) ? f(Pj), then Pi ? Pj. Cost Cij is the cost of assigning Pi to Jj. We want to find a feasible assignment with the minimum cost. i.e. Xij = 1 if Pi is assigned to Jj Xij = 0 otherwise. Minimize ?i,j CijXij e.g. A partial ordering of jobs After topological sorting, one of the following topologically sorted sequences will be generated: One of feasible assignments: P1→J1, P2→J2, P3→J3, P4→J4 A Solution Tree All possible solutions can be represented by a solution tree. Cost Matrix Cost matrix Reduced Cost Matrix Cost matrix A reduced cost matrix can be obtained: subtract a constant from each row and each column respectively such that each row and each column contains at least one zero. Total cost subtracted: 12+26+3+10+3 = 54 This is a lower bound of our solution. Branch-and-Bound for the Personnel Assignment Problem Bounding of subsolutions: The traveling salesperson optimization problem Given a graph, the TSP Optimization problem is to find a tour, starting from any vertex, visiting every other vertex and returning to the starting vertex, with minimal cost. It is NP-complete We try to avoid n! exhaustive search b