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The knapsack problem and the minimum spanning tree problem are both fundamental in operations research and computer science. We are concerned with a combination of these two problems. That is, we are given a knapsack of a fixed capacity, as well as an undirected graph where each edge is associated with profit and weight. The problem is to fill the knapsack with a feasible spanning tree such that the tree profit is maximized. We prove this problem NP-hard, present upper and lower bounds, develop a branch-and-bound algorithm to solve the problem to optimality and propose a shooting method to accelerate computation. We evaluate the developed algorithm through a series of numerical experiments for various types of test problems.

AbsttuctThere are many studies which consider an optimal tour on a given graph including the well-known

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We are concerned with a variation of the knapsack problem, the bi-objective max-min knapsack problem (BKP), where the values of items differ under two possible scenarios. We give a heuristic algorithm and an exact algorithm to solve this problem. In particular, we introduce a surrogate relaxation to derive upper and lower bounds very quickly, and apply the pegging test to reduce the size of BKP. We also exploit this relaxation to obtain an upper bound in the branch-andbound algorithm to solve the reduced problem. To further reduce the problem size, we propose a 'virtual pegging' algorithm and solve BKP to optimality. As a result, for problems with up to 16000 items we obtain a very accurate approximate solution in less a few seconds. Except for some instances, exact solutions can also be obtained in less than a few minutes on ordinary computers. However, the proposed algorithm is less effective for strongly correlated instances.

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