We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.
Johnson and Niemi @cite_7 give an FPTAS for knapsack problems on dependency graphs that are in-arborescences (these are directed trees in which every arc is directed toward a single root). In their problem formulation, the constraints are given as out-arborescences---directed trees in which every arc is directed away from a single root---and feasible solutions are subsets of vertices that are closed under the predecessor operation. This problem can be viewed as an instance of the general, directed 1-neighbour knapsack problem.
@cite_7
2105773506 2157952725 1989921461 2060645964
In the classic k-center problem, we are given a metric graph, and the objective is to open k nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of k-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows: 1 We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version. 2 We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of 1+e. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by 1+e. Let G be an acyclic directed graph with weights and values assigned to its vertices. In the partially ordered knapsack problem we wish to find a maximum-valued subset of vertices whose total weight does not exceed a given knapsack capacity, and which contains every predecessor of a vertex if it contains the vertex itself. We consider the special case where G is an out-tree. Even though this special case is still NP-complete, we observe how dynamic programming techniques can be used to construct pseudopolynomial time optimization algorithms and fully polynomial time approximation schemes for it. In particular, we show that a nonstandard approach we call “left-right” dynamic programming is better suited for this problem than the standard “bottom-up” approach, and we show how this “left-right” approach can also be adapted to the case of in-trees and to a related tree partitioning problem arising in integrated circuit design. We conclude by presenting complexity results which indicate that similar success cannot be expected with either problem when the restriction to trees is lifted. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows. For multiobjective optimization problems, it is meaningful to compute a set of solutions covering all possible trade-offs between the different objectives. The multiobjective knapsack problem is a generalization of the classical knapsack problem in which each item has several profit values. For this problem, efficient algorithms for computing a provably good approximation to the set of all nondominated feasible solutions, the Pareto frontier, are studied.For the multiobjective one-dimensional knapsack problem, a practical fully polynomial-time approximation scheme (FPTAS) is derived. It is based on a new approach to the single-objective knapsack problem using a partition of the profit space into intervals of exponentially increasing length. For the multiobjectivem-dimensional knapsack problem, the first known polynomial-time approximation scheme (PTAS), based on linear programming, is presented.