Combinatorial algorithms for solving the constrained knapsack problems with divisible item sizes and penalties

Abstract

In this paper, we address the constrained knapsack problem with divisible item sizes and penalties (the CK-DSP problem, for short), which is modelled as follows. Given a set $A=\{a_{1},a_{2},\ldots ,a_{n}\}$ of n items and a knapsack K with capacity L, where each item $a_{i}\in A$ has a size $s_{i}\in Z^{+}$, a profit $c_{i}\in R^{+}$, a penalty $p_{i}\in R^{+}$, and these n item sizes are divisible, i.e., either $s_{i}|s_{j}$ or $s_{j}|s_{i}$ for any two distinct items $a_{i}$ and $a_{j}$ in A, each item $a_{i}\in A$ must be either put into K under the constraint that the summation of sizes of items in K does not exceed L, or rejected with its penalty $p_{i}$ that we must pay for, it is asked to find a subset $X\subseteq A$ to satisfy the constraint $s(X)=\sum _{a_i\in X}s_i \le L$. We consider three versions of the CK-DSP problem, respectively. (1) The constrained knapsack problem with divisible item sizes and total penalties (the CK-DSTP problem, for short) is asked to find a subset $X\subseteq A$ to satisfy the constraint $s(X) \le L$, the objective is to maximize the value of total profits of the items in X minus total penalties paid for the rejected items not in X; (2) The constrained knapsack problem with divisible item sizes and maximum penalty (the CK-DSMP problem, for short) is asked to find a subset $X\subseteq A$ to satisfy the constraint $s(X) \le L$, the objective is to maximize the value of total profits of the items in X minus maximum penalty paid for the rejected items not in X; (3) The penalized knapsack problem with divisible item sizes (the PK-DS problem, for short) is asked to find a subset $X\subseteq A$ to satisfy the constraint $s(X) \le L$, the objective is to maximize the value of total profits of the items in X minus maximum penalty paid for the items in X. As our contributions, we design three exact combinatorial algorithms to solve the CK-DSTP problem, the CK-DSMP problem and the PK-DS problem, and these three algorithms run in time $O(n\log n)$, $O(n^{2}\log n)$ and $O(n^{2}\log n)$, respectively, where n is the number of items with divisible sizes.