An effective dynamic programming algorithm for the minimum-cost maximal knapsack packing problem

Highlights

• A new and effective dynamic programming algorithm is presented.

• We studied the two best-performing formulations and proved their equivalence.

• An extensive computational study on instances with up to 5000 items is provided.

• Dynamic programming outperforms the previous approaches by orders of magnitude.

• The covering counterpart of the problem is also studied.

Abstract

Given a set of items with profits and weights and a knapsack capacity, we study the problem of finding a maximal knapsack packing that minimizes the profit of the selected items. We propose an effective dynamic programming (DP) algorithm which has a pseudo-polynomial time complexity. We demonstrate the equivalence between this problem and the problem of finding a minimal knapsack cover that maximizes the profit of the selected items. In an extensive computational study on a large and diverse set of benchmark instances, we demonstrate that the new DP algorithm outperforms a state-of-the-art commercial mixed-integer programming (MIP) solver applied to the two best performing MIP models from the literature.

Introduction

One of the most famous and most studied problems in combinatorial optimization is the classical Knapsack Problem (KP) which is defined as follows: Given a knapsack capacity C > 0 and a set $I=\{1,\dots ,n\}$ of items, with profits p_i ≥ 0 and weights w_i ≥ 0, KP asks for a maximum profit subset of items whose total weight does not exceed the capacity. The problem can be formulated using the following mixed-integer program (MIP):

$$\max \{\sum _{i\in I}{p}_i{x}_i:\sum _{i\in I}{w}_i{x}_i\le C,x_i\in \{0,1\},i\in I\},$$

where each variable x_i takes value 1 if and only if item i is inserted in the knapsack. Without loss of generality, we can assume that all input parameters take integer values. In the following, we will refer to (I, p, w, C) as a knapsack instance. KP is NP-hard, but it is well-known that fairly large instances can be solved to optimality within short computing time (see, e.g., Kellerer, Pferschy, Pisinger, 2004, Martello, Pisinger, Toth, 2000, Martello, Toth, 1990, for comprehensive surveys on applications and variants of KP).

Optimization problems related to KP search for solutions satisfying one of the following two properties:

Definition 1 Maximal Knapsack Packing

Given a knapsack instance (I, p, w, C), a packing $\mathcal{P}\subset I$ is a subset of items which does not exceed the capacity C, i.e., ${\sum }_{i\in \mathcal{P}}{w}_i\le C$. A packing $\mathcal{P}$ is called maximal if and only if $\mathcal{P}\cup \left\{i\right\}$ is not a packing for any $i\notin \mathcal{P}$.

Definition 2 Minimal Knapsack Cover

Given a knapsack instance (I, p, w, C), a cover $\mathcal{K}\subset I$ is a subset of items which exceeds or it is equal to the capacity C, i.e., ${\sum }_{c\in \mathcal{K}}{w}_c\ge C$. A cover $\mathcal{K}$ is called minimal if and only if $\mathcal{K}\setminus \left\{i\right\}$ is not a cover for any $i\in \mathcal{K}$.

In this article we study weighted versions of the knapsack covering and knapsack packing problems which are defined below.

Definition 3 The Minimum-Cost Maximal Knapsack Packing Problem (MCMKP)

Given a knapsack instance (I, p, w, C), the goal of MCMKP is to find a maximal knapsack packing S* ⊂ I that minimizes the profit of selected items, i.e.:

$$\begin{array}{c}\hfill S^{*}=\arg \underset{S\subset I}{\min }\left\{\sum _{i\in S}{p}_i\right|\sum _{i\in S}{w}_i\le C\text{and}\sum _{i\in S}{w}_i+w_j>C,\forall j\notin S\}.\end{array}$$

Definition 4 The Maximum-Profit Minimal Knapsack Cover Problem (MPMKC)

Given a knapsack instance (I, p, w, C), the goal of MPMKC is to find a minimal knapsack cover S*⊆I that maximizes the profit, i.e.:

$$\begin{array}{c}\hfill S^{*}=\arg \underset{S\subset I}{\max }\left\{\sum _{i\in S}{p}_i\right|\sum _{i\in S}{w}_i\ge C\text{and}\sum _{i\in S\setminus \left\{j\right\}}{w}_i<C,\forall j\in S\}.\end{array}$$

In both MCMKP and MPMKC, we assume that the problem instance (I, p, w, C) is non-trivial, i.e., the corresponding optimal solution S* is a proper subset of I.

In the classical KP, one tries to pack items while maximizing their (non-negative) profits, hence, any optimal KP solution is a maximal knapsack packing. On the contrary, the objective function of MCMKP goes in the opposite direction – one searches for a subset of items to pack, while minimizing the profit of the selected items. Hence, solving the standard KP in which the profit-maximization objective is replaced by profit-minimization results in a trivial (empty) solution. This is why the maximal packing property has to be explicitly imposed when searching for an optimal MCMKP solution.

MCMKP models applications related to scheduling jobs on a single machine with common arrival times and a common deadline. The problem consists of selecting a subset of jobs to schedule (before the given deadline) and accordingly the order in which the jobs are scheduled becomes irrelevant. The set I refers to all possible jobs, job durations are given as w_i and p_i is the cost for executing job i, i ∈ I. The common deadline C corresponds to the capacity of the knapsack. The goal is to minimize the cost for executing scheduled jobs while at the same time ensuring that the maximal subset of jobs is scheduled (i.e., a maximal packing is preserved). Consequently, the scheduling task boils down to solving MCMKP. The problem has been called the Lazy Bureaucrat Problem (LBP) with common arrivals and deadlines (see, e.g., Gourvès, Monnot, & Pagourtzis, 2013), motivated by the special case in which $p_i=w_i,$ for all i ∈ I. We deliberately refrain from using this name since in this article we study a more general setting in which no correlation is assumed to exist between the item profits and item weights.

Similarly, in case of MPMKC, the goal is to maximize the profit of scheduled jobs while ensuring that a minimal possible subset of jobs exceeding the deadline is scheduled (i.e., to preserve a minimal cover). MPMKC has been called the Greedy Boss Problem in Gourvès et al. (2013), but again, since this name is also motivated by the setting $p_i=w_i,$ for all i ∈ I, we will not use this name in the remainder of the article.

MCMKP has been introduced in Arkin, Bender, Mitchell, and Skiena (2003) where it has been shown that a more general problem variant with individual arrival times and deadlines is NP-hard. Following this seminal work, the problem has been extensively studied in the Computer Science literature and results concerning the complexity of special classes of the problem and approximation algorithms have been presented (Gai, Zhang, 2008, Gourvès, Monnot, Pagourtzis, 2013, Hepner, Stein, 2002). Only the recent preliminary work (Furini, Ljubić, & Sinnl, 2015) studies the development of exact solution methods for MCMKP.

According to the definition of the coefficients p_i of the objective function, the following special cases are known in the (scheduling) literature:

• min-number-of-jobs: In this case, $p_i=1$ for all i ∈ I and the goal is to minimize the number of scheduled jobs. By reduction from the subset-sum problem, it has been shown in Gai and Zhang (2008) that this problem variant is weakly NP-hard.

• min-time-spent: This variant corresponds to the problem in which $p_i=w_i$ for all i ∈ I. Since job profits are equal to job durations, this explains the name LBP, i.e., the goal of a “lazy employee” is to go home as early as possible, while having an excuse that there is no more time left to start a new job. This variant is also equivalent to the min-makespan variant, since all jobs have common arrival times and the goal is to minimize the time spent on executing jobs. In Gourvès et al. (2013), two greedy heuristics and an FPTAS have been proposed for the min-time-spent variant with common arrivals and deadlines.

• min-weighted-sum: This variant considers the most general case, with no restrictions on the values of p_i, and hence it covers all previous cases. We consider this variant in our article. In Furini et al. (2015), an O(n²C) dynamic programming algorithm is proposed, thus showing that MCMKP is weakly NP-hard in this general setting. In addition, three MIP formulations and their constraint programming (CP) counterparts are developed and tested, showing that the best computational performance is obtained by two of the proposed MIP models (these models are revised in Section 4).

This article aims at providing an in-depth study of exact methods (based on MIP and DP techniques) for solving MCMKP and MPMKC to provable optimality. Our contribution is as follows: First, we design a dynamic programming algorithm for MCMKP that runs in O(nC) time and $O(n+C)$ space. It improves upon the previous DP from Furini et al. (2015) which has been shown to run in O(n²C) time and O(nC) space. Second, we compare the quality of the linear programming (LP) relaxations of the two best performing MIP models from the literature. In an extensive computational study on a large and diverse set of benchmark instances, we demonstrate that the new DP algorithm outperforms a state-of-the-art commercial MIP solver applied to these two best performing models.

Finally, we prove that MCMKP and MPMKC are two equivalent problems and that there exists a linear transformation between them. This implies that MPMKC is weakly NP-hard as well, and that there exists a DP procedure which runs in $O\left(n\right(W-C\left)\right)$ time and requires $O(n+(W-C\left)\right)$ space, where W is the sum of all item weights.

The remainder of this article is organized as follows: in the following section, we present structural properties of optimal MCMKP solutions that can be used to derive DP algorithms and MIP models. Section 3 is devoted to our new DP procedure for MCMKP. In Section 4, we present and discuss new results for two MIP models for the MCMKP. In Section 5 we prove the equivalence between MCMKP and MPMKC. Computational results are provided in Section 6 and conclusions are drawn in Section 7.