Optimal Macroitem Sequences in the Precedence Constrained Knapsack Problem

2026-06-20

AI Summary

This work addresses the knapsack problem with precedence constraints and proposes an efficient method for computing the optimal solution to its linear programming (LP) relaxation. By introducing “macro-items” and their feasible sequences, the item set is structurally partitioned, and items are packed in non-increasing order of profit-to-weight ratio per macro-item—allowing at most one macro-item to be fractionally selected—while respecting precedence constraints. The key contributions include a complete characterization of the LP optimal solution structure, an explicit correspondence between Lagrangian dual breakpoints and macro-item ratios, and a combinatorial flow interpretation of the dual solution. Building on these insights, an O(n²) algorithm is developed for forest-structured precedence graphs, with further improvements to O(n log n) for in-trees or out-trees, substantially enhancing computational efficiency.

Abstract

The Precedence Constrained Knapsack Problem (PCKP) asks for a maximum-profit subset of items, subject to a knapsack capacity constraint and precedence constraints encoded by a directed acyclic graph. We study the structure of optimal solutions of the Linear Programming (LP) relaxation of the natural Integer Linear Programming formulation of the PCKP. We introduce the notion of macroitem and of feasible sequence of macroitems, which partitions the item set while respecting the precedence structure. We establish that an optimal LP solution is fully characterized by the optimal sequence of macroitems: items are packed in nonincreasing order of the profit-to-weight ratio of their macroitem, with at most one macroitem fractionally included. We further show that the breakpoints of the parametric Lagrangian function of the capacity constraint coincide with the profit-to-weight ratios of the macroitems in the optimal sequence, and provide a complete combinatorial characterization of optimal dual solutions in terms of a feasible flow within each macroitem. Finally, for the special case in which the precedence graph is a forest, we devise an O(n^2) algorithm to compute the optimal sequence, which improves to O(n log n) for in-trees or out-trees, where n denotes the number of items.

Problem

• Precedence Constrained Knapsack Problem
• Linear Programming relaxation
• macroitem
• optimal solution structure
• precedence constraints

Innovation

• macroitem
• precedence constrained knapsack problem
• linear programming relaxation
• combinatorial characterization
• parametric Lagrangian