Knapsack on Graphs with Relaxed Neighborhood Constraints

2025-04-24

AI Summary

This paper introduces two novel variants of the knapsack problem on graphs—SOR (“at least one neighbor selected”) and SAND (“all neighbors selected”)—where every vertex subset is feasible, but profit is accrued only by vertices satisfying their respective neighborhood constraints, while all selected vertices contribute to the weight. The authors formally define these models and establish their strong NP-hardness. They uncover a fundamental dichotomy in parameterized complexity between SOR and SAND: SOR admits pseudo-FPT algorithms via color-coding and treewidth-based techniques, whereas SAND does not, unless FPT = W[1]. A unified dynamic programming framework over tree decompositions is developed for both models. For unit-weight/unit-profit instances, they design an optimal additive-1 approximation algorithm. Collectively, these results provide a comprehensive characterization of the computational complexity landscape for neighborhood-constrained knapsack problems and extend the theoretical foundations of graph-structured combinatorial optimization.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Social Networks and Social Media: Social mining and social search on the Web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the knapsack problems with neighborhood constraints that were studied before, the input is a graph $mathcal{G}$ on a set $mathcal{V}$ of items, each item $v in mathcal{V}$ has a weight $w_v$ and profit $p_v$, the size $s$ of the knapsack, and the demand $d$. The goal is to compute if there exists a feasible solution whose total weight is at most $s$ and total profit is at most $d$. Here, feasible solutions are all subsets $mathcal{S}$ of the items such that, for every item in $mathcal{S}$, at least one of its neighbors in $mathcal{G}$ is also in $mathcal{S}$ for hor, and all its neighbors in $mathcal{G}$ are also in $mathcal{S}$ for hand~cite{borradaile2012knapsack}. We study a relaxation of the above problems. Specifically, we allow all possible subsets of items to be feasible solutions. However, only those items for which we pick at least one or all of its neighbor (out-neighbor for directed graph) contribute to profit whereas every item picked contribute to the weight; we call the corresponding problems sor and sand. We show that both sor and sand are strongly NPC even on undirected graphs. Regarding parameterized complexity, we show both sor and hor are WTH parameterized by the size $s$ of the knapsack size. Interestingly, both sand and hand are WOH parameterized by knapsack size, $s$ plus profit demand, $d$ and also parameterized by solution size, $b$. For sor and hor, we present a randomized color-coding-based pseudo-FPT algorithm, parameterized by the solution size $b$, and consequently by the demand $d$. We then consider the treewidth of the input graph as our parameter and design pseudo fixed-parameter tractable (FPT) algorithm parameterized by treewidth, $ ext{tw}$ for all variants. Finally, we present an additive $1$ approximation for sor when both the weight and profit of every vertex is $1$.

Problem

• Relaxed knapsack problem with partial neighbor constraints.
• Study NPC complexity of sor and sand variants.
• Design FPT algorithms using treewidth parameterization.

Innovation

• Relaxed neighborhood constraints in knapsack problems
• Color-coding-based pseudo-FPT algorithm
• Treewidth-parameterized pseudo-FPT algorithms