Knapsack on Graphs with Relaxed Neighborhood Constraints

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, $\text{tw}$ for all variants. Finally, we present an additive $1$ approximation for \sor when both the weight and profit of every vertex is $1$.

Overview of "Knapsack on Graphs with Relaxed Neighborhood Constraints"

The paper "Knapsack on Graphs with Relaxed Neighborhood Constraints" presents novel variants of the classical knapsack problem, applied to graphical settings with relaxed neighborhood constraints. This research contributes to both theoretical and practical aspects by devising algorithms for these complex formulations and understanding their computational implications.

Problem Formulation

Traditionally, knapsack problems involve selecting items to maximize profit while staying within weight constraints. In graphical settings, these items are vertices of a graph, and additional neighborhood constraints are imposed. Previous studies on graph-based knapsack problems required each vertex in the selection to have neighbors included, either partially or fully, with stringent constraints limiting solution space exploration.

The authors introduce relaxed variants termed SOR (Soft OR) and SAND (Soft AND) which allow greater flexibility. In SOR, a vertex contributes to profit if at least one neighbor is included. In SAND, a vertex contributes only when all neighbors are included, yet selection isn't mandatory for feasibility.

Key Findings

Complexity Results

The paper delves deeply into the classical and parameterized complexity of these problems. On a theoretical level, both variants are proven to be strongly NP-complete on various graph structures. For effective solution discovery:

• SOR and SAND are NP-complete even in constrained settings like bipartite and directed acyclic graphs.
• Parameterized complexity is explored, showing W[2]-hardness of SOR with respect to the knapsack size and W[1]-hardness of SAND with respect to combined parameters of knapsack size and profit demand.

Algorithmic Contributions

For graphs with bounded treewidth, novel dynamic programming approaches present fixed-parameter tractable (FPT) solutions:

• Pseudo-polynomial algorithms are introduced specifically for treewidth-bounded graphs.
• A color-coding approach provides FPT algorithms based on solution size and breaks down complex graph structures into manageable components.

The paper details linear-time algorithms for solving uniform, undirected versions of these problems, proving practicality in simpler graph settings. Additionally, randomized and deterministic solutions are offered for general instances, ensuring that approximations yield near-optimal results reliably.

Implications and Future Research

The implications of this work are broad, touching on efficient planning in network design, resource allocation, and computational graph theory. By proving complexity on various models and providing practical algorithms, it opens pathways for tackling real-world applications where dependency and interconnectedness define success criteria.

Potential future research avenues include refining these algorithms, exploring heuristic methods under specific constraints, and extending the range of graph classes for which efficient solutions can be obtained. The complexity exploration suggests further investigation into computational boundaries and algorithmic optimizations, providing a fertile ground for ongoing discovery.

This paper emphasizes the importance of considering relaxed constraints for knapsack-like problems and establishes a foundational basis for further exploration in graph algorithmics, especially under real-world likeness with interdependent systems.