insight - Algorithms and Data Structures - # Partitioning Problems with Splittings and Interval Targets

Core Concepts

The authors study three variants of the classic n-way number partitioning problem that relax the constraints of the original problem. The first two variants allow a bounded number of split items or splittings, while the third variant requires the largest bin sum to be within a pre-specified interval. The authors provide a complete picture of the computational complexity of these variants, showing that they can be solved efficiently in polynomial time in certain parameter regimes, while remaining NP-complete in others.

Abstract

The authors consider three variants of the classic n-way number partitioning problem:

- Dec-SplitItemn, s: Decide if there exists a partition of the items X among n bins with at most s split items, such that the maximum bin sum is at most S.
- Dec-Splittingn, t: Decide if there exists a partition of the items X among n bins with at most t splittings, such that the maximum bin sum is at most S.
- Dec-Intern, u: Decide if there exists a partition of the items X among n bins such that the maximum bin sum is between S and S + u * M, where S is the average bin sum and M is the size of the largest item.

For the Dec-SplitItem and Dec-Splitting variants, the authors show that:

- When the number of split items or splittings is less than n-2 or n-1 respectively, the problems are NP-complete.
- When the number of split items or splittings is at least n-2 or n-1 respectively, the problems can be solved in polynomial time.

For the Dec-Inter variant, the authors show that:

- When u < n-2, the problem is NP-complete.
- When u ≥ n-2, the problem can be solved in polynomial time.

The authors also provide a fully polynomial-time approximation scheme (FPTAS) for the case where the number of split items is lower than n-2.

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by Samuel Bismu... at **arxiv.org** 09-12-2024

Deeper Inquiries

The algorithms and techniques presented in this paper can be extended to handle more general partitioning problems by incorporating additional parameters that account for item values and constraints. For instance, in scenarios where items have different values, one could modify the objective functions to not only minimize the maximum bin sum but also to maximize the total value allocated to each bin. This could involve adapting the min-max and max-min optimization frameworks to include weighted sums, where each item contributes differently based on its value.
Moreover, the introduction of constraints such as minimum or maximum bin capacities could be integrated into the existing algorithms. Techniques such as dynamic programming or greedy approaches could be employed to ensure that these constraints are respected while still aiming for an optimal partition. The use of approximation algorithms, as discussed in the paper, could also be beneficial in scenarios where exact solutions are computationally infeasible due to the NP-hard nature of the problems.
Additionally, the reduction techniques used in the paper could be generalized to connect different partitioning problems, allowing for the transfer of insights and solutions from one problem domain to another. For example, the reduction from the interval target problem to the split item problem could be adapted to handle cases where items have varying values or additional constraints, thereby broadening the applicability of the algorithms.

The bounded splitting and interval target variants have significant practical implications in real-world applications such as fair division and machine scheduling. In fair division scenarios, where items need to be allocated among multiple agents, the ability to allow a limited number of splits can facilitate a more equitable distribution of resources. For instance, in the case of dividing assets like real estate or inheritance, allowing a fixed number of splits can help achieve a fair allocation without necessitating complex joint ownership arrangements, which can be cumbersome and lead to disputes.
In machine scheduling, the bounded splitting variant allows for the efficient allocation of jobs to machines while minimizing the overall processing time. By permitting a limited number of splits, jobs can be divided among multiple machines, leading to reduced makespan and improved resource utilization. This is particularly relevant in environments where machines have varying capacities or processing speeds, as it enables a more flexible and responsive scheduling approach.
The interval target variant also plays a crucial role in applications where deviations from an ideal allocation are acceptable within certain limits. For example, in resource allocation for projects with budget constraints, stakeholders may prefer solutions that fall within a specified range of costs rather than exact figures. This flexibility can lead to more feasible and practical solutions that better reflect real-world constraints and preferences.

Yes, there are notable connections between the partitioning problems studied in this paper and other combinatorial optimization problems such as knapsack and bin packing. Both knapsack and bin packing problems involve the allocation of items into bins or containers with the goal of optimizing some objective, which aligns closely with the objectives of the partitioning problems discussed.
For instance, the knapsack problem, which seeks to maximize the total value of items placed in a knapsack without exceeding its capacity, shares similarities with the min-max and max-min objectives in partitioning. Insights from the knapsack problem, particularly regarding dynamic programming approaches and greedy algorithms, could be adapted to enhance the efficiency of partitioning algorithms, especially in scenarios with bounded splits or interval targets.
Similarly, the bin packing problem, which focuses on minimizing the number of bins used to pack a set of items, can inform the development of algorithms for the partitioning problems. Techniques such as first-fit or best-fit strategies used in bin packing could be integrated into the partitioning algorithms to improve their performance in practical applications.
Furthermore, the reduction techniques employed in the paper could be utilized to establish new relationships between these problems, potentially leading to novel algorithmic strategies that leverage the strengths of each problem domain. By exploring these connections, researchers can develop more robust and versatile algorithms that address a wider range of combinatorial optimization challenges.

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