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Core Concepts

Given n positive integers with total sum less than 2^n-1, the algorithm efficiently finds two distinct subsets with equal subset sums.

Abstract

The paper presents an improved algorithm for solving the Pigeonhole Equal Sums problem, which is a variant of the Subset Sum problem.
The key insights are:
Structural characterization of instances with few non-subset-sums: The algorithm exploits a rigid structure of the input numbers when the number of non-subset-sums (parameter d) is small. This allows improving the naive meet-in-middle approach.
Subsampling approach for instances with many solutions: When d is large, there are many solutions, which enables a subsampling technique combined with modular dynamic programming to speed up the search.
The paper provides two algorithms:
For small d ≤ Δ ≤ 2^n/(3n^2), the algorithm runs in O*(√Δ) time deterministically.
For large 2^n/2 ≤ Δ < 2^n and d ≥ Δ, the algorithm runs in O*((2^2n/Δ)^(1/3)) time using randomization.
Combining these two algorithms gives an overall O*(2^0.4n) time randomized algorithm for Pigeonhole Equal Sums.
The paper also presents a polynomial-space algorithm that runs in O*(2^0.75n) time.

Stats

w([i]) ≥ 2^i - 1 for all i ∈ [n-1]
w([n]) < 2^n - 1

Quotes

None

Key Insights Distilled From

by Ce Jin,Hongx... at **arxiv.org** 03-29-2024

Deeper Inquiries

The techniques developed in the paper can potentially be applied to solve other problems in the PPP complexity class more efficiently. The key lies in understanding the structural properties of the specific problem at hand and devising algorithms that leverage these properties to reduce the time complexity. By identifying patterns or characteristics unique to the problem, similar optimization strategies can be employed to develop faster algorithms. Additionally, the use of randomized algorithms, dynamic programming, and subsampling techniques showcased in the paper can serve as a foundation for tackling other problems within the PPP complexity class.

If the pigeonhole promise is relaxed, meaning the total sum of the n integers is not necessarily less than 2^n-1, the algorithms for solving Pigeonhole Equal Sums would need to be adjusted. Without the constraint imposed by the pigeonhole promise, the search space for potential solutions would increase, leading to a broader range of possible subset sums. This expansion in the solution space could impact the efficiency of the algorithms, potentially requiring more sophisticated techniques to handle the larger input instances. Strategies such as adaptive sampling, refined dynamic programming approaches, or enhanced data structures may be necessary to address the relaxed pigeonhole promise condition.

There are likely additional structural properties of Pigeonhole Equal Sums instances that could be exploited to further improve the time complexity of the algorithms. One potential avenue for optimization could involve identifying specific subsets or combinations of integers that lead to equal sums more frequently. By focusing on these high-frequency subsets, algorithms could be tailored to prioritize the search in these areas, potentially reducing the overall computational effort required to find solutions. Additionally, exploring the distribution of subset sums and leveraging any inherent symmetry or regularity in the problem instances could lead to novel algorithmic enhancements. Further research into the inherent characteristics of Pigeonhole Equal Sums instances could unveil new insights for optimizing algorithm performance.

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