Random Subset Sum Problem (RSSP) Overview

• RSSP is a probabilistic generalization of the classical subset sum problem that seeks a subset of random variables whose sum approximates a target value within a specified error tolerance.
• It employs elementary concentration techniques and dynamic programming to achieve high-probability ε-coverage with O(log(1/ε)) samples in one dimension and polynomially in higher dimensions.
• RSSP has practical applications in cryptography, neural network universality, and coding theory, providing deep insights into average-case complexity and algorithmic efficiency.

The Random Subset Sum Problem (RSSP) is a probabilistic and algorithmic generalization of the classical Subset Sum Problem, in which the goal is to approximate or achieve a given target value using subset sums of independently sampled random variables. RSSP is central to analyses in average-case complexity, probabilistic combinatorics, cryptography, statistical mechanics, and has seen recent connections to neural network universality. Its complexity and solution properties depend critically on the distribution of the underlying variables, the dimension, and the approximation error tolerance.

1. Formal Definition and Classical Regimes

The RSSP requires, for given $n \in \mathbb{N}$, random variables $X_1, \ldots, X_n$ (typically i.i.d., e.g., uniform on $[-1,1]$ or standard normal), error parameter $\varepsilon > 0$, and target $z$ (in $[-1,1]$ or $[-1,1]^d$ for $d$-dimensional variants), the identification of a subset $S \subseteq \{1, 2, \ldots, n\}$ such that

$$\left| \sum_{i \in S} X_i - z \right| \leq \varepsilon$$

in one dimension, or

$$\left\| \sum_{i \in S} X_i - z \right\|_\infty \leq \varepsilon$$

in $d$ dimensions (Cunha et al., 2022, Becchetti et al., 2022). A sample $(X_1, \ldots, X_n)$ is called $\varepsilon$-good if this property holds for all $z$ in the designated range.

A central question is to determine the minimal $n$ (as a function of $\varepsilon$ and $d$) such that, with high probability, a single random draw of $(X_1, \ldots, X_n)$ is $\varepsilon$-good.

2. Average-Case Guarantees and Concentration Phenomena

The core theoretical insight, following Lueker (1998) and further simplified by Da Cunha et al., is that for i.i.d. variables $X_i$ with suitable density (e.g., uniform on $[-1,1]$ with density bounded below on a subinterval), there exists an absolute constant $C > 0$ such that if

$n \geq C \log \frac{1}{\varepsilon}$

then, with probability at least $1 - \varepsilon$, for all $z \in [-1,1]$, there is a subset sum approximating $z$ to error $\varepsilon$ (Cunha et al., 2022). The proof utilizes an explicit volume-tracking sequence

$v_t = \frac{1}{2} \int_{-1}^1 f_t(z) \, dz$

with indicator $f_t(z) = 1$ if $z$ can be approximated by subset sums of the first $t$ variables, and leverages a two-phase argument: (1) exponential growth of the covered fraction while $v_t < 1/2$, and (2) exponential decay of the uncovered fraction once $v_t > 1/2$.

No martingale or non-elementary inequalities are needed in the new proof; classical concentration tools like Markov’s and Hoeffding’s inequalities, along with basic properties of integration, suffice. The approach is remarkably elementary and provides direct insight into why $O(\log(1/\varepsilon))$ samples suffice for high-probability $\varepsilon$-coverage.

3. Constructive Algorithms and Complexity

Though the existence result is probabilistic, given a fixed sequence $(X_1, \ldots, X_n)$, an explicit subset approximating an arbitrary $z$ can be constructed by dynamic programming. The algorithm proceeds as follows:

1. Discretize $[-1,1]$ into a grid of mesh $\varepsilon/2$.
2. Maintain a Boolean table $A_t[y]$ storing which grid points are achievable via subset sums of the first $t$ variables.
3. Initialize $A_0[0] = \text{true}$; all other entries false.
4. Iterate $A_t[y] = A_{t-1}[y] \lor A_{t-1}[y - X_t]$.
5. For a given $z$, find the closest $y$ with $A_n[y] = \text{true}$; backtrack to recover the responsible subset.

This procedure runs in $O((\log(1/\varepsilon))/\varepsilon)$ time and leverages the small $n = O(\log(1/\varepsilon))$ regime (Cunha et al., 2022).

4. High-Dimensional Extensions

In $d$ dimensions, the RSSP asks for $n$ i.i.d. random vectors $X_i \in [-1,1]^d$ such that for each $z \in [-1,1]^d$, there exists $S \subseteq [n]$ with

$$\left\|\sum_{i \in S} X_i - z \right\|_\infty \leq \varepsilon.$$

The main theorem establishes that

$$n \geq C d^{3} \log \frac{1}{\varepsilon} \left(\log \frac{1}{\varepsilon} + \log d\right)$$

suffices to guarantee, with high probability, the $\varepsilon$-approximation property for all $z \in [-1,1]^d$ (Becchetti et al., 2022). The proof employs $\varepsilon$-nets for $[-1,1]^d$, the second-moment method over carefully selected combinatorial families of subsets with bounded pairwise intersection, and Gaussian volume estimates.

This higher-dimensional dependence is optimal up to cubic factors and reflects the exponential complexity introduced by the covering number of the $d$-dimensional unit cube.

5. Algorithmic and Cryptographic Regimes

RSSP has fundamental implications in cryptographic security and algorithm analysis. In classical settings, for samples $a_1, \ldots, a_n \in \mathbb{Z}_{2^n}$ and a target $t$, heuristic (random instance) algorithms based on the “representation method” and search trees have achieved significant progress. For instance, enumerative algorithms (e.g., Becker-Coron-Joux) yield heuristic time $2^{0.291n}$, while sampling-based search tree approaches improve this to $2^{0.255n}$ for depth at least $13$ (Esser et al., 2019). In addition to subset sum, these techniques impact decoding algorithms for random linear codes, reducing the half distance decoding runtime from $2^{0.048n}$ down to $2^{0.042n}$.

Quantum algorithms further improve upon these bounds. The state-of-the-art quantum algorithm based on an EM(4)-type sampling strategy and quantum walks achieves heuristic time and space $\widetilde{O}(2^{0.209n})$ by carefully balancing initial sampling parameters, representation tree depth, and quantum-walk costs (Li et al., 2019). These algorithms assume concentration of the number of valid representations and require that truncation of quantum-walk updates does not degrade the effective marked fraction or spectral gap.

The key algorithmic regimes are summarized in the following table:

Algorithm Type	Heuristic Time Complexity	Techniques Used
Classical (BCJ)	$2^{0.291n}$	Enumerative, search trees
Classical (Sampling)	$2^{0.255n}$	Sampling, deep search trees
Quantum (EM(4))	$\widetilde{O}(2^{0.209n})$	Sampling, quantum walk

6. Applications and Theoretical Significance

RSSP has been leveraged in a diverse array of theoretical and applied contexts:

• Average-case analysis: Establishes striking separation between random and worst-case subset sum, with random instances solvable/approximable with exponentially fewer elements for given accuracy (Cunha et al., 2022).
• Multidimensional signal and neural network representations: The high-dimensional extension of RSSP underpins recent universality theorems for neural network models. For example, in the Neural-Net-Evolution (NNE) model, the existence of a subset of “gene tensors” (random weight matrices) that approximate any target network up to $\varepsilon$ in weight sup-norm is guaranteed, with the number of genes bounded polynomially in network size and $O(\log(1/\varepsilon))$ (Becchetti et al., 2022). This demonstrates that random sum architectures are, with high probability, universal approximators.
• Cryptography and coding theory: The hardness (or average-case easiness) of RSSP underpins the security and efficiency of cryptographic systems and algorithms for code-based cryptography (Esser et al., 2019, Li et al., 2019).

7. Extensions, Limitations, and Open Directions

Principal extensions of RSSP theory include:

• Non-uniform distributions: The approximation results hold under any distribution with density bounded below on a subinterval of $[-1,1]$.
• Integer and constrained problems: The framework accommodates integer-valued random variables or additional constraints (e.g., knapsack structure).
• Improvements in quantum and classical algorithms: Reducing the quantum walk exponent below $0.209$ or designing better trade-offs between memory and time for both quantum and hybrid algorithms are prominent open questions (Li et al., 2019).
• Generalization to further statistical and learning problems: The framework of random subset sums and their covering properties is potentially applicable to problems in randomized numerical integration, randomized control, and learning theory.

A notable insight is the sharp contrast between random and worst-case input regimes: whereas the worst-case subset sum is NP-hard and requires $2^n$ subsets to ensure full coverage, in the random regime, only $O(\log(1/\varepsilon))$ samples suffice for arbitrary approximation accuracy. This phenomenon, and its high-dimensional and algorithmic extensions, continue to motivate applications in theoretical computer science, cryptography, and applied mathematics.