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Lower bound of computational complexity of knapsack problems

This paper claims to determine the lower bound of computational complexity for knapsack problems by applying quantum statistics to reveal that nontrivial topological structures arising from dimensional contradictions create an NP-intermediate region, thereby preventing these problems from collapsing directly into the P-class and guiding the development of subexponential algorithms.

Original authors: Zhidong Zhang

Published 2026-06-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhidong Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Impossible" Puzzle

Imagine you have a massive, incredibly difficult puzzle. In the world of computer science, this is called the Knapsack Problem. It's like trying to pack a suitcase with the most valuable items possible without exceeding the weight limit. You have thousands of items, and you need to figure out the perfect combination.

For decades, computers have struggled with this. The time it takes to solve it grows so fast that even the fastest supercomputers would take longer than the age of the universe to solve a large version of the puzzle. This class of problems is known as NP-complete.

The author of this paper, Zhidong Zhang, claims to have found a "lower bound" for how hard this puzzle really is. In other words, he wants to know the absolute fastest possible time a computer could ever solve this, no matter how smart the algorithm gets.

The Secret Ingredient: Spins and Frustration

To solve this, the author doesn't just look at the suitcase; he looks at a completely different field: Physics, specifically the study of magnets and "spin glasses."

The Analogy: Imagine a room full of people (spins) holding hands. Some want to face North, others South. But here's the catch: they are all connected randomly. Person A wants to face North, but their neighbor wants to face South. This creates a "frustration" where no one can be happy at the same time.
The Connection: The author shows that packing a suitcase (Knapsack Problem) is mathematically identical to finding the most stable arrangement of these frustrated magnets (Spin Glass Model). If you can solve the magnet puzzle, you can solve the suitcase puzzle.

The "3D vs. 2D" Clash

The core of the author's discovery lies in a clash between dimensions.

The 3D Reality: The magnets (or the items in the suitcase) exist in a 3-dimensional space. They are connected in all directions.
The 2D Tool: When physicists try to calculate the answer, they use a mathematical tool called a "transfer matrix," which is essentially a flat, 2-dimensional sheet.

The Metaphor: Imagine trying to flatten a crumpled, tangled ball of yarn (the 3D reality) onto a flat piece of paper (the 2D tool) without cutting any strings. Because the yarn is 3D, when you flatten it, the strings have to cross over each other in impossible ways. These "crossings" create non-trivial topological structures.

The author argues that these crossings are the source of the difficulty. You cannot simply "flatten" the problem to make it easy (a "P" problem) because the 3D nature of the connections forces these complex tangles to exist.

The "Absolute Minimum Core" (AMC)

The paper introduces a concept called the Absolute Minimum Core (AMC) model.

The Analogy: Think of the Knapsack problem as a giant, multi-story building. To solve the whole building, you don't need to look at every single floor. The author claims there is a specific "core" section—just two layers of the building—that contains the essential difficulty.
The Finding: This "core" is the smallest version of the problem that still retains all the hard, tangled features. The author proves that you cannot simplify this core any further into an easy problem. It sits right on the border between "hard" and "easy."

The "Middle Ground" (NPI)

For a long time, computer scientists thought problems were either:

Easy (P): Solvable quickly.
Hard (NP-complete): Solvable only by checking every possibility (brute force).

The author proposes a third category called NP-Intermediate (NPI).

The Metaphor: Imagine a staircase. At the bottom is "Easy." At the top is "Hard." The author claims there is a landing in the middle. The "Core" model sits right on the edge of this landing.
The Result: The Knapsack problem cannot be collapsed all the way down to "Easy." It lives in this middle zone. It is harder than a polynomial problem but potentially easier than the worst-case brute-force scenario.

The New Speed Limit

The paper concludes with a claim about how fast we can solve these problems in the future.

Current Status: Current best algorithms take a time that grows exponentially (like $1.3^N$ , where $N$ is the number of items). This is very slow.
The Claim: The author suggests that by understanding the "Core" and using a specific parallel computing strategy (solving layers of the problem simultaneously), we can improve the speed to something like $(1 + \epsilon)^N$ .
What this means: The time required would still grow, but much, much slower than before. It would move from "impossible" to "sub-exponential" (very fast, but not instant).

Summary of Claims

The Origin of Hardness: The difficulty comes from the clash between the 3D nature of the problem and the 2D tools used to solve it, creating unavoidable "knots" or crossings.
The Core: There is a minimal "core" version of the Knapsack problem that cannot be made easier.
The Middle Zone: There is a "middle ground" (NPI) between easy and hard problems where the Knapsack problem lives.
The Solution: By targeting this core and using parallel processing, we can theoretically develop algorithms that solve these problems much faster than current methods, though they will still be complex.

The author states this applies to physics, biology, finance, and information technology, but strictly within the context of solving these specific optimization puzzles.

Technical Summary of "Lower bound of computational complexity of knapsack problems"

Problem Statement
The paper addresses the fundamental question of determining the lower bound of the computational complexity for the knapsack problem (specifically the 0-1 knapsack problem, MKP), which is a classic NP-complete optimization problem. The author seeks to understand the origin of the nontrivial topological structures that render such problems computationally hard and to establish whether an "Absolute Minimum Core" (AMC) model exists that defines the boundary between polynomial (P) and NP-complete problems. The work posits that current algorithms (such as dynamic programming and genetic algorithms) often rely on approximations or heuristics and fail to identify the exact solution for large-scale systems because they do not address the fundamental topological nature of the problem.

Methodology
The author employs a theoretical framework bridging statistical physics and computer science, specifically utilizing the quantum statistics mechanism and the spin-glass Ising model as an interdisciplinary platform. The methodology proceeds through the following steps:

Mapping: The 0-1 knapsack problem is mapped to the spin-glass Ising model. In this correspondence:
- Binary decision variables ( $x_i \in \{0,1\}$ ) in the knapsack problem correspond to Ising spins ( $s_i \in \{-1, 1\}$ ).
- Object weights ( $w_i$ ) map to random interactions (couplings) between spins.
- Object costs ( $c_i$ ) map to effective magnetic fields (bias terms).
- Maximizing total cost in the knapsack problem is equivalent to minimizing the free energy (or finding the ground state) in the spin-glass model.
Topological Analysis: The author analyzes the 3D Edwards-Anderson spin-glass Ising model ( $M_{SGI}^{3D}$ ). A key focus is the "nontrivial topological structure" arising from the contradiction between the 3D character of the lattice and the 2D character of the transfer matrices used in quantum statistics. The paper argues that this contradiction generates long-range spin entanglements and non-planar graphs, which are the source of computational hardness.
Definition of the AMC Model: The concept of an Absolute Minimum Core (AMC) model is introduced. This is defined as the simplest construct (e.g., a two-level grid lattice for the spin-glass or knapsack problem) that retains the essential physical properties and topological structures of the full 3D system. The author asserts that the computational complexity of the full problem is bounded by the complexity of this core model.
Complexity Derivation: By analyzing the transfer matrices and the entanglement of spins in the AMC model, the author derives the lower bound of the computational complexity. The analysis suggests that the complexity is not strictly exponential ( $O(2^N)$ ) but falls into a specific class between polynomial and exponential.

Key Contributions and Results

Origin of Hardness: The paper identifies the origin of nontrivial topological structures in NP-complete problems as the conflict between the 3D lattice structure and the 2D transfer matrix representation in quantum statistics. This leads to non-local effects and long-range entanglements that cannot be eliminated without changing the fundamental nature of the problem.
Existence of NP-Intermediate (NPI) Problems: The author proposes a phase diagram for computational complexity that includes a distinct NP-intermediate (NPI) region situated between P-problems and NP-complete problems.
- P-problems: Represented by 2D spin-glass models or 2D knapsack problems.
- NP-complete: Represented by 3D spin-glass models or 3D knapsack problems.
- NPI: A region containing problems with complexity greater than polynomial but less than the full NP-complete bound (e.g., quasi-polynomial times like $O(N^{\log N})$ ).
The AMC Boundary: The paper claims that the AMC model sits precisely on the border between the NPI region and the NP-complete region. Crucially, the author argues that the AMC model cannot collapse directly into a P-problem; it retains a complexity that is strictly higher than polynomial.
Lower Bound of Complexity: The computational complexity of the knapsack problem ($CL(MKP)$) is determined to be subexponential and superpolynomial. Specifically, the lower bound is characterized as:
$O((1 + \epsilon)^N)$
where $\epsilon \to 0$ and $\epsilon \neq 1/N$ . This is presented as an improvement over the current status of $O(1.3^N)$ found in existing literature.
Optimum Algorithm Strategy: Based on these findings, the author proposes a strategy for an optimum algorithm. This involves fixing $z$ -layers of the AMC model as an algorithmic element and performing parallel computations on $l/z$ layers. This approach aims to find the exact solution with sufficient accuracy in the shortest possible time, theoretically achieving the $O((1 + \epsilon)^N)$ bound.

Significance and Claims
The paper claims to illuminate the "road" for solving various optimum problems across physics, biology, finance, and information technology. By identifying the AMC model and the NPI region, the author suggests that:

The fundamental barrier to solving NP-complete problems is not merely the lack of better heuristics but a misunderstanding of the topological core (AMC) of these problems.
It is possible to develop algorithms that solve NP-complete problems (like the knapsack problem, Traveling Salesman Problem, and K-SAT) in subexponential time, significantly improving upon current methods like dynamic programming and genetic algorithms, which often rely on approximations or fail for large scales.
The results provide a unified framework for understanding the complexity of diverse NP-complete problems, asserting they all belong to the same universality class regarding their lower computational bounds.

The author explicitly states that under the guidance of these results, one may develop the "best algorithms" for solving various optimum problems, moving from current limitations to a theoretical limit defined by the AMC model's complexity.