Strongly-polynomial time¶
A complexity notion requiring polynomially many arithmetic operations independent of numeric magnitudes.
Core Idea¶
A strongly polynomial-time algorithm has an arithmetic-operation count bounded by a polynomial in the combinatorial size of its input—such as the number of variables, constraints, or numeric entries—independently of the magnitudes and bit lengths of those numbers, while also keeping every intermediate value representable with polynomially many bits. The first condition removes dependence on numerical scale from the number of arithmetic steps; the second prevents a unit-cost arithmetic model from hiding exponentially large numbers. Together they imply a polynomial implementation in the ordinary bit-cost model.
Scope of Application¶
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Network flow. Algorithms can be distinguished from capacity-dependent and scaling methods by whether arithmetic steps depend on encoded values.
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Matching and assignment. Strong bounds expose when numerical weights affect only comparisons rather than iteration count.
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Linear optimization. The open status of general linear programming illustrates the distinction from ordinary polynomial solvability.
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Combinatorial optimization. Runtime comparisons separate number of variables and constraints from the bit length or magnitude of costs and capacities.
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Arithmetic-model analysis. Claims must name permitted operations and show polynomially bounded intermediate numbers.
Clarity¶
Strongly polynomial time distinguishes algorithms whose number of arithmetic operations depends polynomially only on combinatorial input dimensions from those whose iteration count also grows with numerical magnitudes or bit lengths. Requiring polynomially bounded intermediate representations closes the loophole of unit-cost operations on exponentially large numbers. The term therefore sharpens ‘efficient’ for numerical optimization.
Manages Complexity¶
Strong polynomiality strips away numerical scale and asks whether arithmetic work depends only on combinatorial dimensions, while separately bounding the bit size of intermediate values. This two-part test compresses many implementation concerns into operation count and representation growth. Algorithms then split into strongly polynomial, weakly or bit-polynomial, pseudo-polynomial, and nonpolynomial branches. An analyst can rescale coefficients or lengthen their encodings and read whether the iteration structure should remain unchanged.
Abstract Reasoning¶
Scale-independence move. Rescale or lengthen numerical coefficients while holding combinatorial dimensions fixed; if arithmetic-step count can grow, the algorithm is not strongly polynomial. Bit-growth move. Inspect intermediate values to rule out hidden exponential representation even when operation count is small. Classification move. Separate strongly polynomial from weakly or bit-polynomial and pseudo-polynomial behavior using both tests. Design move. Search for progress measures based on variables, constraints, or graph structure rather than coefficient magnitude. Boundary move. Ordinary polynomial-time complexity in total encoded length does not by itself imply strong polynomiality.
Knowledge Transfer¶
Within the home domain. Strongly polynomial time transfers across combinatorial optimization, linear programming variants, flows, matchings, and algorithms over numeric data when operation count is polynomial in structural input dimensions and intermediate numbers remain polynomially bounded independent of value magnitudes. Arithmetic model and encoding distinctions retain exact roles. Beyond the home domain (C — complexity criterion). The criterion applies literally to algorithms satisfying its formal definition, regardless of application subject. Its limit is over-reading: ordinary polynomial bit complexity, good practical speed, and scale-free behavior are not equivalent.
Relationships to Other Abstractions¶
Current abstraction Strongly-polynomial time Domain-specific
Parents (1) — more general patterns this builds on
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Strongly-polynomial time is a kind of Complexity Class Domain-specific
Strongly-polynomial time is a domain-specific kind of Complexity Class: A complexity notion requiring polynomially many arithmetic operations independent of numeric magnitudes.
Hierarchy paths (6) — routes to 5 parentless roots
- Strongly-polynomial time → Complexity Class → Classification
- Strongly-polynomial time → Complexity Class → Complexity (Time/Space) → Complexity
- Strongly-polynomial time → Complexity Class → Complexity (Time/Space) → Constraint
- Strongly-polynomial time → Complexity Class → Complexity (Time/Space) → Scaling and Scale Dependence → Scale
- Strongly-polynomial time → Complexity Class → Complexity (Time/Space) → Asymptotic Behavior → Scaling and Scale Dependence → Scale
- Strongly-polynomial time → Complexity Class → Complexity (Time/Space) → Asymptotic Behavior → Approximation → Representation → Abstraction
Neighborhood in Abstraction Space¶
Strongly-polynomial time sits in a sparse region of the domain-specific corpus (66th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Computational Complexity & Hardness (17 abstractions)
Nearest neighbors
- Pseudo-polynomial transformation — 0.85
- Perfect digit-to-digit invariant — 0.85
- Computational hardness assumption — 0.84
- Cryptographic Hash Function — 0.84
- Element distinctness problem — 0.84
Computed from structural-signature embeddings · 2026-10-08