SETH: Strong Exponential Time Hypothesis

Updated 19 July 2025

SETH is a conjecture in computational complexity that asserts for every ε > 0, there exists a clause width k such that k-SAT cannot be solved in O((2–ε)^n) time.
The hypothesis underpins fine-grained reductions, providing tight conditional lower bounds for problems like Edit Distance, LCS, and various graph problems.
SETH influences algorithm design by delineating the limits of improving brute-force search methods, impacting both classical complexity and emerging quantum applications.

The Strong Exponential Time Hypothesis (SETH) is a central conjecture in computational complexity theory asserting tight lower bounds on the time complexity of satisfiability (SAT) and, by extension through fine-grained reductions, on a wide array of combinatorial and algorithmic problems. SETH posits that for every constant ε > 0, there exists a clause width k such that no algorithm can solve k-SAT on n variables in O((2–ε)ⁿ) time. Over the last decade, SETH has emerged as a foundational assumption not only for NP-hardness but also for explaining the pervasive difficulty in obtaining faster algorithms for numerous polynomial-time problems, fine-tuning the limits of algorithmic improvement across complexity theory, combinatorial optimization, parameterized complexity, circuit lower bounds, and practical algorithm design.

1. Formal Statement and Basic Properties

SETH extends the Exponential Time Hypothesis (ETH) by asserting that not only is 3-SAT unlikely to be solved faster than in exponential time, but the trivial O(2ⁿ) search is essentially optimal for all large enough k:

$(\forall\ \varepsilon > 0)(\exists\ k\ \geq 3) \text{ such that $k $-SAT on$ n$ variables cannot be solved in time } O((2-\varepsilon)^n)\,.$

As a corollary, any algorithm that would decide k-SAT in time O((2–ε)ⁿ) for arbitrarily small ε > 0 and all k would refute SETH. The hypothesis thus asserts that for Boolean formulas, improvements over brute-force enumeration of assignments are limited to the polynomial overhead and the subexponential error term, but not in the constant base of the exponent (1112.2275). SETH also admits formalizations via growth rates and infima over exponents achievable by families of algorithms:

$\lim_{k\rightarrow\infty} \sigma(\#1\text{-}k\text{-CNF-SAT}/n) = 1\,,$

where σ denotes the infimum of achievable exponents (1112.2275).

2. Conditional Lower Bounds and Fine-Grained Reductions

SETH underpins tight conditional lower bounds for dozens of important algorithmic problems, both within NP-hard optimization (such as Hitting Set, Set Splitting, NAE‑SAT, Max‑SAT, Set Cover) and for canonical "hardness of P" problems, including Edit Distance, Longest Common Subsequence (LCS), Dynamic Time Warping (DTW), and various graph problems (1007.5450, 1112.2275, Backurs et al., 2014, Abboud et al., 2015, Polak, 2017, Abboud et al., 2017). A core methodology is fine-grained reductions, which preserve the relevant hardness constant in the exponent and transfer improvements or impossibility results from SAT to the target problem.

A quintessential example is the set of SETH-based lower bounds for algorithms parameterized by structural measures (treewidth, pathwidth) on graphs:

Problem	Lower Bound (assuming SETH)	Best Known Algorithm	Tightness
Independent Set	O^{*((2–ε)^tw(G))}	O^{*(2^tw(G))}	tight
Dominating Set	O^{*((3–ε)^tw(G))}	O^{*(3^tw(G))}	tight
Max Cut	O^{*((2–ε)^tw(G))}	O^{*(2^tw(G))}	tight
Odd Cycle Transversal	O^{*((3–ε)^tw(G))}	O^{*(3^tw(G))}	tight
q-Coloring (q ≥ 3)	O^{*((q–ε)^tw(G))}	O^{*(q^tw(G))}	tight (for q ≥ 3)
Partition into Triangles	O^{*((2–ε)^tw(G))}	O^{*(2^tw(G))}	tight

Here, tw(G) is the treewidth of the graph G, and O^* suppresses polynomial factors (1007.5450). An improvement over the exponential base in these algorithms would yield a breakthrough in SAT solving and thus contradict SETH.

3. Scope, Equivalences, and Hierarchy

SETH’s influence extends beyond standard SAT formulations. Multiple equivalence classes and hierarchies have been established, showing that SETH is, in a precise sense, equivalent to versions of SAT or Max-SAT parameterized by backdoors, modulators, or circuit restrictions (Lampis, 12 Jul 2024).

The hierarchy detailed in (Lampis, 12 Jul 2024) classifies several shades of SETH-equivalent hypotheses:

Standard SETH: Hardness for SAT/Max-SAT even when parameterized by the size of a modulator to constant tree-depth or small backdoor.
Max-SAT/Weight-parameterized: Hardness for parameterized Max-SAT algorithms relates directly to SETH.
2-SAT Backdoors/Logarithmic Pathwidth: Hardness equivalence for SAT when parameterized by backdoor size or modulator to logarithmic pathwidth.
Linear Depth Circuits and W[SAT]: Hardness for SAT/Circuit-SAT with bounded depth circuits equivalents to algorithms for small treewidth/pathwidth modulators or for weight-k assignments.
General Circuits/Horn Backdoors/W[P]: Hardness for arbitrary circuit SAT is equivalent to breaking brute-force for strong Horn backdoor parameterizations and to W[P]-complete weighted SAT.

Formally, if one could devise

$\text{an algorithm for SAT solving in time }(2–\varepsilon)^m | \varphi |^{O(1)}$

where m is the modulator size to constant tree-depth, this would refute standard SETH (Lampis, 12 Jul 2024).

4. Practical Consequences in Algorithmic Design

SETH has become the central explanatory tool for the observed computational intractability of improving upon O(2ⁿ), O(2^tw(G)), or quadratic time for major classes of problems (1007.5450, Backurs et al., 2014, Abboud et al., 2015, Polak, 2017). Examples include:

Edit Distance and LCS: No algorithm computes Edit Distance or LCS of two n-length sequences in strongly subquadratic O(n^{2–δ}) time unless SETH is false (Backurs et al., 2014, Abboud et al., 2015).
Graph Diameter Approximation: Exact computation of the diameter in sparse graphs requires time m^{2–o(1)} (where m is the number of edges), and 2-approximation algorithms are conditionally tight under SETH (Dalirrooyfard et al., 2020, Bonnet, 2021).
Subset Sum: BeLLMan’s O^*(T)-time pseudo-polynomial algorithm is essentially optimal; any T^{{1–ε}-time} algorithm would refute SETH (Abboud et al., 2017).

These results delimit the boundary of what is achievable by current algorithmic paradigms. Improvements over the base constant in the exponential parameterization or quadratic base would not only be algorithmic breakthroughs but would fundamentally alter the accepted fine-grained landscape of complexity theory.

5. Links to Circuit Complexity and Evidence Barriers

Recent research demonstrates a profound connection between SETH-based lower bounds and circuit lower bounds: proving even mild (e.g., n^{1+ε}) SETH-based hardness for various natural problems would require or imply circuit lower bounds that have remained open for decades (Belova et al., 2022, Belova et al., 2023). For instance:

Problems with efficient "polynomial formulations" (allowing reduction to evaluating bounded-degree polynomials) resist SETH-based superlinear lower bounds unless new circuit lower bounds are established.
For many problems in P (including k-SUM, triangle detection), the existence of efficient polynomial formulations means that any meaningful superlinear SETH-based lower bound would break major barriers in Boolean or arithmetic circuit complexity (Belova et al., 2023).
Nondeterministic analogues (NSETH) and oracle-based complexity further clarify which problems fundamentally admit SETH-based lower bounds, and which do not without unforeseen circuit complexity breakthroughs.

This creates formal "barriers" to extending SETH-based fine-grained reductions, explaining both why matching lower bounds cannot yet be proved for certain problems and why SETH remains plausible for those with matching reductions.

6. SETH in Quantum Information and Physics

SETH’s interpretive reach extends to fundamental principles in quantum mechanics and black hole physics. If SETH holds, it places a computational limit on quantum determinism: predicting the exact quantum state of large or macroscopic physical systems governed by local Hamiltonians encoding NP-complete instances requires time exponential in the number of degrees of freedom (Bolotin, 2014). In particular:

The inability, under SETH, to systematically "compute" the evolution of arbitrary systems within realistic timeframes, even given complete knowledge of the initial quantum state, limits quantum determinism as a practical principle.
This has been proposed as a computational underpinning for the black hole information loss paradox, suggesting that computational intractability—not physical law—may account for apparent departures from determinism (Bolotin, 2014).

A plausible implication is that computational assumptions such as SETH may have physical consequences, influencing the boundaries of predictability in natural systems.

7. Broader Impact, Limitations, and Perspectives

SETH underlies a broad range of research in fine-grained complexity, parameterized algorithms, and conditional lower bounds. Its practical impact includes:

Explaining the tightness of dynamic programming on bounded-treewidth graphs, subset sum, and sequence alignment.
Justifying why many long-standing algorithmic attempts to "shave logs" or improve exponents have failed in the absence of major complexity-theoretic breakthroughs.
Structuring the foundations of parameterized complexity theory by revealing equivalence classes among various seemingly disparate "brute force" hypotheses (Lampis, 12 Jul 2024).

At the same time, recent work shows that SETH-based reductions face obstacles for many problems (e.g., those with succinct polynomial formulations or easy verification models) unless long-standing open problems in circuit complexity are resolved (Belova et al., 2022, Belova et al., 2023).

Ongoing research continues to clarify the precise boundaries of SETH’s relevance, the relationships among various strengthened or weakened forms of the hypothesis (e.g., via backdoors, circuit depth, weighed variants), and its ramifications for both classical and quantum computing, as well as for broader computational and physical systems.

In summary, the Strong Exponential Time Hypothesis is not only a central conjecture regarding the hardness of SAT but has established itself as a unifying principle governing the fine-grained and parameterized complexity of a wide array of problems. It tightly connects advances in algorithm design to foundational questions in complexity and circuit theory, providing a powerful explanatory paradigm for the observed resilience of exponential and quadratic time barriers across computational domains.