Gap-Preserving Reductions Overview

• Gap-Preserving Reductions are structural transformations that map YES/NO instances while maintaining the key completeness-soundness gap in optimization and promise problems.
• They employ combinatorial, algebraic, and spectral techniques—such as assignment refined gadgets and expander graphs—to control gap loss across classical, quantum, and coding frameworks.
• These reductions yield strong inapproximability results by ensuring a constant-factor preservation of the gap, significantly impacting areas like CSPs, lattice problems, quantum protocols, and cryptography.

Gap-preserving reductions are structural transformations between computational problems that maintain or control the completeness-soundness gap of associated optimization or promise problems. They are central to the paper of hardness of approximation, complexity of constraint satisfaction, and the propagation of inapproximability phenomena through reduction chains. In their most fundamental form, a gap-preserving reduction from problem $A$ (with gap parameters $c_A,s_A$) to problem $B$ (with gap parameters $c_B,s_B$) ensures that a YES-instance for $A$ maps to a YES-instance for $B (c_B \ge c_A)$, and likewise for NO-instances ($s_B \le s_A)$, but more subtly, the reduction guarantees that the difference $c_B-s_B$ is at least a fixed fraction ($\alpha(c_A-s_A)$) of the original gap, with $\alpha>0$ independent of instance size. This preserves the meaningfulness of hardness: inapproximability for $A$ is transferred to $B$ at a controlled quantitative rate. Gap-preserving reductions have been formalized and exploited across classical, quantum, coding-theoretic, lattice, and reconfiguration settings.

1. Formal Definitions and Gap Structure

A gap-preserving reduction transforms instances of one promise problem $(L_1, c_1, s_1)$ to another $(L_2, c_2, s_2)$ such that YES/NO instances are mapped appropriately and the completeness-soundness gap behaves as: $c_2 - s_2 \ge \alpha (c_1 - s_1)$ for some absolute constant $\alpha > 0$ (Mančinska et al., 8 May 2025). A key property is constant gap blow-up: the loss in gap is bounded by an absolute constant, independent of instance parameters.

In CSP, SAT, and reconfiguration settings, the completeness/soundness gap reflects the fraction of constraints that remain satisfied in YES/NO cases. For reconfiguration problems (e.g., Maxmin SAT Reconfiguration), the gap is measured in the minimal fraction of constraints maintained during a solution path from $\psi^{\mathsf{ini}}$ to $\psi^{\mathsf{tar}}$ (Ohsaka, 2022). In quantum nonlocal games (MIP$^*$-protocols), the gap refers to the difference between maximal achievable entangled value in the YES-case and the maximal value attainable in the NO-case (Mančinska et al., 8 May 2025).

Gap-preserving reductions differ fundamentally from standard NP-reductions by quantifying the preservation of metric structure (e.g., approximation value or entangled game value) rather than merely solution existence.

2. Methodologies for Gap-Preserving Reductions

Gap-preserving reductions are engineered via a suite of combinatorial, algebraic, and analytic techniques:

• Assignment Refined Gadgets: For codes and CSP, reductions are constructed via gadgets that encode every local constraint as a block and enforce local consistencies; e.g., each NAND constraint is translated to a collection of codeword coordinates with explicit linear consistency checks (Austrin et al., 2010).
• Expander Graphs and Mixing Lemmas: Degree reduction in CSP and reconfiguration settings is performed via explicit families of near-Ramanujan expanders, leveraging the expander mixing lemma to control soundness (Ohsaka, 2022).
• Spectral Gap Amplification: In Small-Set Expansion and related graph cut problems, powering the random-walk operator amplifies expansion gaps without increasing instance size, distinguishing it from parallel repetition methods that blow up the complexity exponentially (Raghavendra et al., 2013).
• Quantum Operator Stability: In the quantum regime, reductions enforce approximate commutation among measurement operators and then "round" nearly commuting operators to commuting projectors, with polynomial error dependence, resulting in robust transfer of value gaps (Mančinska et al., 8 May 2025).
• Alphabet Reduction via Coding Theory: Hadamard codes are used to encode large alphabets down to small, fixed alphabets in CSP reconfiguration, enabling robust soundness/completeness guarantees and avoiding excessive blow-up in parameter sizes (Ohsaka, 16 Feb 2024).
• Partial Assignment Systems (PAS): Combinatorial gap theorems are established by analyzing partial assignment chains, showing that unsatisfiable CSPs are always bounded away from full satisfiability, and reducing general PCSPs via canonical long-code or repetition templates (Barto et al., 2021).

Table: Selected Gap-Preserving Reduction Paradigms

Methodology	Application Domain	Core Reduction Technique
Expander graphs	CSP/Reconfiguration	Degree reduction, alphabet squaring
Spectral graph ops	Small-Set Expansion	Random-walk matrix powering
Coding gadgets	Code Min Distance, CSP	Explicit local linear encodings
Quantum rounding	MIP$^*$ Nonlocal Games	Stability of nearly commuting PVMs
PAS/Minion maps	Promise CSP/PCSP	Repetition template, minor chains

3. Principal Results and Notable Chains

Gap-preserving reductions produce strong inapproximability results with constant-factor loss across extensive classes of problems.

• SAT to Minimum Distance of Code: Austrin–Khot demonstrate a deterministic reduction from SAT (Max NAND) to Gap-Min-Dist($q$), preserving constant gap even for asymptotically good codes—without recourse to randomized or high-cost gadgets. The reduction encodes each constraint with four variables and imposes carefully constructed moment equations (Austrin et al., 2010).
• SVP and GapSVP: Dimension-preserving search-to-decision reductions for SVP and GapSVP show that, for small gaps $(\gamma = 1 + O(\log n/n))$, randomized and deterministic reductions yield close alignment of search and decision complexity in lattice problems. Sparsification and guided Babai techniques preserve gap (Stephens-Davidowitz, 2015).
• Reconfiguration Problems (RIH): Under the Reconfiguration Inapproximability Hypothesis (RIH), a gap version of Maxmin CSP Reconfiguration is shown to be PSPACE-hard, and via degree reduction, alphabet squaring, and expander mixing, this inapproximability is transferred to a wide range of reconfiguration problems (e.g., Vertex Cover, Clique) (Ohsaka, 2022, Ohsaka, 16 Feb 2024).
• Quantum Interactive Proofs: A quantum framework for gap-preserving reduction compresses general MIP$^*$ gapped promise problems to independent set games on graphs, demonstrating a constant gap blow-up and establishing undecidability (RE-completeness) for the value problem in quantum nonlocal games, in stark contrast to classical decidability (Mančinska et al., 8 May 2025).
• Small-Set Expansion: Raghavendra–Schramm use lazy random walks to amplify expansion gaps, showing that spectral powering achieves near-perfect soundness with only linear completeness degradation, maintaining the vertex set (Raghavendra et al., 2013).
• Promise CSPs and Layered Label Cover: Combinatorial gap theorems ensure that unsatisfiable CSP instances are always bounded away from full solution by combinatorial value measures, allowing for gap hard reductions between PCSPs through minion homomorphisms (Barto et al., 2021).

4. Robustness, Technical Obstacles, and Soundness Analysis

Achieving gap-preservation demands rigorous control of both completeness and soundness through each transformation:

• Perfect Completeness Maintenance: In reconfiguration reductions, the move from "equality" to "containment" relations in clouded expanders (via alphabet squaring) ensures that fully satisfying sequences in the source are mapped to fully satisfying sequences in the target (Ohsaka, 2022).
• Soundness Degradation Bounds: At each reduction stage, the loss in soundness is quantified; e.g., in the Hadamard code alphabet reduction, any $\varepsilon$-fraction of edge violations in the source induces at least $\kappa \varepsilon$ fraction in the target (Ohsaka, 16 Feb 2024).
• Avoidance of High-Cost Gadgets: Algebraic and combinatorial constructions are preferred to randomized high-expansion gadgets, resulting in transparent, polynomial-time algorithms with explicit parameter control (Austrin et al., 2010).
• Operator Stability Issues in Quantum Setting: Earlier reductions suffered exponential loss when rounding nearly commuting measurement operators; new joint diagonalization techniques establish only polynomial ($O(\text{gap})$) loss (Mančinska et al., 8 May 2025).
• Instance Size Control: Graph powering for gap amplification in Small-Set Expansion avoids exponential blow-up in vertex set size by acting on edge weights and spectral operators (Raghavendra et al., 2013).

5. Applications and Impact Across Complexity Theory

Gap-preserving reductions unify and propagate hardness results and inapproximability phenomena across diverse domains:

• Hardness of Approximation: These reductions formalize the infeasibility of approximating solution values within specified factors in both classical (NP-hard) and quantum (RE-hard) regimes.
• Constraint Satisfaction and PCSPs: By establishing canonical reductions and layered templates, gap-hardness arguments demonstrate that most NP-hard PCSPs are "universal" for inapproximability (Barto et al., 2021).
• Reconfiguration Problems: Via robust gap-preserving chains and alphabet reductions, problems such as Maxmin SAT, Vertex Cover Reconfiguration, and Independent Set Reconfiguration inherit strong PSPACE-hardness of approximation, even for fixed alphabet and bounded degree (Ohsaka, 2022, Ohsaka, 16 Feb 2024).
• Coding Theory: Deterministic reductions from SAT to minimum distance in codes capture constant-factor NP-hardness on families with constant rate and distance, fundamentally impacting coding-theoretic security proposals (Austrin et al., 2010).
• Lattice Problems and Cryptography: Search-to-decision gap reductions in SVP and CVP underpin worst-case to average-case hardness in lattice-based cryptography for problems admitting approximation factors slightly greater than one (Stephens-Davidowitz, 2015).
• Quantum Complexity: Gap-preserving reductions compress complex quantum protocols to combinatorial games, establishing the undecidability and RE-completeness even in simple graph games under quantum entanglement (Mančinska et al., 8 May 2025).
• Expander Graph Theory: Spectral gap amplification plays a critical role in transfer of hardness in graph partitioning and expansion problems (Raghavendra et al., 2013).

6. Open Problems and Future Directions

Key unresolved questions pertain to expanding the scope and precision of gap-preserving reductions:

• Improving Inapproximability Factors: Current deterministic reductions produce only small constant-factor gaps (e.g., $g > 1$); sharper algebraic or combinatorial techniques may yield stronger separations (Austrin et al., 2010).
• Transfer to Lattice SVP Hardness: While code minimum distance admits elementary deterministic reductions, the analogous results for Gap-SVP on lattices remain open (Austrin et al., 2010).
• Extension Beyond Binary CSPs: Alphabet reduction for reconfiguration and gap amplification without a blow-up in domain size remains an active topic (Ohsaka, 16 Feb 2024).
• Quantum-to-Classical Gap-Preservation: Whether other classical CSPs (graph coloring, Hamiltonian cycle) admit quantum gap-preserving reductions akin to independent set games is unresolved (Mančinska et al., 8 May 2025).
• Scaling and Efficiency: The complexity and polynomial-time feasibility of gap-preserving reductions for larger gaps or higher approximation factors, especially in cryptographic applications, present ongoing challenges (Stephens-Davidowitz, 2015).
• Layered Template Generalization: The algebraic structure of PAS and minion homomorphisms underlying canonical gadget reductions points toward broader universal reductions across PCSPs (Barto et al., 2021).

A plausible implication is that gap-preserving reduction designs will continue to play a foundational role in classifying approximation complexity and in establishing robust equivalences between search, decision, and reconfiguration frameworks in both classical and quantum regimes.