Gap-Bounding Algorithms

• Gap-bounding algorithms are a family of methods that certify upper or lower bounds on gap quantities, such as spectral gaps and duality gaps, in various technical settings.
• They employ techniques like convex relaxations, compactness arguments, and probabilistic reductions to transform complex, intractable problems into verifiable certificates.
• Applications range from stochastic eigenvalue problems and nonconvex optimization to edit distance decision-making and conformal bootstrap, highlighting their versatile practical utility.

Searching arXiv for papers on “gap-bounding algorithm” and closely related gap-bounding methods across domains. arxiv.search query: "all:gap-bounding algorithm OR ti:\"Bounding\" gap algorithm" arxiv.search({"query":"all:\"gap-bounding algorithm\" OR ti:\"Bounding\" AND all:gap","max_results":10,"sort_by":"submittedDate"}) arxiv.search({"query":"all:\"Bounding the spectral gap\"","max_results":5}) In current arXiv usage, “gap-bounding algorithm” does not denote a single canonical procedure. The literature surveyed here suggests a domain-dependent umbrella term for constructive methods that certify upper or lower bounds on a quantity called a gap: the spectral gap $\lambda_2(y)-\lambda_1(y)$ in stochastic elliptic eigenproblems, the duality gap $p^\star-p$ in separable nonconvex optimization, the commitment gap in Costly Information Combinatorial Selection, the relaxation gap in multistage stochastic AC OPF, the spacing between adjacent zeros of $\Phi(a,b,z)$, or the decision gap in edit distance (Gilbert et al., 2019, Udell et al., 2014, Chawla et al., 27 Aug 2025, Grangereau et al., 2021, Langel, 6 Apr 2026, Goldenberg et al., 2019). In each case, the central object is not merely a heuristic estimate but a certified inequality tied to a variational, probabilistic, geometric, or combinatorial construction.

1. Terminological scope and representative gap quantities

A useful way to organize the topic is by the quantity being bounded and the mechanism used to obtain the bound.

Gap quantity	Setting	Representative construction
$\lambda_2(y)-\lambda_1(y)$	Stochastic elliptic eigenvalue problem	Uniform lower bound over all $y\in U$
$p^\star-p$	Separable nonconvex optimization with linear constraints	Convex-envelope relaxation plus randomized extreme-point selection
$ComGap(I)$	Costly Information Combinatorial Selection	Reduction to Bayesian Combinatorial Selection and ex ante free-order prophet inequalities
$$\operatorname{val}(P)-\operatorname{val}(P_{\mathrm{SOC}})$$	Multistage stochastic AC OPF with storage	A posteriori upper bound via two convex problems
$\Delta a=a_{k-1}^*-a_k^*$	Zeros of $\Phi(a,b,z)$	Explicit lower bound from zero-motion ODE comparison
$p^\star-p$0 versus $p^\star-p$1	Edit-distance gap decision	Sublinear adaptive distinguisher

These examples show that the word “gap” may denote a distance between eigenvalues, a discrepancy between primal and relaxed optima, a worst-case performance ratio, a zero spacing, or a promise gap in a decision problem. This suggests that the unifying feature of a gap-bounding algorithm is methodological rather than semantic: it converts a hard object into a tractable certificate.

2. Optimization, relaxation, and stochastic-control constructions

In nonconvex separable optimization with linear coupling constraints, the gap-bounding construction of "Bounding Duality Gap for Separable Problems with Linear Constraints" identifies a convexified problem in which each $p^\star-p$2 is replaced by its convex envelope $p^\star-p$3, and then proves that an extreme point of the convexified optimal set satisfies

$p^\star-p$4

The constructive step is a randomized linear optimization over the convexified optimal set,

$p^\star-p$5

with $p^\star-p$6 drawn uniformly at random from the unit sphere in $p^\star-p$7. With probability one, this selects a unique extreme point, and only at most $p^\star-p$8 blocks can contribute nonconvexity to the gap bound (Udell et al., 2014). The significance of the construction is that the bound depends on the number of simultaneously active coupling constraints rather than on ambient dimension.

In Costly Information Combinatorial Selection, the gap object is a policy-performance ratio,

$p^\star-p$9

"Commitment Gap via Correlation Gap" reduces this quantity to Bayesian Combinatorial Selection by amortizing MDP costs into surrogate terminal values and then relates the resulting bound to ex ante free-order prophet inequalities. The master bound is

$\Phi(a,b,z)$0

with instantiated bounds $\Phi(a,b,z)$1 for matroids, $\Phi(a,b,z)$2 for $\Phi(a,b,z)$3-systems, and $\Phi(a,b,z)$4 for knapsack constraints (Chawla et al., 27 Aug 2025). Here the gap-bounding step is neither direct simulation nor local dynamic programming; it is a reduction to a simpler stochastic benchmark in which only the feasibility family $\Phi(a,b,z)$5 remains.

In multistage stochastic AC OPF on radial networks, the gap is the discrepancy between the original nonconvex problem and its SOC relaxation. "Multi-stage Stochastic Alternating Current Optimal Power Flow with Storage: Bounding the Relaxation Gap" defines the relaxation gap as

$\Phi(a,b,z)$6

proves exactness under a priori sign and monotonicity conditions, and gives the a posteriori upper bound

$\Phi(a,b,z)$7

The practical procedure is to solve $\Phi(a,b,z)$8 and the restricted SOC problem $\Phi(a,b,z)$9, then compute their value difference as a certificate-like upper bound (Grangereau et al., 2021). The paper further states that a null or low relaxation gap may be expected for applications with light reverse power flows or if sufficient storage capacities with low cost are available.

Across these optimization settings, the recurring pattern is a certified surrogate: convex envelopes, ex ante reductions, or SOC relaxations replace an intractable original object with one whose gap to the original can itself be bounded.

3. Spectral and analytic lower-bound constructions

In stochastic elliptic eigenvalue problems, the gap quantity is the fundamental spectral gap. "Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients" studies

$\lambda_2(y)-\lambda_1(y)$0

with

$\lambda_2(y)-\lambda_1(y)$1

uniform ellipticity, and the decay assumption

$\lambda_2(y)-\lambda_1(y)$2

Its main theorem states that there exists $\lambda_2(y)-\lambda_1(y)$3, independent of $\lambda_2(y)-\lambda_1(y)$4, such that

$\lambda_2(y)-\lambda_1(y)$5

The proof combines Lipschitz continuity of eigenvalues with a weighted reparametrization that turns the infinite-dimensional parameter set into a compact subset of $\lambda_2(y)-\lambda_1(y)$6, so that the continuous positive gap attains a positive minimum (Gilbert et al., 2019). In this setting, the gap-bounding mechanism is a compactness argument made available by summability of the coefficient modes.

"Bounding the Gap Between Zeros of the Variable-Parameter Confluent Hypergeometric Function" addresses an analytic spacing problem rather than an optimization gap. For fixed positive $\lambda_2(y)-\lambda_1(y)$7 and $\lambda_2(y)-\lambda_1(y)$8, the zeros in the variable $\lambda_2(y)-\lambda_1(y)$9 satisfy

$y\in U$0

and the paper studies the adjacent-zero spacing

$y\in U$1

Its method begins with an exact ODE for zero motion derived from the implicit function theorem and Buchholz’s integral identity, then upper-bounds the denominator integral to obtain a differential inequality, and finally combines that inequality with a known zero-ratio estimate in the $y\in U$2-domain. The resulting theorem gives an explicit lower bound on $y\in U$3 under the condition $y\in U$4, together with a monotonicity theorem showing that for $y\in U$5 the lower bound is a monotonically decreasing function of $y\in U$6 (Langel, 6 Apr 2026). The application is to residue-tail control for first passage probabilities of a Wiener process.

These two examples illustrate two distinct analytic archetypes. One uses continuity and compactness to prove a uniform positive lower bound over an infinite-dimensional parameter space; the other derives a pointwise explicit lower bound from an exact trajectory equation and comparison inequalities.

4. Decision-gap algorithms and the GAP dynamic-programming recurrence

A different use of gap bounding appears in promise problems, where the goal is to distinguish a small regime from a much larger one. "Sublinear Algorithms for Gap Edit Distance" studies the quadratic gap problem for edit distance and gives an algorithm with query and time complexity

$y\in U$7

which outputs close with probability $y\in U$8 when $y\in U$9 and far with probability at least $p^\star-p$0 when $p^\star-p$1 (Goldenberg et al., 2019). Its main technical departure from earlier work is adaptive switching between uniform sampling and reading contiguous blocks. The paper emphasizes that previous sublinear edit-distance algorithms chose queried coordinates non-adaptively, whereas the new algorithm exploits periodicity and local violations of periodicity to control candidate diagonals.

The term GAP also names the edit-distance-with-gaps dynamic-programming recurrence

$p^\star-p$2

"Nested Dataflow Algorithms for Dynamic Programming Recurrences with more than $p^\star-p$3 Dependency" answers an open question of Galil and Park by giving the first work-efficient and sublinear-time algorithm for the general GAP problem. The final bound is

$p^\star-p$4

with $p^\star-p$5 space and $p^\star-p$6 cache misses (Tang, 2019). The construction combines the closure method with Nested Dataflow and the partial-dependency operator $p^\star-p$7 to remove artificial control dependencies while preserving work efficiency.

These two papers use “gap” in different but related senses. In the edit-distance promise problem, the gap is a decision threshold separation. In the GAP recurrence, the gap is a model feature of the alignment dynamic program. Both settings nevertheless fit the broader pattern of bounding or exploiting separation structure to obtain stronger algorithmic guarantees.

5. Functional reformulations and bootstrap gap maximization

In the conformal bootstrap, gap bounding is recast as an optimization over linear functionals. "Bounding 3d CFT correlators" studies Euclidean four-point functions of identical scalar primaries and shows that the usual gap-maximization bootstrap can be reproduced by a numerically easier optimization problem. Rather than performing a repeated binary search over a candidate gap $p^\star-p$8, the paper optimizes a functional $p^\star-p$9 subject to positivity constraints on conformal blocks, obtaining upper and lower bounds on correlator values and recovering the gap-maximizing solution from the small-radius limit of correlator minimization (Paulos et al., 2021).

The paper’s formulation converts gap maximization into a single linear optimization problem over functionals acting on the crossing equation. It further reports that the 3d Ising spin correlator takes the minimal possible allowed values on the Euclidean section, and that for $ComGap(I)$0 there are gap-independent maximal bounds on CFT correlators. Under certain conditions, the maximizing correlator is given by the generalized free boson for general Euclidean kinematics (Paulos et al., 2021).

This functional perspective is structurally close to other gap-bounding methods discussed above. A difficult feasibility or extremal problem is replaced by a dual certificate whose sign conditions encode admissibility. In the bootstrap setting, the certificate is a functional; in relaxation theory it is an auxiliary convex program; in stochastic selection it is an ex ante benchmark; in spectral theory it is a compactness argument.

6. Ambiguities, adjacent usages, and non-examples

The surveyed literature also shows that “gap” is heavily overloaded. Some algorithms named Gap are not gap-bounding procedures at all. "From H&M to Gap for Lightweight BWT Merging" introduces Gap as a lightweight algorithm for merging Burrows-Wheeler transforms and LCP arrays; the name refers to skipping irrelevant monochrome blocks during phase-based refinement, not to bounding any optimality or spectral quantity (Manzini, 2016). "Mend the gap: A smart repair algorithm for noisy polygonal tilings" treats gaps as geometric defects between perturbed polygons and repairs them using shortest-path convexification, strong mutual visibility, and recursive subdivision to preserve adjacency relations (Clelland, 2023). "Combined Sieve Algorithm for Prime Gaps" accelerates concurrent sieving in prime-gap search; its contribution is a batch modular-arithmetic method and a modular solver $ComGap(I)$1, not a proof of numerical gap bounds (Troisi, 2020).

Other nearby uses are closer in spirit but remain domain-specific. "The Gap Number of the T-Tetromino" studies the least number of monominos needed in a tiling and introduces the fringe digraph with Bellman–Ford negative-cycle detection to obtain lower bounds on monomino density and the global result that if $ComGap(I)$2, then $ComGap(I)$3 (Hochberg, 2014). "Bounding the gap between a free group (outer) automorphism and its inverse" defines complexity functions $ComGap(I)$4 and $ComGap(I)$5 to measure the maximal possible gap between the norm of an automorphism and the norm of its inverse, proving exact behavior in rank $ComGap(I)$6 and polynomial lower bounds, plus a polynomial upper bound for $ComGap(I)$7, in higher rank (Ladra et al., 2012). "Greedy-like bases for sequences with gaps" treats gaps as quotient gaps in an increasing subsequence $ComGap(I)$8 and proves that bounded quotient gaps are exactly the condition under which $ComGap(I)$9-quasi-greediness collapses to ordinary quasi-greediness (Berasategui et al., 2020).

A bibliographic caution also arises in the supplied record for "Bounding Optimality Gap in Stochastic Optimization via Bagging: Statistical Efficiency and Stability." The associated details explicitly state that the provided document is a satirical, mock-technical article about margarine and butter and contains no stochastic optimization content, no sample average approximation, no bootstrap bagging method, and no optimization theory (Lam et al., 2018). This is not a substantive gap-bounding source.

Taken together, these cases suggest that “gap-bounding algorithm” is best treated as a family resemblance term. What unifies the family is the production of a rigorous certificate for a named separation quantity; what varies is the meaning of the gap itself, which may be spectral, variational, probabilistic, analytic, combinatorial, geometric, or purely terminological.