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 ^^^^^^^^11query11 "^^^^^^^^11all:gap-bounding algorithm OR ti:\11^^^^^^^^"Bounding\" gap algorithm11^^^^^^^^" arxiv.search({"^^^^^^^^11query11 algorithm\"^^^^^^^^11 OR ti:\11^^^^^^^^"Bounding\" AND all:gap11^^^^^^^^"," gap algorithm11all:gap-bounding algorithm OR ti:\11^^^^,"^^^^^^^^11sort_by11 arxiv.search({"^^^^^^^^11query11 the spectral gap\"","^^^^^^^^11max_results11 AND all:gap11^^^^}) 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 PRESERVED_PLACEHOLDER_^^^^11all:gap-bounding algorithm OR ti:\11^^^^ in stochastic elliptic eigenproblems, the duality gap PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11^^^^ 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 PRESERVED_PLACEHOLDER_^^^^11query11^^^^, or the decision gap in edit distance (&&&^^^^11all:gap-bounding algorithm OR ti:\11^^^^&&&, &&&^^^^11 gap algorithm11^^^^&&&, &&&^^^^11query11^^^^&&&, &&&^^^^11all:\11^^^^&&&, &&&^^^^11 OR ti:\11^^^^&&&, &&&^^^^11 AND all:gap11^^^^&&&). 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.

^^^^11 gap algorithm11^^^^. 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
PRESERVED_PLACEHOLDER_^^^^11all:\11^^^^	Stochastic elliptic eigenvalue problem	Uniform lower bound over all PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11^^^^
PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11^^^^	Separable nonconvex optimization with linear constraints	Convex-envelope relaxation plus randomized extreme-point selection
PRESERVED_PLACEHOLDER_^^^^11max_results11^^^^	Costly Information Combinatorial Selection	Reduction to Bayesian Combinatorial Selection and ex ante free-order prophet inequalities
PRESERVED_PLACEHOLDER_^^^^11sort_by11^^^^	Multistage stochastic AC OPF with storage	A posteriori upper bound via two convex problems
PRESERVED_PLACEHOLDER_^^^^11submittedDate11^^^^	Zeros of PRESERVED_PLACEHOLDER_^^^^11query11^^^^	Explicit lower bound from zero-motion ODE comparison
PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11all:gap-bounding algorithm OR ti:\11^^^^ versus PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11 gap algorithm11^^^^	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.

^^^^11query11^^^^. 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 PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11query11^^^^ is replaced by its convex envelope PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11all:\11^^^^, and then proves that an extreme point of the convexified optimal set satisfies

PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11 OR ti:\11^^^^

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

PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11 AND all:gap11^^^^

with PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11max_results11^^^^ drawn uniformly at random from the unit sphere in PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11sort_by11^^^^. With probability one, this selects a unique extreme point, and only at most PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11submittedDate11^^^^ blocks can contribute nonconvexity to the gap bound (&&&^^^^11 gap algorithm11^^^^&&&). 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,

PRESERVED_PLACEHOLDER_^^^^11 gap algorithm11query11^^^^

"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

PRESERVED_PLACEHOLDER_^^^^11query11all:gap-bounding algorithm OR ti:\11^^^^

with instantiated bounds PRESERVED_PLACEHOLDER_^^^^11query11 gap algorithm11^^^^ for matroids, PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^ for PRESERVED_PLACEHOLDER_^^^^11query11all:\11^^^^-systems, and PRESERVED_PLACEHOLDER_^^^^11query11 OR ti:\11^^^^ for knapsack constraints (&&&^^^^11query11^^^^&&&). 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 PRESERVED_PLACEHOLDER_^^^^11query11 AND all:gap11^^^^ 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

PRESERVED_PLACEHOLDER_^^^^11query11max_results11^^^^

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

PRESERVED_PLACEHOLDER_^^^^11query11sort_by11^^^^

The practical procedure is to solve PRESERVED_PLACEHOLDER_^^^^11query11submittedDate11^^^^ and the restricted SOC problem PRESERVED_PLACEHOLDER_^^^^11query11query11^^^^, then compute their value difference as a certificate-like upper bound (&&&^^^^11all:\11^^^^&&&). 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.

^^^^11all:\11^^^^. 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

PRESERVED_PLACEHOLDER_^^^^11all:\11all:gap-bounding algorithm OR ti:\11^^^^

with

PRESERVED_PLACEHOLDER_^^^^11all:\11 gap algorithm11^^^^

uniform ellipticity, and the decay assumption

PRESERVED_PLACEHOLDER_^^^^11all:\11query11^^^^

Its main theorem states that there exists PRESERVED_PLACEHOLDER_^^^^11all:\11all:\11^^^^, independent of PRESERVED_PLACEHOLDER_^^^^11all:\11 OR ti:\11^^^^, such that

PRESERVED_PLACEHOLDER_^^^^11all:\11 AND all:gap11^^^^

The proof combines Lipschitz continuity of eigenvalues with a weighted reparametrization that turns the infinite-dimensional parameter set into a compact subset of PRESERVED_PLACEHOLDER_^^^^11all:\11max_results11^^^^, so that the continuous positive gap attains a positive minimum (&&&^^^^11all:gap-bounding algorithm OR ti:\11^^^^&&&). 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 PRESERVED_PLACEHOLDER_^^^^11all:\11sort_by11^^^^ and PRESERVED_PLACEHOLDER_^^^^11all:\11submittedDate11^^^^, the zeros in the variable PRESERVED_PLACEHOLDER_^^^^11all:\11query11^^^^ satisfy

PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11all:gap-bounding algorithm OR ti:\11^^^^

and the paper studies the adjacent-zero spacing

PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11 gap algorithm11^^^^

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 PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11query11^^^^-domain. The resulting theorem gives an explicit lower bound on PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11all:\11^^^^ under the condition PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11 OR ti:\11^^^^, together with a monotonicity theorem showing that for PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11 AND all:gap11^^^^ the lower bound is a monotonically decreasing function of PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11max_results11^^^^ (&&&^^^^11 OR ti:\11^^^^&&&). 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.

^^^^11 OR ti:\11^^^^. 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 ^^^^^^^^11query11^^^^^^^^ and time complexity

PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11sort_by11^^^^

which outputs close with probability PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11submittedDate11^^^^ when PRESERVED_PLACEHOLDER_^^^^11 OR ti:\11query11^^^^ and far with probability at least PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11all:gap-bounding algorithm OR ti:\11^^^^ when PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11 gap algorithm11^^^^ (&&&^^^^11 AND all:gap11^^^^&&&). 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

PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11query11^^^^

"Nested Dataflow Algorithms for Dynamic Programming Recurrences with more than PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11all:\11^^^^ 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

PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11 OR ti:\11^^^^

with PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11 AND all:gap11^^^^ space and PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11max_results11^^^^ cache misses (&&&^^^^11 gap algorithm11query11^^^^&&&). The construction combines the closure method with Nested Dataflow and the partial-dependency operator PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11sort_by11^^^^ 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.

^^^^11 AND all:gap11^^^^. Functional reformulations and bootstrap gap maximization

In the conformal bootstrap, gap bounding is recast as an optimization over linear functionals. "Bounding ^^^^11all:\11^^^^d 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 PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11submittedDate11^^^^, the paper optimizes a functional PRESERVED_PLACEHOLDER_^^^^11 AND all:gap11query11^^^^ 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 (&&&^^^^11 gap algorithm11all:\11^^^^&&&).

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 ^^^^11all:\11^^^^d Ising spin correlator takes the minimal possible allowed values on the Euclidean section, and that for PRESERVED_PLACEHOLDER_^^^^11max_results11all:gap-bounding algorithm OR ti:\11^^^^ 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 (&&&^^^^11 gap algorithm11all:\11^^^^&&&).

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.

^^^^11max_results11^^^^. 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 (&&&^^^^11 gap algorithm11 AND all:gap11^^^^&&&). "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 (&&&^^^^11 gap algorithm11max_results11^^^^&&&). "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 PRESERVED_PLACEHOLDER_^^^^11max_results11 gap algorithm11^^^^, not a proof of numerical gap bounds (&&&^^^^11 gap algorithm11sort_by11^^^^&&&).

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 PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^, then PRESERVED_PLACEHOLDER_^^^^11max_results11all:\11^^^^ (&&&^^^^11 gap algorithm11submittedDate11^^^^&&&). "Bounding the gap between a free group (outer) automorphism and its inverse" defines complexity functions PRESERVED_PLACEHOLDER_^^^^11max_results11 OR ti:\11^^^^ and PRESERVED_PLACEHOLDER_^^^^11max_results11 AND all:gap11^^^^ to measure the maximal possible gap between the norm of an automorphism and the norm of its inverse, proving exact behavior in rank PRESERVED_PLACEHOLDER_^^^^11max_results11max_results11^^^^ and polynomial lower bounds, plus a polynomial upper bound for PRESERVED_PLACEHOLDER_^^^^11max_results11sort_by11^^^^, in higher rank (&&&^^^^11 gap algorithm11query11^^^^&&&). "Greedy-like bases for sequences with gaps" treats gaps as quotient gaps in an increasing subsequence PRESERVED_PLACEHOLDER_^^^^11max_results11submittedDate11^^^^ and proves that bounded quotient gaps are exactly the condition under which PRESERVED_PLACEHOLDER_^^^^11max_results11query11^^^^-quasi-greediness collapses to ordinary quasi-greediness (&&&^^^^11query11all:gap-bounding algorithm OR ti:\11^^^^&&&).

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 (&&&^^^^11query11 gap algorithm11^^^^&&&). 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.