Discrete Min-Max Violation (DMMV) Overview

• Discrete Min-Max Violation (DMMV) is a discrete optimization formulation that minimizes the worst-case constraint violation using an ℓ∞ criterion.
• It employs a MILP reformulation with an auxiliary variable to explicitly model and control maximum deviations, and is proven NP-hard.
• Recent advances include a GPU-accelerated heuristic (AMVM) that enhances performance in applications like quantized regression, discrete tomography, and FIR filter design.

Discrete Min-Max Violation (DMMV) denotes, in its most explicit recent formulation, a discrete optimization problem that seeks an assignment of discrete values to variables minimizing the largest constraint violation. Given a real matrix $A \in \mathbb{R}^{m \times n}$, a real target vector $b \in \mathbb{R}^m$, and a discrete feasible set $\mathbb{V} \subset \mathbb{R}$, the task is to choose $\mathbf{x} \in \mathbb{V}^n$ so as to minimize $\|A\mathbf{x}-\mathbf{b}\|_\infty$, equivalently $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$ (Ahmed et al., 19 Aug 2025). The literature also uses the label more broadly for discrete minimization of metric or ultrametric violations in $\ell_0$, and for settings where discrete min-max behavior diverges from continuous or classical max-min principles; these uses are related by a common focus on worst-case or obstruction-type violations, but they are not identical definitions (Cohen-Addad et al., 2022, Jost et al., 2021, Tsaknakis et al., 2021).

1. Definition and formal optimization model

In the context-free formulation, DMMV is specified by the triplet $(A,b,\mathbb{V})$ and asks for

$$\min_{\mathbf{x} \in \mathbb{V}^n} \|A\mathbf{x} - \mathbf{b}\|_\infty$$

or, equivalently,

$$\min_{\mathbf{x} \in \mathbb{V}^n} \max_{i=1,\ldots,m} | (A\mathbf{x} - \mathbf{b})_i |.$$

The objective is a worst-case criterion: it minimizes the maximum absolute deviation over all constraints, rather than an average or total error (Ahmed et al., 19 Aug 2025).

The same problem admits a standard MILP reformulation through an auxiliary variable $b \in \mathbb{R}^m$0: $b \in \mathbb{R}^m$1 This makes explicit that DMMV combines linear constraints, discrete decision variables, and an $b \in \mathbb{R}^m$2 objective (Ahmed et al., 19 Aug 2025).

A defining feature of this formulation is that it is described as context-free: it is not tied to any specific application area, and only assumes linear constraints, discrete variable domains, and minimization of the worst-case constraint violation. The cited applications include quantized regression, robust combinatorial design, digital filtering, and tomography with quantized pixel values, which illustrate how the same abstract model can encode heterogeneous discrete design tasks (Ahmed et al., 19 Aug 2025).

2. Complexity and computational status

The explicit DMMV formulation is NP-hard. More precisely, the decision problem

$b \in \mathbb{R}^m$3

is NP-complete, via reduction from Subset-Sum (Ahmed et al., 19 Aug 2025). Accordingly, exact solution is generally intractable at large scale, especially when $b \in \mathbb{R}^m$4 is large or the discretization $b \in \mathbb{R}^m$5 is fine.

This hardness is consistent with a broader pattern in discrete min-max optimization. In a related but distinct line of work, minimax problems with coupled linear constraints are shown to violate the classical max-min inequality: for

$b \in \mathbb{R}^m$6

there is no deterministic ordering; any of $b \in \mathbb{R}^m$7, $b \in \mathbb{R}^m$8, or $b \in \mathbb{R}^m$9 can occur. The same paper proves NP-hardness even when $\mathbb{V} \subset \mathbb{R}$0 is strongly convex in $\mathbb{V} \subset \mathbb{R}$1 and strongly concave in $\mathbb{V} \subset \mathbb{R}$2 (Tsaknakis et al., 2021). Although that problem is not the same as the $\mathbb{V} \subset \mathbb{R}$3 DMMV model, it sharpens the general observation that discrete min-max formulations can behave very differently from their classical convex-concave archetypes.

A further computational distinction appears in online learning. For many polynomial-time solvable offline min-max discrete problems, efficient vanishing regret or even vanishing approximate $\mathbb{V} \subset \mathbb{R}$4-regret is impossible for some $\mathbb{V} \subset \mathbb{R}$5, unless standard complexity collapses occur; min-max vertex cover is the principal positive exception discussed, with a matching $\mathbb{V} \subset \mathbb{R}$6-approximate vanishing regret upper bound via online gradient descent (Bampis et al., 2019). This suggests that even when a static min-max objective is algorithmically manageable, dynamic or repeated variants may inherit additional hardness.

3. Algorithmic solution methods for the explicit DMMV problem

To address practical DMMV instance sizes, a GPU-accelerated heuristic called the Accelerated Maximum Violation Minimizer (AMVM) has been proposed. AMVM is based on Adaptive Large Neighborhood Search and uses an iterative destruction-repair-local-search workflow (Ahmed et al., 19 Aug 2025).

Initialization obtains an initial discrete solution either from a relaxed LP plus rounding or from a baseline such as round-to-nearest. Each iteration then applies destruction, repair, and local search. Destruction may be Random Destroy or Worst-Remove Destroy based on impact scores. Repair may be Random Repair or Greedy Repair, with the latter selecting the neighbor value minimizing the current objective. Local search includes both 1-OPT, which moves one variable to another discrete level if the objective improves, and Swap, which exchanges two variable values subject to a strictly improving condition (Ahmed et al., 19 Aug 2025).

For variables $\mathbb{V} \subset \mathbb{R}$7, the swap criterion is stated as follows: a swap strictly improves the maximum violation iff

$\mathbb{V} \subset \mathbb{R}$8

where $\mathbb{V} \subset \mathbb{R}$9, $\mathbf{x} \in \mathbb{V}^n$0, and $\mathbf{x} \in \mathbb{V}^n$1. A corollary then restricts attention to swap pairs that are likely to alter the worst violated constraints and that pass necessary quick checks on a subset of worst residuals, reducing the need to evaluate all $\mathbf{x} \in \mathbb{V}^n$2 swaps (Ahmed et al., 19 Aug 2025).

The acceleration strategy is explicitly architectural as well as algorithmic. All destructive, repair, and improvement moves are implemented as fused CUDA kernels in PyTorch; matrix-vector products, norm calculations, impact scoring, candidate evaluations, and reductions are executed in batch on GPU; and keeping tensor operations on-device minimizes CPU-GPU synchronization overhead. Reported iteration rates are up to $\mathbf{x} \in \mathbb{V}^n$3 faster than single-CPU implementations (Ahmed et al., 19 Aug 2025).

4. Metric and ultrametric violation distance as $\mathbf{x} \in \mathbb{V}^n$4-type DMMV

A second major usage of the DMMV idea appears in metric repair. Given pairwise distances $\mathbf{x} \in \mathbb{V}^n$5, the Metric Violation Distance problem is

$\mathbf{x} \in \mathbb{V}^n$6

where $\mathbf{x} \in \mathbb{V}^n$7 is the set of metrics on $\mathbf{x} \in \mathbb{V}^n$8 points and $\mathbf{x} \in \mathbb{V}^n$9 counts the number of modified entries. The Ultrametric Violation Distance problem is

$\|A\mathbf{x}-\mathbf{b}\|_\infty$0

where $\|A\mathbf{x}-\mathbf{b}\|_\infty$1 is the set of ultrametrics, characterized by

$\|A\mathbf{x}-\mathbf{b}\|_\infty$2

The paper explicitly states that both MVD and UMVD can be interpreted as discrete minimization of the number of violations of (ultra)metricity and are essentially the DMMV problem in the Hamming, i.e. $\|A\mathbf{x}-\mathbf{b}\|_\infty$3, norm (Cohen-Addad et al., 2022).

Algorithmically, the minimization versions admit strong approximations. For unweighted MVD, the cited result is an $\|A\mathbf{x}-\mathbf{b}\|_\infty$4-approximation running in $\|A\mathbf{x}-\mathbf{b}\|_\infty$5 time. For UMVD, a pivot-based method gives an $\|A\mathbf{x}-\mathbf{b}\|_\infty$6-approximation, more precisely $\|A\mathbf{x}-\mathbf{b}\|_\infty$7 when $\|A\mathbf{x}-\mathbf{b}\|_\infty$8 is the number of distinct distances, and a separate constant-factor approximation is obtained by interpreting UMVD as hierarchical correlation clustering. Weighted UMVD admits an $\|A\mathbf{x}-\mathbf{b}\|_\infty$9-approximation (Cohen-Addad et al., 2022).

The maximization versions behave very differently. When the objective is to select the largest subset of entries already forming a metric or ultrametric, both Max-Metric Violation Distance and Max-Ultrametric Violation Distance are NP-hard to approximate within any constant factor assuming the Unique Games Conjecture, even for unweighted and complete instances (Cohen-Addad et al., 2022). This sharp contrast between minimization and maximization is one of the clearest complexity-theoretic fault lines in the DMMV-related literature.

5. Terminological range and theoretical interpretations

The term “Discrete Min-Max Violation” is used across several nearby research programs in partially overlapping senses. The following summary captures the main usages present in the cited literature.

Usage	Core content	Source
Explicit optimization problem	$\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$0	(Ahmed et al., 19 Aug 2025)
Hamming-norm violation repair	MVD/UMVD minimize $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$1 changes to satisfy metric or ultrametric constraints	(Cohen-Addad et al., 2022)
Discrete–continuous discrepancy	discrete and continuous min-max problems may disagree	(Jost et al., 2021)
Failure of classical max-min ordering	coupled constraints can make either min-max or max-min larger	(Tsaknakis et al., 2021)

In the discrete-to-continuous min-max framework based on homogeneous and piecewise multilinear extensions, DMMV refers to cases where the discrete and continuous saddle point problems disagree: the continuous saddle point problem may admit a solution while the original discrete problem may not. This appears in the discussion surrounding the non-converse of the discrete–continuous saddle point equivalence theorem (Jost et al., 2021). The conceptual role of “violation” here is not residual fitting, but a failure of exact transfer between combinatorial and continuous formulations.

Earlier work on discrete min-max optimization also provides an algorithmic lineage for DMMV-style problems. For instances consisting of a discrete structure together with $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$2 comparables from a totally ordered set $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$3, where the solution must be one of the comparables, the ordered version with runtime $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$4 yields an unordered algorithm running in $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$5 time and $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$6 comparisons. The paper characterizes such problems via min-max circuits, continuous order statistics, and commutation with every monotone function (0801.4130). This suggests a precursor framework in which discrete worst-case objectives are exploited without full sorting.

6. Applications and empirical performance

In post-training quantization of LLMs, DMMV is instantiated row-wise as

$\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$7

For Meta OPT-125M at 3-bit precision and without outlier separation, AMVM reports perplexities $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$8 and $\max_{i=1,\ldots,m} |(A\mathbf{x}-\mathbf{b})_i|$9 and is described as outperforming SqueezeLLM, GPTQ, OmniQuant, ApiQ, and RTN on C4 and WikiText2 by approximately $\ell_0$0 average improvement (Ahmed et al., 19 Aug 2025).

In discrete tomography, the objective is to reconstruct an image from projection data while constraining pixel values to a finite set and minimizing worst-case residuals. On four-level segmented brain images with strong uniform noise, AMVM on GPU reduces reconstruction error by $\ell_0$1 over DART, yields the lowest worst-case $\ell_0$2 errors among compared methods, and runs $\ell_0$3 faster than CPU (Ahmed et al., 19 Aug 2025).

In discrete FIR filter design, the min-max objective corresponds to minimizing ripple under fixed-point coefficient constraints. On moderate instances, AMVM matches Gurobi’s optimal ripple $\ell_0$4 in seconds; on a larger instance with $\ell_0$5, it is reported as $\ell_0$6 faster and achieves ripple $\ell_0$7 versus $\ell_0$8; and on an Anti-Hum $\ell_0$9Hz filter, GPU AMVM attains ripple $(A,b,\mathbb{V})$0 compared with Gurobi’s $(A,b,\mathbb{V})$1 in $(A,b,\mathbb{V})$2s versus $(A,b,\mathbb{V})$3s (Ahmed et al., 19 Aug 2025).

Related discrete min-max objectives also arise outside linear residual fitting. In swarm robotics on the infinite grid, the min-max gathering problem seeks a gathering node minimizing the maximum Manhattan distance any robot must travel. The cited work characterizes all initial configurations where the optimal gathering problem is unsolvable, proves that the subgraph of min-max nodes is always a step-graph, a disconnected step-graph, or a $(A,b,\mathbb{V})$4-cycle, and gives a deterministic distributed algorithm for gathering $(A,b,\mathbb{V})$5 robots to a designated min-max node (Chakraborty et al., 2024). This suggests that DMMV belongs to a broader family of discrete worst-case optimization problems in which the central object is not average efficiency but the control of the most adverse local deviation.