Column Number Problem: A Multi-Context View

• The Column Number Problem is a multifaceted concept addressing matrix column-related challenges in optimization, enumeration, and extremal analysis.
• It encompasses distinct research programs, from NP-complete column subset selection and MDP-based column generation to holonomic recurrences in balanced matrices and asymptotic bounds in Δ-modular theory.
• Applications span numerical linear algebra, large-scale linear programming, combinatorics, and sensing-matrix design, highlighting its practical and theoretical significance.

“Column Number Problem” is a polysemous term rather than a single canonical problem. In the literature represented here, it denotes at least four distinct research programs: the exact optimization of column subset selection in numerical linear algebra, the decision of how many columns to add during column generation for large-scale linear programming, the enumeration of balanced binary matrices as the number of columns varies, and the extremal question of how many columns a $\Delta$-modular integer matrix can have. A related usage appears in sensing-matrix design, where the number and placement of columns govern worst-case conditioning of small submatrices (Shitov, 2017, Hu et al., 2024, Dougherty-Bliss et al., 2024, Kriepke et al., 21 Feb 2025, Achanta et al., 2012).

1. Terminological scope

Across these literatures, the common object is a matrix column, but the role of the column changes fundamentally from one context to another. In numerical linear algebra, a column is an observable feature vector and the problem is to choose a subset of actual columns that best reconstruct the full matrix. In column generation, a column is a variable of a master linear program, often representing a route, pattern, or zone; the issue is not how many columns a matrix possesses in principle, but which columns should be added to the restricted master and in what quantity. In enumerative combinatorics, the “column number” is the symbolic parameter $2n$ in a family of $2k\times 2n$ balanced $0$-$1$ matrices. In the $\Delta$-modular literature, the problem is extremal: determine the maximum number of nonzero or pairwise non-parallel columns compatible with bounded full-rank minors (Dougherty-Bliss et al., 2024, Paat et al., 2022, Paat et al., 16 Sep 2025).

This semantic divergence matters because the associated notions of difficulty are different. In CSSP, hardness is NP-completeness of an exact decision problem. In column generation, the central difficulty is state-space explosion or pricing complexity under an exponential implicit variable set. In balanced-matrix counting, the difficulty is symbolic and numeric computation of P-recursive sequences. In $\Delta$-modular theory, the difficulty is the asymptotic extremal structure of integer matrices and their matroids. A frequent misconception is therefore to treat “Column Number Problem” as a single standardized term; the surveyed literature does not support that identification.

2. Column subset selection in numerical linear algebra

In numerical linear algebra, the problem appears as the Column Subset Selection Problem (CSSP), described in older language as related to the “Column Number Problem.” For a real matrix $M\in\mathbb R^{r\times c}$ and integer $k\le c$, one seeks

$\delta_k(M)=\min_{S,A}\|M-SA\|,$

where $2n$0 ranges over all $2n$1 submatrices formed by selecting $2n$2 columns of $2n$3, and $2n$4 is arbitrary. The paper uses the Frobenius norm throughout, and for fixed $2n$5 the optimal coefficient matrix is the least-squares solution $2n$6, so the objective can be written equivalently as

$2n$7

The exact decision problem asks: given $2n$8, $2n$9, and positive rational $2k\times 2n$0, is $2k\times 2n$1? This decision version is NP-complete in the Turing model for rational inputs (Shitov, 2017).

The hardness proof is by reduction from graph 3-coloring. For a graph $2k\times 2n$2 with $2k\times 2n$3 and $2k\times 2n$4, the construction defines $2k\times 2n$5 and builds a structured rational matrix $2k\times 2n$6 with $2k\times 2n$7 rows and $2k\times 2n$8 columns. The reduction is calibrated so that

$2k\times 2n$9

The proof first shows that any feasible $0$0-column subset must choose exactly one of the three copies $0$1 for each vertex $0$2 and no edge column; this decodes the selected columns into a map $0$3. It then proves that the threshold is met exactly when $0$4 is a proper 3-coloring. The result closes an open complexity question that had persisted for decades and provides an unconditional hardness statement stronger than earlier UG-hardness results. A plausible implication is that the prominence of approximation algorithms, randomized methods, and heuristic QR- or CUR-based procedures in the CSSP literature reflects not only practical convenience but genuine worst-case intractability of exact optimization.

3. Variable-number column selection in column generation

In large-scale linear programming, “column number problem” has a different meaning: not the number of columns in the input matrix, but how many improving variables should be added to the restricted master problem at each column-generation iteration. In the standard master problem

$0$5

pricing searches for a column with minimum reduced cost

$0$6

The classical greedy policy adds the single most negative reduced-cost column. The opposing extreme, adding all negative reduced-cost columns, may flood the master with redundant variables. Fast Family Column Generation (FFCG) formalizes this trade-off as an MDP whose action is a nonempty subset $0$7 of candidate columns, with reward

$0$8

To avoid the $0$9 subset action space, FFCG sequentializes selection with a STOP action, reducing the effective action-space complexity to $1$0. On the reported benchmarks, it reduces the number of column-generation iterations by $1$1 for CSP and $1$2 for VRPTW, and total computing time by $1$3 for CSP and $1$4 for VRPTW on average relative to Greedy-S (Hu et al., 2024).

This formulation generalizes an earlier reinforcement-learning view in which CG is treated as sequential decision-making but still selects a single candidate column per iteration. RLCG represents the current restricted master and candidate set as a bipartite graph, uses a GNN-based DQN to score candidate columns, and reports average iteration reductions of $1$5 for CSP and $1$6 for VRPTW against a greedy most-negative-reduced-cost policy (Chi et al., 2022). The distinction between the two papers is substantive: RLCG optimizes the order of single-column additions, whereas FFCG explicitly addresses the variable-number selection problem.

In application-specific CG frameworks, the same issue appears under different column semantics. In the generalized Micro-Transit Zoning Problem, a column is a feasible connected zone $1$7 with cost $1$8, and pricing searches for a new zone with positive reduced cost under a global budget model (Hu et al., 8 Mar 2026). In real-time train dispatching, a column is a complete train path for one service, and pricing is a MIP because clique shadow prices require deciding whether a candidate path belongs to conflict cliques (Schälicke et al., 2023). In the drone-aided blood collection routing problem, a column is a full-day truck–drone tandem tour encoded at the master level by a binary coverage vector over donation batches $1$9 (Abbaszadeh et al., 28 Jan 2026). In all three cases, the “column number problem” is operational rather than extremal: how to manage an exponential implicit column set so that only a small useful subset is ever materialized.

4. The number of columns as a symbolic parameter: balanced binary matrices

In enumerative combinatorics, the phrase refers to a fixed-row, variable-column counting problem. For fixed $\Delta$0, let $\Delta$1 denote the number of $\Delta$2 binary matrices in which each row has exactly $\Delta$3 ones and $\Delta$4 zeros and each column has exactly $\Delta$5 ones and $\Delta$6 zeros. The parity constraints force the natural dimensions $\Delta$7. The central question is then: for fixed row count $\Delta$8, how does the count depend on the symbolic column parameter $\Delta$9? The paper proves that for every fixed $\Delta$0, the sequence $\Delta$1 is holonomic, so there exist $\Delta$2 and polynomials $\Delta$3 with $\Delta$4 such that

$\Delta$5

A constant-term representation is

$\Delta$6

where $\Delta$7 is the elementary symmetric polynomial. This is the paper’s structural answer to the column-number problem in this setting: the dependence on the column parameter is always P-recursive for fixed row count (Dougherty-Bliss et al., 2024).

The result is existential rather than uniformly explicit. For $\Delta$8, the paper gives a rigorous second-order recurrence for the number of $\Delta$9 balanced matrices; for $M\in\mathbb R^{r\times c}$0, it gives a rigorous fourth-order recurrence for the number of $M\in\mathbb R^{r\times c}$1 balanced matrices. For $M\in\mathbb R^{r\times c}$2, the authors extend OEIS data substantially but do not derive recurrences. The paper emphasizes two distinct barriers. Symbolically, creative telescoping and holonomic-systems elimination become computationally prohibitive as $M\in\mathbb R^{r\times c}$3 grows. Numerically, even generating many initial terms is difficult, motivating a dynamic program on exponent vectors

$M\in\mathbb R^{r\times c}$4

together with symmetry reduction, state compression, and modular arithmetic. A restricted-pattern version with forbidden horizontal and vertical words $M\in\mathbb R^{r\times c}$5 and $M\in\mathbb R^{r\times c}$6 is handled at the same structural level: for fixed $M\in\mathbb R^{r\times c}$7, $M\in\mathbb R^{r\times c}$8, and $M\in\mathbb R^{r\times c}$9, the sequence $k\le c$0 is again holonomic. In this literature, “column number problem” therefore means symbolic dependence on the number of columns, not optimization over subsets or variables.

5. Extremal column counts for $k\le c$1-modular integer matrices

A third major usage is extremal and arithmetical. For a rank-$k\le c$2 integer matrix $k\le c$3, let $k\le c$4 be the maximum absolute value of an $k\le c$5 subdeterminant. A matrix is $k\le c$6-modular if every rank-size minor has determinant at most $k\le c$7 in absolute value. The column number problem asks for the maximum number of nonzero, pairwise non-parallel columns of a rank-$k\le c$8, $k\le c$9-modular matrix. In matroid language this is the maximum number of points of a simple representable matroid in the class $\delta_k(M)=\min_{S,A}\|M-SA\|,$0. A general asymptotic upper bound states that for each fixed positive integer $\delta_k(M)=\min_{S,A}\|M-SA\|,$1 and sufficiently large $\delta_k(M)=\min_{S,A}\|M-SA\|,$2, every rank-$\delta_k(M)=\min_{S,A}\|M-SA\|,$3 $\delta_k(M)=\min_{S,A}\|M-SA\|,$4-modular matrix has at most

$\delta_k(M)=\min_{S,A}\|M-SA\|,$5

nonzero, pairwise non-parallel columns. The lower-bound family $\delta_k(M)=\min_{S,A}\|M-SA\|,$6 gives

$\delta_k(M)=\min_{S,A}\|M-SA\|,$7

so the upper bound is tight in the quadratic term and differs only by an additive linear term in $\delta_k(M)=\min_{S,A}\|M-SA\|,$8 (Paat et al., 2022).

For generic rank-2 matrices, the picture is much sharper. Let $\delta_k(M)=\min_{S,A}\|M-SA\|,$9 denote the maximum number of columns of a generic rank-2 $2n$00-modular integer matrix, where generic means that every $2n$01 minor is nonzero. For every $2n$02 and every $2n$03,

$2n$04

equivalently

$2n$05

The paper further shows that, in this regime, $2n$06 is quasi-linear of period $2n$07, nondecreasing, and always even (Kriepke et al., 21 Feb 2025).

For $2n$08, the exact large-rank formula is now known. If $2n$09 is sufficiently large, the maximum number of pairwise non-parallel columns in a rank-$2n$10, $2n$11-modular matrix is

$2n$12

This settles a conjecture of Lee, Paat, Stallknecht, and Xu for the trimodular case. The same paper shows that extremality is not unique: for sufficiently large $2n$13 there are at least three pairwise non-isomorphic extremal $2n$14-modular matroids, and more generally, if $2n$15, the number of $2n$16-modular extremal matroids of size $2n$17 is at least $2n$18, where $2n$19 is the partition number of $2n$20 (Paat et al., 16 Sep 2025). In this literature, “column number” means an exact or asymptotic extremal function, not a constructive column-selection algorithm.

6. Adjacent formulations: conditioning, deterministic selection, and column recurrences

Several nearby problems concern columns centrally without using the phrase in exactly the same sense. In sensing-matrix design, one asks for a $2n$21 matrix with unit columns whose worst $2n$22 submatrix conditioning is as small as possible. Writing $2n$23, the objective reduces to minimizing the worst triple value of

$2n$24

The exact optimal constructions depend on parity: for even $2n$25, uniform spacing is optimal; for $2n$26 or $2n$27, uniform spacing modulo $2n$28 is optimal; for odd $2n$29, the optimum is

$2n$30

so uniform $2n$31-point spacing is not optimal (Achanta et al., 2012).

In deterministic subset extraction from a normalized rectangular matrix $2n$32, another line of work selects columns greedily by a potential

$2n$33

where $2n$34 is the current selected submatrix and $2n$35 are eigenvectors of $2n$36. For a concrete choice $2n$37, the paper proves that one can extract $2n$38 columns with near-isometric behavior provided

$2n$39

The contribution is fully constructive and yields individual eigenvalue bounds for the selected Gram matrix (Chretien et al., 2015).

At much larger scale, CPQR-based column selection for extremely wide matrices is revisited through the deterministic CCEQR algorithm, whose three phases are “collect, commit, expand.” CCEQR maintains a tracked subset of likely pivot columns, applies CPQR only to a small candidate block, and provably recovers a column permutation equivalent to the one computed by the Golub–Businger algorithm. Its stopping criterion is a user-specified target number $2n$40; it accelerates CPQR-based column selection, but does not itself solve the separate model-order question of how $2n$41 should be chosen (Armstrong et al., 29 Jan 2025).

A combinatorial analogue appears in path-counting tables. For lattice paths inside an $2n$42 table with steps $2n$43, $2n$44, and $2n$45, let $2n$46 be the $2n$47-th column sum. The paper proves that $2n$48 satisfies a minimal linear recurrence of order $2n$49, equivalently

$2n$50

with characteristic polynomial $2n$51. Here the “column number problem” is the derivation of minimal recurrences for column sequences and column sums rather than matrix optimization or extremal determinant bounds (Ghouchan, 2019).

The surveyed literature therefore supports no single universal definition of the Column Number Problem. The phrase names a family of column-centered questions whose mathematical cores are quite different: exact subset optimization, online variable injection in decomposition algorithms, symbolic dependence on column count, and extremal column cardinality under arithmetic constraints. What unifies them is not a shared theorem or formulation, but the fact that columns are the primary combinatorial, algebraic, or algorithmic resource.