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arXiv2026

Shortest self-orthogonal embeddings of binary linear codes

There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal C$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $\mathcal H_r$. Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $\mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $\mathcal H_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],\, [98, 8, 46],\,[114, 8, 54]$, and $[191, 8, 94]$.

cs.IT↗
arXiv2026

A penalised Saito functional for heuristic search of free line arrangements

We introduce the penalised Saito functional $\mathfrak S_{λ,β}(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ of $n$ lines and a prescribed pair $d_1+d_2=n-1$. It measures the alignment of a candidate Saito determinant with the defining polynomial while penalising the failure of the candidate derivations to be logarithmic. We prove that the functional takes values in $[0,1]$, vanishes exactly when $\mathcal{A}$ is free with exponents $(1,d_1,d_2)$, and lies strictly between $0$ and $1$ otherwise. For fixed $(d_1,d_2)$, it is upper semicontinuous on the reduced configuration space, continuous at arrangements free with the prescribed pair, and converges as $λ\to\infty$ to the corresponding binary freeness test. We use a numerical approximation of this functional, together with a small $b_2$-shell term, to guide fixed-cardinality line-replacement searches over $\mathbb{Q}$ and selected quadratic extensions. Numerical values are used only to select candidates; every reported arrangement is certified in exact arithmetic using Saito's criterion. At the current snapshot, the certified database contains $6{,}146$ representatives with distinct Weisfeiler--Leman fingerprints and cardinalities up to $n=28$. Among them, $3{,}012$ have multiplicity gap $ε(\mathcal{A})=d_1-m(\mathcal{A})\geq2$, including lower-bound-extremal examples with $ε=7$. These non-supersolvable arrangements provide test cases for studying realisation spaces and the persistence of freeness among realisations of the same intersection lattice, in connection with Terao's conjecture.

math.AG↗
arXiv2026

The Deterministic Hare Core Is Nonempty for Nine-Seat Approval Elections

Core stability gives approval-based committee elections a strong form of coalitional proportionality, but deterministic existence is not known for an arbitrary number of seats. We prove that every finite election with nine seats has a deterministic Hare-quota core committee. Starting from an eight-seat committee that is both a global Proportional Approval Voting (PAV) maximizer and core stable, we suppose that all one-candidate extensions are blocked and extract an inclusion-minimal closed response system. Singleton responses are eliminated by exact equality rigidity. For the remaining responses, rational Farkas certificates bound every nonpositive PAV add-marginal drift below by $-n/189$ and force every successor drift above $n/126$. The uniform response chain is irreducible, so stationarity makes these bounds incompatible. The computer-assisted component comprises an exact pointwise check over 7,356 voter types, 36 one-blocker cells, and a symmetry-complete family of 19 two-blocker motifs. Exact certificate archives and standard-library verifiers support these checks. The theorem settles the nine-seat case but does not decide deterministic core existence for arbitrary committee size.

cs.GT↗
arXiv2026

Geometry-dependent rank defect in $C^1$ cubic spline space

Determining the dimension of the $C^1$ cubic spline space $S_3^1(\mathcal{T})$ on an arbitrary nondegenerate planar triangulation has remained unresolved since the 1970s. Schumaker's lower bound includes a local correction $σ$ for singular interior four-stars, and it was conjectured that this bound is always attained. We disprove this conjecture by constructing a one-parameter family of nondegenerate realizations of a fixed 18-triangle complex, with only the central vertex moving as $v_6(t)=(t,0)$ on the admissible interval $I=(-3/4,24/55)$. The family exhibits three distinct cases. For $t\in I\setminus\{1/5,3/83\}$, the lower bound is attained and $\dim S_3^1(\mathcal{T}(t))=33$. At $t=3/83$, the central four-star is singular, $σ=1$, and the resulting dimension 34 is exactly accounted for by the classical local correction. At $t=1/5$, however, all interior vertices are nonsingular and $σ=0$, yet $\dim S_3^1(\mathcal{T}(1/5))=34>P_{\mathcal{T}(1/5)}(1,3)=33$. The smoothing-cofactor calculation shows that the dependence at $t=3/83$ is confined to the central vertex block, whereas the dependence at $t=1/5$ couples all seven interior vertex cycles even though every individual block has full row rank. A complementary Bernstein--Bézier calculation gives the same dimension profile. Thus the singular-four-star correction does not capture every geometry-dependent contribution to $\dim S_3^1(\mathcal{T})$; genuinely global compatibility must also be taken into account.

math.NA↗
arXiv2026

The Most Malicious Maître D'

We revisit the family of "napkin problems" first discussed by Peter Winkler in his 2004 puzzle book and discussed subsequently in several papers. This family of problems involves diners being seated around a circular table in which napkins are placed between the seats, so it is ambiguous which napkin belongs to which seat. The problems seek to quantify the proportion of diners who end up with no napkin under a variety of assumptions, one of which is that the diners have an adaptive adversary known as "the malicious maitre d'," who seeks to maximize the number of napkinless diners. Winkler's original strategy for the adaptive maitre d' was shown to be suboptimal in 2023 when Acton, Petersen, Shirman, and Toal presented a better (yet also suboptimal) strategy. In this paper we demonstrate an optimal strategy for the malicious maitre d' and compute the expected proportion of napkinless diners as a function of the probability of a diner taking the left napkin. Interestingly, the strategy does not depend on the exact probability; rather, it only depends on which napkin (left or right) the diners tend to prefer.

math.CO↗
arXiv2026

Explicit Rank Extractors and Subspace Designs via Function Fields, with Applications to Strong Blocking Sets

We give new explicit constructions of several fundamental objects in linear-algebraic pseudorandomness and combinatorics, including lossless rank extractors, weak subspace designs, and strong $s$-blocking sets over finite fields. Our focus is on the small-field regime, where the field size depends only on a secondary parameter (such as the rank or codimension) and is independent of the ambient dimension. This regime is central to several applications, yet remains poorly understood from the perspective of explicit constructions. In this setting, we obtain the first explicit constructions of lossless rank extractors and weak subspace designs for $r\ll k$, where $r$ denotes the rank (or codimension), over finite fields $\mathbb{F}_q$ with $q \ge \mathrm{poly}(r)$ and $q$ non-prime, with near-optimal parameters. For other finite fields, including prime fields and small fields, we obtain weaker but still improved bounds. As a consequence, we construct explicit strong $s$-blocking sets in $\mathrm{PG}(k-1,q)$ of size $O(s(k-s)q^s)$ for all sufficiently large non-prime fields $q \ge \mathrm{poly}(s)$, matching the best known non-explicit bounds up to constant factors. This significantly improves the previous best bound $2^{O(s^2 \log s)} q^s k$ of Bishnoi and Tomon (Combinatorica, 2026), which requires $q \ge 2^{Ω(s)}$. Our approach is primarily algebraic, combining techniques from function fields and polynomial identity testing. In addition, we develop a complementary Fourier-analytic framework based on $\varepsilon$-biased sets, which yields improved explicit constructions of strong $s$-blocking sets over small fields.

cs.IT↗
arXiv2026

Dimension Dependent Correlation Gap Bounds under Restricted Independence

The pairwise independent correlation gap is the ratio of the maximum expected value of a set function under arbitrary dependence to that under pairwise independence, measuring the loss from this independence restriction. Under mutual independence, this gap is universally bounded by $e/(e-1)$ for monotone submodular functions. With pairwise independence, a tighter $4/3$ upper bound was established for several special cases, including $n=3$, and conjectured to hold universally. A recent AI-assisted counterexample disproved this conjecture for $n=5$, leaving the validity of the $n=4$ bound and the tight worst case bound open. We resolve both questions. First, for $n=4$, we establish that the $4/3$ bound holds universally and is tight using an AI-assisted proof combining theoretical analysis and computational verification. The proof combines a structural characterization of optimal numerator vertices, permutation symmetry, cone certificate systems, Bernstein polynomial representations, recursive simplex subdivision, and verification of $2,745$ Bernstein coefficient systems. Second, we show that the worst case pairwise independent correlation gap attains $e/(e-1)$ asymptotically by constructing an instance with identical marginal probabilities and a monotone submodular union coverage function on a ground set partitioned into $m$ blocks. The number of blocks grows sublinearly with the ground set size. The result follows by constructing a feasible solution to a scaled asymptotic reduced dual of the pairwise independent linear program and immediately extends to $t$-wise independent random elements ($t\ge2$), since $t$-wise independence implies pairwise independence. Thus, pairwise independence, despite being the least restrictive form of independence in the $t$-wise independence hierarchy, can be as restrictive as mutual independence in the worst case.

math.PR↗
arXiv2026

Refutation of the Non-Cancelling Intersections Conjecture

The Non-Cancelling Intersections (NCI) conjecture of Amarilli, Monet and Suciu [arXiv:2401.16210] states that the union of a finite family of sets can always be built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In Wilhelm [arXiv:2608.19414] the conjecture was shown to fail when the witnessing dot-algebra expression is required to be left-linear. Here we remove that restriction and show that the conjecture is false in general: there is a finite lattice admitting no dot-algebra representation of its top element whatsoever. The counterexample is a lattice $P_{p,\mathfrak{m}}$ as in Wilhelm [arXiv:2608.19414], and the argument differs in only two ways. First, we replace the sequential "toggle game" of Wilhelm [arXiv:2608.19414] by a corresponding tree-shaped object, the plane tree, which stands to dot-algebra trees as the toggle game stands to left-linear ones. Second, we use a marked plane in which there is no admissible set of any size between $2p$ and $4p$, which also removes the need for the Erdős--Beck theorem and for the arithmetic Nullstellensatz. Consequently $p$ need not be astronomically large: every prime $p \ge 10^{5}$ works.

math.CO↗
arXiv2026

Determining a graph from its reconfiguration graph

Given a graph $G$ and a natural number $k$, the $k$-recolouring graph $\mathcal{C}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and whose edges link pairs of colourings which differ at exactly one vertex of $G$. Recently, Hogan et al. proved that $G$ can be determined from $\mathcal{C}_k(G)$ provided $k$ is large enough (quadratic in the number of vertices of $G$). We improve this bound by showing that $k=χ(G)+1$ colours suffice, and provide examples of families of graphs for which $k=χ(G)$ colours do not suffice. We then extend this result to $k$-Kempe-recolouring graphs, whose vertices are again the $k$-colourings of a graph $G$ and whose edges link pairs of colourings which differ by swapping the two colours in a connected component of the subgraph induced by selecting those two colours. We show that $k=χ(G)+2$ colours suffice to determine $G$ in this case. Finally, we investigate the case of independent set reconfiguration, proving that in only a few trivial cases is one guaranteed to be able to determine a graph $G$.

math.CO↗
arXiv2026

Structural Packing and Dyadic Factorization of Sparse Positive Definite Matrices

Efficient inversion of large sparse positive definite matrices requires exploiting sparsity patterns beyond those captured by conventional bandwidth reduction. In this work, we recast nested dissection, a prominent alternative, as a two-stage framework. The matrix was first packed into block-tridiagonal or dyadic form, followed by sparse Gram-Schmidt orthogonalization. This decomposition provided a unified perspective on sparse matrix factorization and inversion and identified dyadic structure as a fundamental component of sparse Cholesky factorization. For the first stage, we introduced a packing algorithm that recovered block-tridiagonal and dyadic patterns using a novel $\ell_1$ criterion. Using approximate distances obtained through classical multidimensional scaling, the method was effective when the target structure was sufficiently represented among the nonzero entries. Iterative application could also remove structural noise and reveal hidden dyadic organization, corresponding to separator identification in nested dissection. For the second stage, we developed the theory of dyadically structured matrices. We derived sparse factorization and inversion procedures, analyzed their computational complexity, and obtained an efficient inversion algorithm. A modified version reduced the cost of inverting block-tridiagonal matrices, demonstrating the benefit of exploiting their structure directly rather than treating them as generic band matrices.

math.NA↗
arXiv2026

The Erdős-Pósa Property for Colorful Minors

A colorful graph relation enhances the minor relation by merging color sets along contractions and by allowing the removal of colors; it generalizes rooted minors and models problems on graphs with several, possibly overlapping, annotated vertex sets. A graph has the Erdős-Pósa property for minors if and only if it is planar, by a classical theorem of Robertson and Seymour. In this work we determine, for the colorful minor relation, exactly which colorful graphs have the Erdős-Pósa property. Our characterization takes three equivalent forms. The first is structural: the colorful graphs with the property are those that can be drawn with all their colored vertices on one face and whose colors are, in a precise sense, laid out along that face without interleaving. The second is given by an obstruction set: they are those excluding every member of an explicit infinite family $\mathcal{O},$ of which only $\mathbf{O}(|I|^{4})$ members have colors that are a subset of $I,$ for every finite set $I$ of colors. The third is grid-like: they are exactly the colorful minors of unions of particular families of segregated grids, the colorful analogues of the grids that drive the classical proof.

math.CO↗
arXiv2026

On the Laplacian spectral gap of generalized pancake graphs

The generalized pancake graph $P(m,n)$ is the Cayley graph of the group of colored permutations $\mathbb{Z}_m\wr S_n=(\mathbb{Z}_m)^n\rtimes S_n$ generated by generalized prefix reversals. In this paper, we establish that, for all $m,n\geq2$, the spectral gap $γ(P(m,n))$ of the normalized Laplacian satisfies $α_m/n\leqγ(P(m,n))\leq1/n$, where $α_m$ is a positive constant that depends only on $m$. As a consequence, for every fixed $m\geq2$, $γ(P(m,n))$ is $Θ_m(1/n)$ as $n\to\infty$. The proof combines Cesi's semi-recursive spectral-gap inequality with a Fourier decomposition of the appropriate operators associated with a coset Schreier graph of color-position pairs. For fixed $n\geq2$, we also establish that $γ(P(m,n))$ is $Θ_n(m^{-2})$ as $m\to\infty$. This disproves a conjecture of Blanco and Buehrle asserting that, for fixed $n$, the corresponding undirected generalized pancake graphs form an expander family. Additionally, we present a counterexample to a recent conjecture of Greaves and Zhu concerning equality between the spectral gaps of the full Cayley graph and the associated coset Schreier graph.

math.CO↗
arXiv2026

Statistical Inference for Privatized Data with Unknown Sample Size

We develop both theory and algorithms to analyze privatized data in unbounded differential privacy (DP), where even the sample size is considered a sensitive quantity that requires privacy protection. We show that the distance between the sampling distributions under unbounded DP and bounded DP goes to zero as the sample size $n$ goes to infinity, provided that the noise used to privatize $n$ is at an appropriate rate; we also establish that Approximate Bayesian Computation (ABC)-type posterior distributions converge under similar assumptions. We further give asymptotic results in regimes where the privacy budgets vary, establishing similarity of sampling distributions as well as showing that the MLE in the unbounded setting converges to the bounded-DP MLE. To facilitate valid, finite-sample Bayesian inference on privatized data under unbounded DP, we propose a reversible jump MCMC algorithm which extends the data augmentation MCMC of Ju et al. (2022). We also propose a Monte Carlo EM algorithm to compute the MLE from privatized data in both bounded and unbounded DP. We apply our methodology to analyze a linear regression model as well as a 2019 American Time Use Survey Microdata File which we model using a Dirichlet distribution.

math.ST↗
arXiv2026

Strengthening Recursive Constructions for Zero-Error Shannon Capacity

The exact Shannon capacity is unknown for every odd cycle beyond the five-cycle $C_5$, making odd cycles a central open problem in zero-error information theory. Improving the known lower bounds requires constructing large independent sets in strong powers of these graphs. Recent AI-assisted work has produced a rapid sequence of improvements: building on the construction of Itty et al., Gao developed a recursive product construction for combining structured independent sets, and Buys, Polak, and Zuiddam (BPZ) subsequently strengthened this through a richer recursion framework. We continue this line of AI-assisted exploration and introduce a heterogeneous refinement of these constructions. The central observation is that the usefulness of an intermediate construction depends not only on the size of its current main independent set, but also on the auxiliary structure it carries into subsequent recursion. Consequently, different parts of that auxiliary structure need not use the same independent set, and different occurrences in a recursion need not use the same intermediate representation. We formalize this for Gao's binary product and derive explicit propagation rules showing how heterogeneous choices strengthen the resulting gadget while leaving its current code size unchanged, then extend the principle to the more general BPZ framework, tailoring constructions to the distinct roles they play within the recursion. Applying these refinements to the seven-cycle $C_7$, we obtain an independent set in $C_7^{\boxtimes 500}$ yielding $Θ(C_7)\ge 3.25883262\ldots$, improving the best known lower bound. Beyond the numerical gain, the results illustrate a general principle for recursive zero-error constructions: intermediate structures with the same dimension and current code size can have different downstream value depending on where and how they are used in the recursion.

cs.IT↗
arXiv2026

Fair Division of Graphs: Beyond Traceability

In this paper, we study fair division problems in which resources are structured as graphs and agents must receive connected bundles. This connectivity requirement fundamentally alters the problem, making it significantly more challenging than its classical counterpart. We focus on the fairness notion of $\mathrm{EF1}_{\mathrm{outer}}$, where envy can be eliminated by removing at most one vertex whose deletion does not disconnect the bundle -- a critical constraint for applications such as land division and network allocation. Our first result extends prior work by establishing the existence of $\mathrm{EF1}_{\mathrm{outer}}$ allocations for an infinite family of non-traceable graphs (that is, graphs that do not admit a Hamiltonian path), answering a central open question and generalizing Bilò et al.'s result for traceable graphs. We then make progress on a conjecture concerning the $\mathrm{EF1}_{\mathrm{outer}}$ spectrum of trees due to Chen and Zwicker. Finally, we complement our structural results with algorithmic insights, showing that deciding the existence of an $\mathrm{EF1}_{\mathrm{outer}}$ allocation is NP-complete even for binary additive valuations, thereby resolving an open complexity question. Taken together, our results deepen the connection between graph theory and fair division, and offer new tools for studying fairness in structured resource environments.

cs.GT↗
arXiv2026

Constructive Characterization and Recognition Algorithm for Grafts with a Connected Minimum Join

Minimum joins in a graft $(G, T)$, also known as minimum $T$-joins of a graph $G$, are said to be connected if they determine a connected subgraph of $G$. Grafts with a connected minimum join have gained interest ever since Middendorf and Pfeiffer showed that they satisfy Seymour's min-max formula for joins and $T$-cut packings; that is, in such grafts, the size of a minimum join is equal to the size of a maximum packing of $T$-cuts. In this paper, we provide a constructive characterization of grafts with a connected minimum join. We also obtain a polynomial time algorithm that decides whether a given graft has a connected minimum join and, if so, outputs one. Our algorithm has two bottlenecks; one is the time required to compute a minimum join of a graft, and the other is the time required to solve the single-source all-sink shortest path problem in a graph with conservative $\pm 1$-valued edge weights. Thus, our algorithm runs in $O(n(m + n\log n) )$ time. In the nondense case, it improves upon the time bound for this problem due to Sebő and Tannier that was introduced as an application of their results on metrics on graphs.

cs.DM↗
arXiv2026

On the Gap of Finite Posets

Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.

math.CO↗
arXiv2026

Bridging the Gap Between Stable Marriage and Stable Roommates: A Parameterized Algorithm for Optimal Stable Matchings

In the Stable Roommates Problem (SR), a set of $2n$ agents rank one another in a linear order. The goal is to find a matching that is stable: one that has no pair of agents who mutually prefer each other over their assigned partners. We consider the problem of finding an optimal stable matching. Agents associate weights with each of their potential partners, and the goal is to find a stable matching that minimizes the sum of the associated weights. Efficient algorithms exist for finding an optimal stable matching in the Stable Marriage Problem (SM), but the problem is NP-hard for general SR instances. In this paper, we define a notion of structural distance between SR instances and SM instances, which we call the minimum crossing distance. When an SR instance has minimum crossing distance $0$, the instance is structurally equivalent to an SM instance, and this structure can be exploited to find an optimal stable matching efficiently. More generally, we show that when an SR instance has minimum crossing distance $k$, an optimal stable matching can be computed in time $2^{O(k)} n^{O(1)}$. Thus, the optimal stable matching problem is fixed-parameter tractable (FPT) with respect to minimum crossing distance.

cs.DS↗