We study uniformity testing for the monomer-dimer model on a known graph $G=(V,E)$. Given samples from an unknown distribution $D_w(M)\propto \prod_{e\in M} w_e$ over matchings $M$ of $G$, we ask whether $D_w$ is the uniform distribution over all matchings of $G$ or is $\varepsilon$-far from it in total variation distance. Our main result is that, for every maximum-degree bound $Δ$, uniformity can be tested with $\widetilde O_Δ(\sqrt m/\varepsilon^2)$ samples, where $m=|E|$. We complement this with an information-theoretic lower bound for general graphs: for every pair of parameters $n\le m\le n^2$, there are graphs with $Θ(n)$ vertices and $Θ(m)$ edges on which every tester requires $Ω(\sqrt m/\varepsilon^2)$ samples. Thus, for bounded-degree graph families with $m=Θ(n)$, testing uniformity over the space of matchings has $\widetilde Θ(\sqrt n/\varepsilon^2)$ sample complexity, in contrast with the $Θ(n/\varepsilon^2)$ samples needed to estimate $D_w$ in total variation distance.
In 1984, Frank Harary introduced the first convexity games in graphs, all of them based on the geodesic convexity, which is the graph convexity related to shortest paths. In 2024, Araújo et al. obtained the first PSPACE-hardness proofs on some of these geodesic games and generalized them to any graph convexity. In this paper, we investigate convexity games on several known induced path convexities: the monophonic $\mathrm{m}$-convexity and the $\ell_k$-convexities, based on induced paths and on induced paths of size at most $k$. We prove that the hull games $\mathrm{CHG}_{\mathrm{m}}$, $\mathrm{CHG}_{\ell_k}$ and their misère variants are PSPACE-complete for every $k\ge2$ even in graphs with diameter at most 3. We also use the Sprague-Grundy Theory to obtain a polynomial time algorithm to decide the winner of the games $\mathrm{CHG}_{\mathrm{m}}$ and $\mathrm{CHG}_{\ell_k}$ for any $k\ge2$ in disjoint unions of paths and cycles. For $k\ge3$ odd, we prove that Alice (1st player) wins $\mathrm{CHG}_{\ell_k}$ in the path $P_n$ if and only if $n$ is odd and she wins in the cycle $C_n$ if and only if $n=3$ or $n=α\cdot (k+1)-1$ with $α\ge2$. For $k\ge2$ even, the only periodic nimber sequences of $\mathrm{CHG}_{\ell_k}$ obtained through extensive computational testing occurred for $k=2^h-4$ with $h\ge3$, e.g, $k\in\{4,12,28,60,\ldots\}$. In this case ($k=2^h-4$ with $h\ge3$), we prove that the nimber sequences of $\mathrm{CHG}_{\ell_k}$ in $P_n$ and in $C_n$ are periodic and Alice loses (resp. wins) in $P_n$ (resp. $C_n$) with $n>k$ only when $n=3k+4$ (resp. $n\in\{2k+1,5k+3\}$). Finally, we show that, for $k=2$, the game $\mathrm{CHG}_{\ell_2}$ in paths $P_n$ is closely related to the classical game \emph{Couples-are-Forever} of J. H. Conway: it is still an open problem if the nimber sequence is periodic or not and Alice loses only for 12 values of $n$ up to $150$ million.
We study the sample complexity of testing $k$-colorability in the dense graph model. A graph is $\varepsilon$-far from being $k$-colorable if at least $\varepsilon n^2$ edge deletions are required to make it $k$-colorable. We prove that a uniformly random induced subgraph on $O(k\log k/\varepsilon)$ vertices is non-$k$-colorable with probability at least $2/3$ whenever the original graph is $\varepsilon$-far from $k$-colorable. The proof combines a palette-based coloring process with a container-style encoding-and-reconstruction argument. This yields $O(k\log k/\varepsilon)$, which is no worse than the previous state of the art $O(k\log (1/\varepsilon)/\varepsilon)$ throughout the nonvacuous parameter range, and is an improvement when $\log k=o(\log(1/\varepsilon))$. For each fixed $k$, combining this upper bound with the known $Ω(\log k/\varepsilon)$ lower bound gives $Θ(1/\varepsilon)$ dependence on $\varepsilon$.
Nicholas J. Cooper, François G. Meyer, Michael L. Roberts, Carlos Zapata-Carratalá, Lijun Chen, Danna Gurari
We introduce a unified theoretical framework for the rigorous analysis and systematic construction of deep neural networks (DNNs). This framework addresses a gap in existing theory by explicitly modeling the structure of tensor operations -- lower level information that is often abstracted. Our framework enables two novel objectives: (1) analysis of the evolution of architectural complexity over deep learning history, and (2) automatic construction of novel architectures based on new types of tensor operations. Our study of DNNs introduced over the past 40 years reveals a connection between groundbreaking architectures and increases in different types of architectural complexity. Moreover, we identify several large classes of higher complexity architectures that have not yet been explored. We then collect a dataset of 3,000+ higher complexity architectures, which we publicly release at: https://github.com/combinatoriallabs/ArchitecturalComplexity.
We study Multi-Level Aggregation with Maximum Delay (MLA-MD) on finite weighted rooted trees. Each service pays for a rooted subtree and for the maximum waiting time among the requests it clears. For the offline problem, we give an exact polynomial-time algorithm that computes both the optimum and an optimal service schedule. For the online problem, we give a deterministic $2$-competitive algorithm and, against an oblivious adversary, a randomized $e/(e-1)$-competitive algorithm. Both competitive ratios are optimal, even on rooted paths. We also extend the offline result and both online guarantees to service systems with submodular costs.
An ordered graph is a graph with a total order over its vertices. A linear layout of an ordered graph is a partition of the edges into sets of either non-crossing edges, called stacks, or non-nesting edges, called queues. The stack (queue) number of an ordered graph is the minimum number of required stacks (queues). Mixed linear layouts combine these layouts by allowing each set of edges to form either a stack or a queue. The minimum number of stacks plus queues is called the mixed page number. It is well known that ordered graphs with small stack number are characterized, up to a function, by the absence of large twists (that is, pairwise crossing edges). Similarly, ordered graphs with small queue number are characterized by the absence of large rainbows (that is, pairwise nesting edges). However, no such characterization via forbidden patterns is known for mixed linear layouts. We address this gap by introducing patterns similar to twists and rainbows, which we call thick patterns; such patterns allow a characterization, again up to a function, of mixed linear layouts of bounded-degree graphs. That is, we show that a family of ordered graphs with bounded maximum degree has bounded mixed page number if and only if the size of the largest thick pattern is bounded. In addition, we investigate an exact characterization of ordered graphs whose mixed page number equals a fixed integer $ k $ via a finite set of forbidden patterns. We show that for every $ k \ge 2 $, there is no such characterization, which supports the nature of our first result.
Let $(T, \circ)$ be a loop of order $k \geq 3$ and $D_S^*$ the loopless central digraph obtained from the associated central digraph $D_S$ by removing the $k$ self-loops at the diagonal vertices. We construct an explicit arc-labeling $f$ of $D_S^*$ ordering all arcs lexicographically by $(\text{tail}, \text{head})$ and assigning labels $1, \ldots, m$ in that order and prove that $f$ is an antimagic labeling. Consequently, $D_S$ is antimagic for every loop of order $k \geq 3$, a property that depends only on $k$ and is invariant across all non-isomorphic loops of the same order.
Hofstadter's G function is recursively defined via G(0) = 0 and then G(n) = n-G(G(n -1)). Following Hofstadter, we vary the number k of nested recursive calls in this equation and obtain a family of functions (F\_k). In a previous article, we proved that this family is ordered pointwise: for all k and n, we have $F_k(n) $le$ F_{k+1}(n)$. Here we propose a much shorter proof of this theorem, without infinite morphic words. The key ingredient for this direct proof is a new formula expressing iterates of the preimages of $F_k$ . Thanks to this formula, we also prove several conjectures mentioned in our previous article. In particular we obtain that $F_k(n) < F_{k+1}(n)$ when n > (k + 1)(k + 6)/2. We provide a certified formalization of all these results in the Rocq proof assistant.
Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{η_1,\ldots,η_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $η_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{η_i,η_j\in S:η_i\neqη_j}|\mathcal N(η_i-η_j)|\] where $\mathcal N$ denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.