Generalized Hockey Stick Theorem
We give a combinatorial proof via a sign-reversing involution for a new identity that generalizes both the Hockey Stick Identity and the Big Hockey Stick and Pucks Identity....
The Jacobs--Keane theorem from the $\cS$-adic viewpoint
In the light of recent developments of the ${\mathcal S}$-adic study of subshifts, we revisit, within this framework, a well-known result on Toeplitz subshifts due to Jacobs--Keane giving a sufficient combinatorial condition to ensure discrete spectr...
Compositional Zero-Shot Learning: A Survey
Compositional Zero-Shot Learning (CZSL) is a critical task in computer vision that enables models to recognize unseen combinations of known attributes and objects during inference, addressing the combinatorial challenge of requiring training data for...
A Hardware Accelerator for the Goemans-Williamson Algorithm
The combinatorial problem Max-Cut has become a benchmark in the evaluation of local search heuristics for both quantum and classical optimisers. In contrast to local search, which only provides average-case performance guarantees, the convex semidefi...
Combinatorial Selection and Least Absolute Shrinkage via the CLASH Algorithm
The least absolute shrinkage and selection operator (LASSO) for linear regression exploits the geometric interplay of the $\ell_2$-data error objective and the $\ell_1$-norm constraint to arbitrarily select sparse models. Guiding this uninformed sele...
Puzzle Ideals for Grassmannians
Puzzles are a versatile combinatorial tool to interpret the Littlewood-Richardson coefficients for Grassmannians. In this paper, we propose the concept of puzzle ideals whose varieties one-one correspond to the tilings of puzzles and present an algeb...
ARM-Explainer -- Explaining and improving graph neural network predictions for the maximum…
Numerous graph neural network (GNN)-based algorithms have been proposed to solve graph-based combinatorial optimization problems (COPs), but methods to explain their predictions remain largely undeveloped. We introduce ARM-Explainer, a post-hoc, mode...
Rudin-Keisler capturing and Mutual Stationairy at successors of Singulars
We introduce a combinatorial notion of measures called Rudin-Keisler capturing and use it to give a new construction of elementary substructures around singular cardinals. The new construction is used to establish mutual stationary results at the fir...
Combinatory completeness in partial groupoids
I characterize the combinatorially complete pargoids (partial applicative systems) by expandability with two constants that satisfy the well-known identities. An example shows that this class contains more than just the reducts of partial combinatory...
Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems, Applications…
For a reduced hyperplane arrangement we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic d...
Nonnegative minors of minor matrices
Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation...
On Formality and Combinatorial Formality for hyperplane arrangements
A hyperplane arrangement is called formal provided all linear dependencies among the defining forms of the hyperplanes are generated by ones corresponding to intersections of codimension two. The significance of this notion stems from the fact that c...
Product-Congruence Games: A Unified Impartial-Game Framework for RSA ($φ$-MuM) and AES (po…
RSA exponent reduction and AES S-box inversion share a hidden commonality: both are governed by the same impartial combinatorial principle, which we call a Product-Congruence Game (PCG). A Product-Congruence Game tracks play via the modular or finite...
Extensions of Karger's Algorithm: Why They Fail in Theory and How They Are Useful in Pract…
The minimum graph cut and minimum $s$-$t$-cut problems are important primitives in the modeling of combinatorial problems in computer science, including in computer vision and machine learning. Some of the most efficient algorithms for finding global...
Additive sink subtraction
Subtraction games are a classical topic in Combinatorial Game Theory. A result of Golomb~(1966) shows that every subtraction game with a finite move set has an eventually periodic nim-sequence, but the known proof yields only an exponential upper bou...
Some Ratio Monotonic Properties of a New Kind of Numbers introduced by Z.-W. Sun
Recently, Z. W. Sun introduced a new kind of numbers $S_n$ and also posed a conjecture on ratio monotonicity of combinatorial sequences related to $S_n$. In this paper, by investigating some arithmetic properties of $S_n$, we give an affirmative answ...
On Ratio Monotonicity of a New Kind of Numbers Conjectured by Z.-W. Sun
Recently, Z. W. Sun put forward a series of conjectures on monotonicity of combinatorial sequences in the form of $\{z_n/z_{n-1}\}_{n=N}^\infty$ and $\{\sqrt[n+1]{z_{n+1}}/\sqrt[n]{z_n}\}_{n=N}^\infty$ for some positive integer $N$, where $\{z_n\}_{n...
Distance graphs having large chromatic numbers and not containing cliques or cycles of giv…
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and...
microsoft/pict
Pairwise Independent Combinatorial Tool (⭐ 1407)
Walrasian's Characterization and a Universal Ascending Auction
We introduce a novel characterization of all Walrasian price vectors in terms of forbidden over- and under demanded sets for monotone gross substitute combinatorial auctions. For ascending and descending auctions we suggest a universal framework fo...