Tverberg's theorem states that for any $k \ge 2$ and any set $P \subset \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points in $d$ dimensions, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class $\text{CLS} = \text{PPAD} \cap \text{PLS}$, but no hardness results are known. In the colorful Tverberg theorem, the points in $P$ have colors, and under certain conditions, $P$ can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Barany, and Mustafa gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of $P$. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any $n$-point set $P \subset \mathbb{R}^d$ and any $k \in \{2, \dots, n\}$ in $O(nd \lceil{\log k}\rceil)$ time a $k$-partition of $P$ such that there is a ball of radius $O\left((k/\sqrt{n})\mathrm{diam(P)}\right)$ that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria's method to reduce the Tverberg problem to the Colorful Caratheodory problem (in the simplified tensor product interpretation of Barany and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem.
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A $c$-labeling $\phi: V(G) \rightarrow \{1, 2, \hdots, c \}$ of graph $G$ is distinguishing if, for every non-trivial automorphism $\pi$ of $G$, there is some vertex $v$ so that $\phi(v) \neq \phi(\pi(v))$. The distinguishing number of $G$, $D(G)$, is the smallest $c$ such that $G$ has a distinguishing $c$-labeling. We consider a compact version of Tyshkevich's graph decomposition theorem where trivial components are maximally combined to form a complete graph or a graph of isolated vertices. Suppose the compact canonical decomposition of $G$ is $G_{k} \circ G_{k-1} \circ \cdots \circ G_1 \circ G_0$. We prove that $\phi$ is a distinguishing labeling of $G$ if and only if $\phi$ is a distinguishing labeling of $G_i$ when restricted to $V(G_i)$ for $i = 0, \hdots, k$. Thus, $D(G) = \max \{D(G_i), i = 0, \hdots, k \}$. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.
Let $\mathbb{F}_q$ be a finite field of characteristic $p$. In this paper we prove that the $c$-Boomerang Uniformity, $c \neq 0$, for all permutation monomials $x^d$, where $d > 1$ and $p \nmid d$, is bounded by $d^2$. Further, we utilize this bound to estimate the $c$-boomerang uniformity of a large class of Generalized Triangular Dynamical Systems, a polynomial-based approach to describe cryptographic permutations, including the well-known Substitution-Permutation Network.
For $1\le p \le \infty$, the Fr\'echet $p$-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean ($p=2$) and median ($p=1$). In this work we prove a collection of limit theorems for Fr\'echet means and related objects, which, in general, constitute a sequence of random closed sets. On the one hand, we show that many limit theorems (a strong law of large numbers, an ergodic theorem, and a large deviations principle) can be simply descended from analogous theorems on the space of probability measures via purely topological considerations. On the other hand, we provide the first sufficient conditions for the strong law of large numbers to hold in a $T_2$ topology (in particular, the Fell topology), and we show that this condition is necessary in some special cases. We also discuss statistical and computational implications of the results herein.
Given a square pencil $A+ \lambda B$, where $A$ and $B$ are complex matrices, we consider the problem of finding the singular pencil nearest to it in the Frobenius distance. This problem is known to be very difficult, and the few algorithms available in the literature can only deal efficiently with pencils of very small size. We show that the problem is equivalent to minimizing a certain objective function over the Riemannian manifold $SU(n) \times SU(n)$, where $SU(n)$ denotes the special unitary group. With minor modifications, the same approach extends to the case of finding a nearest singular pencil with a specified minimal index. This novel perspective is based on the generalized Schur form of pencils, and yields a competitive numerical method, by pairing it with an algorithm capable of doing optimization on a Riemannian manifold. We provide numerical experiments that show that the resulting method allows us to deal with pencils of much larger size than alternative techniques, yielding candidate minimizers of comparable or better quality. In the course of our analysis, we also obtain a number of new theoretical results related to the generalized Schur form of a (regular or singular) square pencil and to the minimal index of a singular square pencil whose nullity is $1$.
The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.
We give an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, in time $2^{O(k)} n$ either outputs a tree decomposition of $G$ of width at most $2k + 1$ or determines that the treewidth of $G$ is larger than $k$. This is the first 2-approximation algorithm for treewidth that is faster than the known exact algorithms, and in particular improves upon the previous best approximation ratio of 5 in time $2^{O(k)} n$ given by Bodlaender et al. [SIAM J. Comput., 45 (2016)]. Our algorithm works by applying incremental improvement operations to a tree decomposition, using an approach inspired by a proof of Bellenbaum and Diestel [Comb. Probab. Comput., 11 (2002)].
The $k$-dimensional Weisfeiler-Leman ($k$-WL) algorithm is a simple combinatorial algorithm that was originally designed as a graph isomorphism heuristic. It naturally finds applications in Babai's quasipolynomial time isomorphism algorithm, practical isomorphism solvers, and algebraic graph theory. However, it also has surprising connections to other areas such as logic, proof complexity, combinatorial optimization, and machine learning. The algorithm iteratively computes a coloring of the $k$-tuples of vertices of a graph. Since F\"urer's linear lower bound [ICALP 2001], it has been an open question whether there is a super-linear lower bound for the iteration number for $k$-WL on graphs. We answer this question affirmatively, establishing an $\Omega(n^{k/2})$-lower bound for all $k$.
The Ramsey number is the minimum number of nodes, $n = R(s, t)$, such that all undirected simple graphs of order $n$, contain a clique of order $s$, or an independent set of order $t$. This paper explores the application of a best first search algorithm and reinforcement learning (RL) techniques to find counterexamples to specific Ramsey numbers. We incrementally improve over prior search methods such as random search by introducing a graph vectorization and deep neural network (DNN)-based heuristic, which gauge the likelihood of a graph being a counterexample. The paper also proposes algorithmic optimizations to confine a polynomial search runtime. This paper does not aim to present new counterexamples but rather introduces and evaluates a framework supporting Ramsey counterexample exploration using other heuristics. Code and methods are made available through a PyPI package and GitHub repository.
Multiple TSP ($\mathrm{mTSP}$) is a important variant of $\mathrm{TSP}$ where a set of $k$ salesperson together visit a set of $n$ cities. The $\mathrm{mTSP}$ problem has applications to many real life applications such as vehicle routing. Rothkopf introduced another variant of $\mathrm{TSP}$ called many-visits TSP ($\mathrm{MV\mbox{-}TSP}$) where a request $r(v)\in \mathbb{Z}_+$ is given for each city $v$ and a single salesperson needs to visit each city $r(v)$ times and return back to his starting point. A combination of $\mathrm{mTSP}$ and $\mathrm{MV\mbox{-}TSP}$ called many-visits multiple TSP $(\mathrm{MV\mbox{-}mTSP})$ was studied by B\'erczi, Mnich, and Vincze where the authors give approximation algorithms for various variants of $\mathrm{MV\mbox{-}mTSP}$. In this work, we show a simple linear programming (LP) based reduction that converts a $\mathrm{mTSP}$ LP-based algorithm to a LP-based algorithm for $\mathrm{MV\mbox{-}mTSP}$ with the same approximation factor. We apply this reduction to improve or match the current best approximation factors of several variants of the $\mathrm{MV\mbox{-}mTSP}$. Our reduction shows that the addition of visit requests $r(v)$ to $\mathrm{mTSP}$ does $\textit{not}$ make the problem harder to approximate even when $r(v)$ is exponential in number of vertices. To apply our reduction, we either use existing LP-based algorithms for $\mathrm{mTSP}$ variants or show that several existing combinatorial algorithms for $\mathrm{mTSP}$ variants can be interpreted as LP-based algorithms. This allows us to apply our reduction to these combinatorial algorithms as well achieving the improved guarantees.
For a state $\rho_{A_1^n B}$, we call a sequence of states $(\sigma_{A_1^k B}^{(k)})_{k=1}^n$ an approximation chain if for every $1 \leq k \leq n$, $\rho_{A_1^k B} \approx_\epsilon \sigma_{A_1^k B}^{(k)}$. In general, it is not possible to lower bound the smooth min-entropy of such a $\rho_{A_1^n B}$, in terms of the entropies of $\sigma_{A_1^k B}^{(k)}$ without incurring very large penalty factors. In this paper, we study such approximation chains under additional assumptions. We begin by proving a simple entropic triangle inequality, which allows us to bound the smooth min-entropy of a state in terms of the R\'enyi entropy of an arbitrary auxiliary state while taking into account the smooth max-relative entropy between the two. Using this triangle inequality, we create lower bounds for the smooth min-entropy of a state in terms of the entropies of its approximation chain in various scenarios. In particular, utilising this approach, we prove an approximate version of entropy accumulation and also provide a solution to the source correlation problem in quantum key distribution.
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