Max-policy iteration is an approach to computing precise numeric program invariants by successive attempts at resolving maximum operators and reduction to mathematical optimization. Mathematical optimization, though, may be expensive. Here, we show, for max-policy iteration on systems of equations over integers as well as over floating point numbers, that mathematical optimization can be replaced by plain value iteration -- which is still guaranteed to terminate. As an application, a precise bound analysis for integer or floating point variables is obtained, avoiding widening operators altogether. We also consider max-policy iteration over the rational numbers, where the right-hand sides are maxima of minima of affine combinations of unknowns. We propose min-policy iteration as an alternative to linear programming for solving the optimization problems posed by max-policy iteration. We prove that max-min policy iteration is guaranteed to return the least solution for bounded systems. We also show how to extend this algorithm to unbounded systems, and how to construct certificates of soundness as well as of optimality of the computed results.
The MAX BISECTION problem seeks a maximum-size cut that evenly divides the vertices of a given undirected graph. An open problem raised by Austrin, Benabbas, and Georgiou is whether MAX BISECTION can be approximated as well as MAX CUT, i.e., to within ${α_{GW}}\approx 0.8785672\ldots$, which is the approximation ratio achieved by the celebrated Goemans-Williamson algorithm for MAX CUT, which is best possible assuming the Unique Games Conjecture (UGC). They conjectured that the answer is yes. The current paradigm for obtaining approximation algorithms for MAX BISECTION, due to Raghavendra and Tan and Austrin, Benabbas, and Georgiou, follows a two-phase approach. First, a large number of rounds of the Sum-of-Squares (SoS) hierarchy is used to find a solution to the ``Basic SDP'' relaxation of MAX CUT which is $\varepsilon$-uncorrelated, for an arbitrarily small $\varepsilon > 0$. Second, standard SDP rounding techniques (such as ${\cal THRESH}$) are used to round this $\varepsilon$-uncorrelated solution, producing with high probability a cut that is almost balanced, i.e., a cut that has at most $\frac12+\varepsilon$ fraction of the vertices on each side. This cut is then converted
We unconditionally prove that it is NP-hard to compute a constant multiplicative approximation to the QUANTUM MAX-CUT problem on an unweighted graph of constant bounded degree. The proof works in two stages: first we demonstrate a generic reduction to computing the optimal value of a quantum problem, from the optimal value over product states. Then we prove an approximation preserving reduction from MAX-CUT to PRODUCT-QMC the product state version of QUANTUM MAX-CUT. More precisely, in the second part, we construct a PTAS reduction from MAX-CUT$_k$ (the rank-k constrained version of MAX-CUT) to MAX-CUT$_{k+1}$, where MAX-CUT and PRODUCT-QMC coincide with MAX-CUT$_1$ and MAX-CUT$_3$ respectively. We thus prove that Max-Cut$_k$ is APX-complete for all constant $k$.
In this paper, we introduce and characterize max-doubly stochastic matrices within the framework of max algebra, where the operations are defined as $x \oplus y = \max(x, y)$ and $x \otimes y = xy$. We explore the fundamental properties of max-doubly stochastic matrices and their role in vector majorization. Specifically, we establish that for vectors $x$ and $y$ in max algebra, $x$ is majorized by $y$ if there exists a max-doubly stochastic matrix $D$ such that $x = D \otimes y$. This provides a new approach to majorization theory within tropical mathematics and enhances the understanding of vector relations in max algebra.
We show that the distribution of the spectral maximum of monotonically independent self-adjoint operators coincides with the classical max-convolution of their distributions. In free probability, it was proven that for any probability measures $σ,μ$ on $\mathbb{R}$ there is a unique probability measure $\mathbb{A}_σ(μ)$ satisfying $σ\boxplus μ= σ\triangleright \mathbb{A}_σ(μ)$, where $\boxplus$ and $\triangleright$ are free and monotone additive convolutions, respectively. We recall that the reciprocal Cauchy transform of $\mathbb{A}_σ(μ)$ is the subordination function for free additive convolution. Motivated by this analogy, we introduce subordination functions for free max-convolution and prove their existence and structural properties.
Let $\mathcal{D}$ be a set family that is the solution domain of some combinatorial problem. The \emph{max-min diversification problem on $\mathcal{D}$} is the problem to select $k$ sets from $\mathcal{D}$ such that the Hamming distance between any two selected sets is at least $d$. FPT algorithms parameterized by $k+\ell $, where $\ell=\max_{D\in \mathcal{D}}|D|$, and $k+d$ have been actively studied recently for several specific domains. This paper provides unified algorithmic frameworks to solve this problem. Specifically, for each parameterization $k+\ell $ and $k+d$, we provide an FPT oracle algorithm for the max-min diversification problem using oracles related to $\mathcal{D}$. We then demonstrate that our frameworks provide the first FPT algorithms on several new domains $\mathcal{D}$, including the domain of $t$-linear matroid intersection, almost $2$-SAT, minimum edge $s,t$-flows, vertex sets of $s,t$-mincut, vertex sets of edge bipartization, and Steiner trees. We also demonstrate that our frameworks generalize most of the existing domain-specific tractability results. Our main technical breakthrough is introducing the notion of \emph{max-distance sparsifier} of $\mathca
Min-max optimization problems, also known as saddle point problems, have attracted significant attention due to their applications in various fields, such as fair beamforming, generative adversarial networks (GANs), and adversarial learning. However, understanding the properties of these min-max problems has remained a substantial challenge. This study introduces a statistical mechanical formalism for analyzing the equilibrium values of min-max problems in the high-dimensional limit, while appropriately addressing the order of operations for min and max. As a first step, we apply this formalism to bilinear min-max games and simple GANs, deriving the relationship between the amount of training data and generalization error and indicating the optimal ratio of fake to real data for effective learning. This formalism provides a groundwork for a deeper theoretical analysis of the equilibrium properties in various machine learning methods based on min-max problems and encourages the development of new algorithms and architectures.
We present a novel multi-agent simulator named Multi-Agent eXperimenter (MAX) that is designed to simulate blockchain experiments involving large numbers of agents of different types acting in one or several environments. The architecture of MAX is highly modular, enabling easy addition of new models.
The Quantum Focusing Conjecture (QFC) lies at the foundation of holography and semiclassical gravity. The QFC implies the Bousso bound and the Quantum Null Energy Condition (QNEC). The QFC also ensures the consistency of the quantum extremal surface prescription and bulk reconstruction in AdS/CFT. However, the central object in the QFC -- the expansion of lightrays -- is not defined at points where geodesics enter or leave a null congruence. Moreover, the expansion admits three inequivalent quantum extensions in terms of the conditional max, min, and von Neumann entropies. Here we formulate a discrete notion of nonexpansion that can be evaluated even at non-smooth points. Moreover, we show that a single conjecture, the discrete max-QFC, suffices for deriving the QNEC, the Bousso bound, and key properties of both max and min entanglement wedges. Continuous numerical values need not be assigned, nor are the von Neumann or min-versions of the quantum expansion needed. Both our new notion of nonexpansion, and also the properties of conditional max entropies, are inherently asymmetric and outward directed from the input wedge. Thus the framework we develop here reduces and clarifies the
Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-$k$AND problem. This generalizes the definition by Feige and Jozeph (Algorithmica '15) of oblivious algorithms for Max-DICUT, a special case of Max-$2$AND. Oblivious algorithms round each variable with probability depending only on a quantity called the variable's bias. For each oblivious algorithm, we design a so-called "factor-revealing linear program" (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all $k$, oblivious algorithms for Max-$k$AND provably outperform a special subclass of algorithms we call "superoblivious" algorithms. Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every $k$, $O(\log n)$-space sketching algorithms for Max-$k$AND known to be optima
We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems, by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS) based SDP bounds and an efficient parser within a branch & bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems, and show that a member of this family provides the rank two guarantee. We also provide a parametric family of semidefinite relaxations for the MAX-SAT, and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an elegant way. Moreover, we relate our SOS-SDP relaxations for the partial MAX-SAT to the known SAT relaxations.
We introduce the Membership Degree Min-Max (MD-Min-Max) localisation algorithm as a precise and simple lateration algorithm for indoor localisation. MD-Min-Max is based on the well-known Min-Max algorithm that computes a bounding box to estimate the position. MD-Min-Max uses a Membership Function (MF) based on an estimated error distribution of the distance measurements to improve the precision of Min-Max. The algorithm has similar complexity to Min-Max and can be used for indoor localisation even on small devices, e.g., in Wireless Sensor Networks (WSNs). To evaluate the performance of the algorithm, we compare it with other improvements of the Min-Max algorithm and maximum likelihood estimators, both in simulations and in a large real-world deployment of a WSN. Results show that MD-Min-Max achieves the best performance in terms of average positioning accuracy while keeping computational cost low compared to the other algorithms.
Assuming the Unique Games Conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the MAX CUT problem is $α_{\text{CUT}}\simeq 0.87856$, obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The currently best approximation algorithm for MAX DI-CUT, i.e., the MAX CUT problem in directed graphs, achieves a ratio of about $0.87401$, leaving open the question whether MAX DI-CUT can be approximated as well as MAX CUT. We obtain a slightly improved algorithm for MAX DI-CUT and a new UGC-hardness result for it, showing that $0.87446\le α_{\text{DI-CUT}}\le 0.87461$, where $α_{\text{DI-CUT}}$ is the best approximation ratio that can be obtained in polynomial time for MAX DI-CUT under UGC. The new upper bound separates MAX DI-CUT from MAX CUT, resolving a question raised by Feige and Goemans. A natural generalization of MAX DI-CUT is the MAX 2-AND problem in which each constraint is of the form $z_1\land z_2$, where $z_1$ and $z_2$ are literals, i.e., variables or their negations (In MAX DI-CUT each constraint is of the form $\bar{x}_1\land x_2$, where $x_1$ and $x_2$ are variables.) Austrin separated MAX 2-AND from MAX C
Given a real inner product space $V$ and a group $G$ of linear isometries, we construct a family of $G$-invariant real-valued functions on $V$ that we call max filters. In the case where $V=\mathbb{R}^d$ and $G$ is finite, a suitable max filter bank separates orbits, and is even bilipschitz in the quotient metric. In the case where $V=L^2(\mathbb{R}^d)$ and $G$ is the group of translation operators, a max filter exhibits stability to diffeomorphic distortion like that of the scattering transform introduced by Mallat. We establish that max filters are well suited for various classification tasks, both in theory and in practice.
A new nature-inspired membrane uses perfectly uniform one-nanometer pores to filter molecules with remarkable precision。 The technology could transform industries such as pharmaceuticals and textiles by reducing energy consumption, improving water reuse, and delivering separation performance far beyond current filters
Tests of age-verification technology show the risks of life-altering errors
Researchers discovered that hydrogen radicals generated by intense UV light can break down stubborn PFAS “forever chemicals” without added chemicals。 The breakthrough reveals a key mechanism that could lead to greener and more effective technologies for permanently destroying these pollutants
A new theory suggests the universe is constantly recording its own history in the fabric of spacetime。 If correct, this cosmic memory could help solve some of the biggest puzzles in physics, from black holes to dark matter and the universe’s ultimate fate