We study the zero-visibility cops and robbers game, where the robber is invisible to the cops until they are caught. This differs from the classic game where full information about the robber’s location is known at any time. A previously known solution for capturing a robber in the zero-visibility case is based on the path decomposition. We provide an alternative solution based on a separation hierarchy, improving capture time and space complexity without asymptotically increasing the zero-visibility cop number in most cases. In addition, the alternative approach leads to a better bound on the approximate zero-visibility cop number for various classes of graphs, where approximate refers to the restriction to polynomial time computable strategies.

In this article, a new formula for computing Cayley's first hyperdeterminant in terms of the Levi-Civita symbol is given. It is then shown that this formula can be used to compute the hyperdeterminant of symmetric tensors in polynomial time with respect to their order (assuming fixed side length). Applications to quantifying the entanglement of states of bosonic quantum systems are then discussed. Additionally, in order to obtain the fast calculation of the hyperdeterminant on symmetric tensors, generalized elimination and duplication matrices are defined, and their explicit formulas are derived.

Abstract A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B . A conformal cross over some cycle C is a pair of disjoint paths $$P_1$$ P 1 , $$P_2$$ P 2 which are internally disjoint from C , the endpoints of each path separate the endpoints of the other path on C , and both $$C\cup P_1\cup P_2$$ C ∪ P 1 ∪ P 2 and $$B-(V(C)\cup V(P_1)\cup V(P_2))$$ B - ( V ( C ) ∪ V ( P 1 ) ∪ V ( P 2 ) ) have a perfect matching. We show that if C is a 4-cycle in a brace B , then C has a a conformal cross if and only if B contains $$K_{3,3}$$ K 3 , 3 as a matching minor. This result implies a polynomial time algorithm which solves the 2-linkage problem for alternating paths in bipartite graphs with perfect matchings.

Explainable artificial intelligence (XAI) is increasingly required for anomaly detection in high-dimensional sensor systems operating in safety-critical and resource-constrained environments. While existing post-hoc explanation methods provide useful insights, they often suffer from high computational cost, unstable attributions, and limited applicability in unlabeled or unsupervised settings. This paper proposes KFASL, a variance-stable and computationally efficient XAI framework that approximates Shapley-based feature attributions using a variance-optimized weighting strategy. The framework integrates local and global explanations with causality-aware regularization to improve attribution stability and interpretability under limited labeling conditions. The proposed approach reduces the computational complexity of Shapley approximation from exponential to polynomial time, enabling scalable deployment for high-dimensional telemetry data. KFASL is evaluated using a combination of real and synthetic datasets, with spacecraft telemetry used as a representative safety-critical case study. Experimental results demonstrate improved attribution stability, explanation fidelity, and runtime efficiency compared to existing XAI techniques, including SHAP, Kernel SHAP, LIME, and Anchors. These results indicate that KFASL provides a general and practical solution for explainable anomaly detection in complex sensor-driven systems.

Given polynomials f 0 , f 1 , …, f k the Ideal Membership Problem, IMP for short, asks if f 0 belongs to the ideal generated by f 1 , …, f k . In the search version of this problem, the task is to find a proof of this fact. The IMP is a well-known fundamental problem with numerous applications. For instance, it underlies many proof systems based on polynomials such as Nullstellensatz, Polynomial Calculus, and Sum-of-Squares. Although the IMP is in general intractable, in many important cases it can be efficiently solved. Mastrolilli [SODA’19] initiated a systematic study of IMPs for ideals arising from Constraint Satisfaction Problems (CSPs), parameterized by constraint languages, denoted IMP(Γ). The ultimate goal of this line of research is to classify all such IMPs accordingly to their complexity. Mastrolilli achieved this goal for IMPs arising from CSP(Γ) where Γ is a Boolean constraint language, while Bulatov and Rafiey [STOC’22] advanced these results to several cases of CSPs over finite domains. In this article, we consider IMPs arising from CSPs over “affine” constraint languages, in which constraints are subgroups (or their cosets) of direct products of Abelian groups. This kind of CSPs include systems of linear equations and are considered one of the most important types of tractable CSPs. Some special cases of the problem have been considered before by Bharathi and Mastrolilli [MFCS’21] for linear equations modulo 2, and by Bulatov and Rafiey [STOC’22] for systems of linear equations over GF ( p ), p prime. Here, we prove that if Γ is an affine constraint language then IMP(Γ) is solvable in polynomial time assuming the input polynomial has bounded degree.

The approximate calculation of iterated nested expectations is a recurring challenging problem in applications. Nested expectations appear, for example, in the numerical approximation of solutions of backward stochastic differential equations (BSDEs), in the numerical approximation of solutions of semilinear parabolic partial differential equations (PDEs), in statistical physics, in optimal stopping problems such as the approximate pricing of American or Bermudan options, in risk measure estimation in mathematical finance, or in decisionmaking under uncertainty. Nested expectations which arise in the above-mentioned applications often consist of a large number of nestings. However, the computational effort of standard nested Monte Carlo approximations for iterated nested expectations grows exponentially in the number of nestings and it has remained an open question whether it is possible to approximately calculate multiply iterated high-dimensional nested expectations in polynomial time. In this article we tackle this problem by proposing and studying a new class of full-history recursive multilevel Picard (MLP) approximation schemes for iterated nested expectations. Specifically, we prove under suitable assumptions that these MLP approximation schemes can approximately calculate multiply iterated nested expectations with a computational effort growing at most polynomially in the number of nestings K ∈ ℕ = {1,2,3,…}, in the problem dimension d ∈ ℕ, and in the reciprocal 1/ε of the desired approximation accuracy ε ∈ (0, ∞). In particular, the proposed MLP approximation schemes can approximately calculate nested expectations arising in the numerical approximation of solutions of BSDEs and semilinear parabolic PDEs with a computational effort growing at most polynomially in K, d, and 1/ε.

Optimal actuator and sensor attack strategies against controllability and observability.

We generalise the popular cops and robbers game to multi-layer graphs, where each cop and the robber are restricted to a single layer (or set of edges). We show that initial intuition about the best way to allocate cops to layers is not always correct, and prove that the multi-layer cop number is neither bounded from above nor below by any increasing function of the cop numbers of the individual layers. We determine that it is NP-hard to decide if $k$ cops are sufficient to catch the robber, even if every cop layer is a tree and a set of isolated vertices. However, we give a polynomial time algorithm to determine if $k$ cops can win when the robber layer is a tree. Additionally, we investigate a question of worst-case divisions of a simple graph into layers: given a simple graph $G$, what is the maximum number of cops required to catch a robber over all multi-layer graphs where each edge of $G$ is in at least one layer and all layers are connected? For cliques, suitably dense random graphs, and graphs of bounded treewidth, we determine this parameter up to multiplicative constants. Lastly we consider a multi-layer variant of Meyniel's conjecture, and show the existence of an infinite family of graphs whose multi-layer cop number is bounded from below by a constant times $n / \log n$, where $n$ is the number of vertices in the graph.

Because of realistic necessity for timely satisfying destination-stage demand, capacitated two-stage time minimization transportation problem with destination-stage demand specified (C2SD) is crucial and important optimization problem with no research report in literature due to its super intractability. In this paper, by creating C2SD’s mathematical model named C2SDM (i.e. the main model systematically formulating C2SD problem) along with auxiliary models and constructing network, C2SD is reduced to a series of search for feasible flow in the constructed network, and consequently four iterative algorithms, with two derived respectively from other two by applying binary search, are developed to solve C2SD. It is proved that the optimum solution is found for C2SD by each of the four algorithms in a polynomial time. Owing to sufficient exploitation of the network structure of C2SD, the proposed algorithms exhibit high computational efficiency and avoid memory overflow. They are also easy to implement computationally and can be readily extended to capacitated multistage time minimization transportation problems with destination-stage demand specified. Distinct size examples are presented to showcase the application value and practical performance difference of the four algorithms compared to generic solver LINGO applied to C2SDM model, and the causes for incurring the performance difference are analyzed. Computation experiments on different size instances produced at random are conducted to further validate the practical performance of the four iterative algorithms compared with LINGO. It is validated that all the four algorithms overwhelmingly excel LINGO in solving quality and rival LINGO in computation time when applied to C2SD instances, successfully overcoming the defect of LINGO with poor optimization effect. It is indicated that the developed four algorithms are robust and efficient exact solution approach to C2SD, and they are able to serve as a powerful tool to tackle other relevant complex optimization problems.

In multiprocessor systems, reliable multi-path transmission is crucial for ensuring network quality of service. A set of k spanning trees rooted at the same node r in a network is said to be k node-independent spanning trees (k node-ISTs) if for each node v other than r, the k paths from r to v, with one path in each spanning tree, are internally node-disjoint. This paper studies the problem of constructing k node-ISTs with minimal total path weight, which is NP-complete and has significant application value in fault-tolerant routing, load balancing and secure communication. Considering the limitation of existing algorithms that have been proposed on specific network topologies or for the case k = 2 are difficult to extend to the case k ≥ 3 in general networks, we propose a heuristic algorithm that can construct the required min k node-ISTs in polynomial time O ( k n 3 ) in any network with n nodes. Experimental simulations indicate that the algorithm can construct high-quality solutions in various random networks and some important specific networks. In addition, this algorithm can effectively meet the practical requirements for fault-tolerance in low-cost and high-connectivity networks.

We consider an assortment optimization problem for a class of online video games where the in-game virtual store has a unique structure with two sections: Featured and Just For You (JFY). All customers (players) are offered the same Featured section assortment, whereas the JFY section is used for personalized recommendations. We model customer choice under a constrained mixture-of-nested-logit model and propose different solution methods for the resulting assortment optimization problems. First, we introduce a novel mixed-integer nonlinear programming (MINLP) formulation. Numerical experiments show that the MINLP formulation generally obtains optimal solutions efficiently, using a variety of instances derived from conversations with our industry partner to mimic the environment found in their video game stores. In addition, we propose three approximate solution methods with theoretical performance guarantees: a fully polynomial time approximation scheme (FPTAS), a mixed-integer linear programming (MILP) formulation, and a heuristic algorithm. To understand the impact of a shared Featured section, we analyze the distribution of display capacity between the Featured and JFY sections. Our numerical experiments highlight that the Featured section plays a critical role in balancing revenue and customer utility. To validate our use of a mixture-of-nested-logit model, we further conduct a simulation study based on ground-truth instances that are independent of the underlying structure of the consumer choice models we consider. The results indicate that our nested structure yields superior performance in terms of both capturing customer behavior and simulation revenue, compared with the mixture-of-MNL model and the current practice of our industry partner. Overall, our paper is the first to study assortment optimization for the gaming industry under discrete choice models; it is also the first to devise both exact and approximate solution approaches for the constrained mixture-of-nested-logit model. Our results provide guidance for effective management of assortments in online video game stores and offer an “assortment” of solution approaches, allowing practitioners to choose one that best suits their environment.

In this paper, we address the edge station deployment problem (ESDP), which aims to determine optimal sites for deploying edge stations so as to maximize user coverage and minimize deployment cost. We first formulate the ESDP as a binary linear programming model and prove its NP-hardness by reducing the set covering problem to a specialized instance of ESDP. To solve the ESDP in polynomial time, we propose a novel heuristic algorithm that prioritizes covering users who are within range of the fewest candidate sites. Our algorithm iteratively selects the site that can cover the most users from among the candidate sites capable of covering those least-covered users. To evaluate the performance of our algorithm, we conduct simulation experiments based on a real-world dataset. Experimental results demonstrate that our algorithm achieves 100% user coverage with lower deployment cost compared to several classical and state-of-the-art algorithms.

This paper formulates the "Minimum Vertex Cut with Reachable Set" (MVCRS) problem as an optimization framework to suppress botnet propagation in networked systems, and clarifies its computational complexity and algorithmic solutions. Building a firewall to minimize damage is essential for addressing botnet propagation in Internet of Things (IoT) networks. We define the basic MVCRS problem as minimizing the sum of the weight of the deployed resources and the resulting propagation scope. While we demonstrate that the constrained version of the problem is NP-complete, we show that the fundamental trade-off optimization model can be solved in polynomial time by reducing it to the maximum flow-minimum cut problem. This provides a theoretical baseline for optimal resource allocation in cybersecurity. Experimental evaluations reveal the limitations of conventional heuristics. In community-structured networks, the degree-based greedy algorithm overlooks critical bridge nodes, yielding an optimality gap of up to 72.6% above the theoretical minimum cost. Conversely, our exact algorithm consistently guarantees the optimal minimum cost (a 0% gap) with high statistical stability across diverse topologies. Furthermore, it scales efficiently to solve 100,000-node IoT networks within practical time limits, proving to be a reliable and efficient foundation for botnet suppression in complex real-world systems.

The hypervolume under the receiver operating characteristic (ROC) manifold, referred to as HUM, is a critical metric for evaluating the performance in multi-classification problems. Nevertheless, its practical applicability is constrained by the polynomial time complexity of the standard implementation. This paper introduces a new fast algorithm for computing HUM value based on probability assessment vectors. Compared with the HUM value based on ordered response with a single marker, it poses unique challenges due to its complicated sample space. Inspired by the method of incomplete U-statistic, we propose a sampling-based algorithm for approximating HUM values. Theoretically, the approximation error diminishes inversely with computational time and becomes asymptotically negligible within quasi-linear computational complexity. Furthermore, we introduce a generalized definition of HUM to accommodate the discrete data, which fits well with our sampling-based method. Additionally, we present inference procedures applicable when probability assessment vectors are either known or estimated. The performance of the proposed procedures is evaluated through simulation studies. We also apply the method to two real-world applications to show its practical utility.

Modern industrial processes generate massive volumes of sensor data requiring automated causal discovery for process optimization and predictive maintenance. However, industrial big data exhibits mixed linear-nonlinear relationships with non-Gaussian noise, challenging existing methods. This paper presents a scalable causal discovery framework leveraging polynomial chaos expansion to handle complex industrial data characteristics. Our approach utilizes a multi-criteria strategy to minimize false discoveries critical for industrial safety. The method processes high-dimensional sensor data in polynomial time while providing uncertainty quantification through bootstrap confidence intervals. Validation on real industrial datasets from chemical processes demonstrates 88.9% F1-score, outperforming baseline methods by 27.7%. Furthermore, experiment on synthetic benchmark confirms robust performance across varying problem dimensions, with improvement margins increasing from 26.7% to 63.9% as dimensionality grows. The framework successfully identifies both linear and nonlinear mechanisms from data, enabling data-driven decision support for operators. This work advances data-driven causal discovery for smart manufacturing applications where understanding complex variable interactions is essential.

• Defines the Riesz Energy Minimization Facility Location Problem • Proves NP-hardness in general metrics via reduction from Independent Set • Analyzes limiting behavior interpolating dispersion and uniform selection • Develops a PTAS for the planar case under minimum-separation condition • Achieves ( 1 + ϵ ) approximation in polynomial time for fixed facility count. This article introduces a geometric facility location problem that minimizes total Riesz interaction energy among p selected sites in the Euclidean plane. The objective generalizes the classical p -dispersion criterion by penalizing spatial proximity through inverse power distances, thereby promoting global dispersion in a smooth manner. The problem is shown to be NP-hard in general metric spaces. A Polynomial-Time Approximation Scheme (PTAS) is developed for the planar case for a truncated Riesz objective (for fixed p ), with the same approximation guarantee extending to the original objective whenever the candidate set satisfies a natural minimum-separation condition. Furthermore, limiting analysis shows that the Riesz objective provides a smooth interpolation between strict dispersion and uniform selection, depending on the strength of the interaction exponent.

Two-stage flowshop scheduling has been extensively studied in the scheduling community. Unlike traditional objectives, which focus on minimizing job completion time objectives such as makespan or total tardiness, this study addresses the minimization of total job rejection costs while ensuring that the makespan remains within a specified threshold. This problem is motivated by outsourcing practices in certain make-to-order scenarios, where a cost is incurred if the manufacturer opts to reject a job and outsource it instead. For the single two-stage flowshop case, a polynomial time approximation scheme is proposed, utilizing a guessing strategy combined with a linear programming rounding technique. For the parallel two-stage flowshops case, when the number of flowshops is a fixed constant, a bicriteria [Formula: see text]-approximation algorithm is introduced, i.e., the total rejection cost does not exceed the minimum possible value, but the schedule is relaxed to possibly exceed the bound on the makespan by a factor of [Formula: see text], where [Formula: see text] is a given arbitrarily small positive constant. The algorithm is derived from a pseudo-polynomial time dynamic programming algorithm coupled with a trimming technique. When the number of flowshops is part of the input, a bicriteria [Formula: see text]-approximation algorithm is proposed. The problem formulation and algorithmic results offer production managers greater flexibility in managing job outsourcing decisions.

ABSTRACT In this paper, we are interested in 4‐colouring algorithms for graphs that do not contain an induced path on six vertices nor an induced bull, that is, the graph with vertex set and edge set . Such graphs are referred to as ‐free graphs. A graph is ‐ vertex‐critical if , and every proper induced subgraph of has . In the current paper, we investigate the structure of 5‐vertex‐critical ‐free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial‐time algorithm to decide if a ‐free graph is 4‐colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4‐colourability or non‐4‐colourability.

Abstract Topology optimization is a key methodology in engineering design for finding efficient and robust structures. Due to the enormous size of the design space, evaluating all possible configurations is typically infeasible. In this work, we present an end-to-end, fault-tolerant quantum algorithm for topology optimization that operates on an exponentially large Hilbert space representing the design space. We demonstrate the algorithm on the two-dimensional Messerschmitt-Bölkow-Blohm beam problem. By restricting design variables to binary values, we reformulate the compliance minimization task as a combinatorial satisfiability problem, solved using Grover’s algorithm. Within Grover’s oracle, the compliance is computed through the finite-element method using established quantum algorithms, including block-encoding of the stiffness matrix, quantum singular value transformation for matrix inversion, Hadamard test, and quantum amplitude estimation. The complete algorithm is implemented and validated using classical quantum-circuit simulations. A detailed complexity analysis shows that the method evaluates the compliance of exponentially many structures in quantum superposition in polynomial time. In the global search, our approach maintains Grover’s quadratic speedup compared to classical unstructured search. Overall, the proposed quantum workflow demonstrates how quantum algorithms can advance the field of computational science and engineering.

A randomized algorithm for a search problem is pseudodeterministic if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser [16] posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an unconditional polynomial-time randomized algorithm B such that, for infinitely many values of n , B(1^n) outputs a canonical n -bit prime p_n with high probability. More generally, we prove that for every dense property Q of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying Q . This improves upon a subexponential-time construction of Oliveira and Santhanam [49]. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell [11], using a variant of the Shaltiel–Umans generator [51].