Minimum Cut Research Articles

Random Tensor Networks with Non-trivial Links

Random tensor networks are a powerful toy model for understanding the entanglement structure of holographic quantum gravity. However, unlike holographic quantum gravity, their entanglement spectra are flat. It has therefore been argued that a better model consists of random tensor networks with link states that are not maximally entangled, i.e., have non-trivial spectra. In this work, we initiate a systematic study of the entanglement properties of these networks. We employ tools from free probability, random matrix theory, and one-shot quantum information theory to study random tensor networks with bounded and unbounded variation in link spectra, and in cases where a subsystem has one or multiple minimal cuts. If the link states have bounded spectral variation, the limiting entanglement spectrum of a subsystem with two minimal cuts can be expressed as a free product of the entanglement spectra of each cut, along with a Marchenko–Pastur distribution. For a class of states with unbounded spectral variation, analogous to semiclassical states in quantum gravity, we relate the limiting entanglement spectrum of a subsystem with two minimal cuts to the distribution of the minimal entanglement across the two cuts. In doing so, we draw connections to previous work on split transfer protocols, entanglement negativity in random tensor networks, and Euclidean path integrals in quantum gravity.

Critical Attacks Set Identification in Attack Graphs for Computer and SCADA/ICS Networks

Supervisory control and data acquisition/industrial control systems (SCADA/ICSs) networks are becoming more vulnerable to attacks that exploit the interdependence of security weaknesses at the atomic level to compromise system-level security. Attack graphs are an effective approach to depict these complex attack scenarios, assisting security administrators in determining how to best safeguard their systems. However, due to time and financial constraints, it is frequently not possible to address all atomic-level flaws at the same time. In this article, we propose a method for automatically detecting a minimal set of critical attacks that, when defended against, render the system secure. Finding a minimal label cut is typically an NP-complete problem. However, we propose a linear complexity approximation that uses the attack graph’s strongly connected components (SCCs) to create a simplified version of the graph in the form of a tree over the SCCs. Then, we perform an iterative backward search over this tree to find a set of backward-reachable SCCs, as well as their outward edges and labels, in order to find a cut of the tree with the fewest labels, which is a critical attack set. We put our proposed method to the test on real-world case studies, such as IT and SCADA networks for a cyber–physical system for water treatment, and outperformed previous state-of-the-art algorithms in terms of approximation accuracy and/or computational speed. Our solution provides security administrators with a practical and efficient method for prioritizing efforts to address vulnerabilities in SCADA/ICS networks.

Integrating stochastic mine planning model with ARDL commodity price forecasting

Commodity price is of paramount importance to mine planning among other stochastic factors such as mining cost and geological uncertainties. Commodity price uncertainty is critical in the determining block values, and it has been incorporated into mine planning by simulation-based methodologies to deal with the effects of price uncertainty during the mining process. In this article, stochastic optimisation was undertaken using parametric minimum cut network flow algorithm for ultimate pit and pushback design, while stochasticity of commodity prices was incorporated into the planning using Autoregressive Distributed Lag (ARDL) simulation model. The push back design and cashflows in two scenarios, namely when the prices are going up and when the prices are going down, were investigated. Moreover, for each scenario, three types of price inputs were considered: (a) actual prices; (b) ARDL forecasted prices; and (c) stochastic prices generated from ARDL forecasted prices. The results of cashflows indicated that when commodity price moving up scenario, the ARDL forecasted prices underestimate the NPV by about 17% and the stochastic minimum cut underestimates the NPV by 72.27% compared to actual price. Similarly, for the price declining scenario, the ARDL forecasted prices underestimated the NPV by about 3% compared to the actual, while the stochastic minimum cut has an NPV value that is 277% of the actual value. Lastly the value of stochastic programming (VSP) was evaluated. The VSP was about 9.2 million dollars when the prices were going up, while the VSP was about 27.1 million dollars, when price was declining. Moreover, declining commodity price produced negative cashflows which indicate need for risk management with a recourse action to evert losses.

Finite-Time Distributed Average Tracking for Multiagent Optimization With Bounded Inputs

In distributed optimization (DO), the designed algorithms are expected to have a fast convergence rate but less computation cost. Moreover, the boundedness of the control inputs is generally required for practical networking agent systems with actuator limitations. Motivated by these observations, we first revisit the well-known finite-time distributed average tracking (FTDAT) problem where a novel sufficient condition on the control gain and the finite settling time estimation are derived based on the minimum cut of the underlying topology. Then, based on FTDAT, three types of discontinuous dynamics with bounded inputs are designed for solving unconstrained and constrained DO problems, respectively. The first algorithm can successfully find the optimal solution for an unconstrained DO in finite time. For DO problems with a common constraint set or separated equality constraints, two projection-based algorithms are designed and utilized to first make the agents' states achieve consensus in finite time, and then drive the common state to the optimum exponentially. Finally, several case studies and extensive numerical simulations are conducted to testify the designed algorithms.

Synthetic lethality in large-scale integrated metabolic and regulatory network models of human cells

Synthetic lethality (SL) is a promising concept in cancer research. A wide array of computational tools has been developed to predict and exploit synthetic lethality for the identification of tumour-specific vulnerabilities. Previously, we introduced the concept of genetic Minimal Cut Sets (gMCSs), a theoretical approach to SL developed for genome-scale metabolic networks. The major challenge in our gMCS framework is to go beyond metabolic networks and extend existing algorithms to more complex protein-protein interactions. In this article, we take a step further and incorporate linear regulatory pathways into our gMCS approach. Extensive algorithmic modifications to compute gMCSs in integrated metabolic and regulatory models are presented in detail. Our extended approach is applied to calculate gMCSs in integrated models of human cells. In particular, we integrate the most recent genome-scale metabolic network, Human1, with 3 different regulatory network databases: Omnipath, Dorothea and TRRUST. Based on the computed gMCSs and transcriptomic data, we discovered new essential genes and their associated synthetic lethal for different cancer cell lines. The performance of the different integrated models is assessed with available large-scale in-vitro gene silencing data. Finally, we discuss the most relevant gene essentiality predictions based on published literature in cancer research.

On Minimal Cut Sets Representation with Binary Decision Diagrams

Since their introduction in form of a canonical representation of logical functions, the Binary Decision Diagrams (BDDs) gained a wide acceptance in numerous industrial applications. This paper summarizes the properties of BDD representation of Minimal Cut Sets (MCS) of Fault Tree (FT) models most typically encountered in nuclear energetics. Cut sets from MCS are defined as paths from the top BDD node to terminal nodes in the BDD, on which a quantitative and qualitative FT analysis (FTA) is performed. The core of the FTA on the BDDs is performed with help of two fundamental algorithms, one for conditional probability evaluation and another for the selection of cut sets. The accuracy of conditional probability evaluation represents an essential feature for an unbiased quantitative analysis, such as the top event probability or the determination of event importance measures. The cut set selection algorithm is shown in a generic version introducing logical predicates for its selection criteria. As it is known, the efficiency of depicted algorithms depends only on the number of BDD nodes used for the FT representation. In order to appraise the compactness of the BDD representation of FT models, their characteristics have herein been evaluated on several real-life models from the Nuclear Power Plant Krško. The extraordinariness of the compactness of the BDD representation reflects in its ability to implement advanced dynamic analysis (i.e. what-if) of FT models. The efficiency of such an approach is recognized by commercial vendors upgrading their FT Tools to new versions by implementing BDD based algorithms.

Single Objective Mayfly Algorithm with Balancing Parameter for Multiple Traveling Salesman Problem

The Multiple Travelling Salesman Problem (MTSP) is a challenging combinatorial problem that involves multiple salesman visiting a set of cities, each exactly once, starting and ending at the same depot. The aim is to determine the optimal route with minimal cost and node cuts for each salesman while ensuring that at least one salesman visits each city. As the problem is NP-Hard, a single-objective metaheuristic algorithm, called the Mayfly Algorithm, inspired by the collective behavior of mayflies, is employed to solve the problem using the TSPlib95 test data. Since the Mayfly Algorithm employs a single fitness function, a balancing parameter is added to perform multiobjective optimization. Three balancing parameters in the optimization process: SumRoute represents the total cost of all salesmen travelling, StdRoute balances each salesman cost, and StdNodes balances the number of nodes for each salesman. The values of these parameters are determined based on the results of various tests, as they significantly impact the MTSP optimization process. With the appropriate parameter values, the single-objective Mayfly Algorithm can produce optimal solutions and avoid premature convergence. Overall, the Mayfly Algorithm shows promise as a practical approach to solving the MTSP problem. Using multiobjective optimization with balancing parameters enables the algorithm to achieve optimal results and avoid convergence issues. The TSPlib95 dataset provides a robust testing ground for evaluating the algorithm’s effectiveness, demonstrating its ability to solve MTSP effectively with multiple salesman.

On the g-extra connectivity of augmented cubes

A g-extra cut of a non-complete graph G,g≥0, is a set of vertices in G whose removal disconnects the graph, while every component in the survival graph contains at least g+1 vertices. The g-extra connectivity of G then refers to the size of a minimum g-extra cut of G.The augmented hypercube, denoted by AQn,n≥3, is a rich variant of the hypercube structure. In this paper, we present a sequence of construction based upper bounds of its g-extra connectivity, study its lower bound via the super connectedness property, and suggest an asymptotically tight bound.

Critical Path Awareness Techniques for Large-Scale Graph Partitioning

Graph partitioning is one of the fundamental problems in many graph-based applications and systems. It enables the division of a graph into smaller sub-graphs for subsequent parallel processing, reducing the processing latency of the graph. The critical path of a graph is the logical path with the longest delay from input to output. The processing time of the graph mainly depends on the delay incurred by the critical path, independent of other paths with small delays. Therefore, it can reduce the processing time of the graph by protecting the critical path of the graph from partition. However, existing approaches to graph partitioning only focus on metrics such as minimum cut and partition balance. As a result, the critical paths of graphs may be destroyed in the partitioning procedure. To address this problem, we present a critical path awareness approach, namely path-metis, to protect the critical paths and alleviate the processing latency after graph partitioning. In path-metis, two efficient strategies, including Slack and critical path fix strategies, are introduced. The Slack strategy, which incorporates critical path information into the weights of DAG, is used as pre-processing before traditional multi-level partitioning methods, like Metis. Then, for the generated partitioning scheme, the critical path fix strategy is proposed to further protect critical paths from being cut. We demonstrate the effectiveness of our approach on both real and synthetic datasets. From the experimental results, compared to Metis, our method improves critical path performance by 17.70%.

Automatic Inference of Fault Tree Models Via Multi-Objective Evolutionary Algorithms

Fault tree analysis is a well-known technique in reliability engineering and risk assessment, which supports decision-making processes and the management of complex systems. Traditionally, fault tree (FT) models are built manually together with domain experts, considered a time-consuming process prone to human errors. With Industry 4.0, there is an increasing availability of inspection and monitoring data, making techniques that enable knowledge extraction from large data sets relevant. Thus, our goal with this work is to propose a data-driven approach to infer efficient FT structures that achieve a complete representation of the failure mechanisms contained in the failure data set without human intervention. Our algorithm, the FT-MOEA, based on multi-objective evolutionary algorithms, enables the simultaneous optimization of different relevant metrics such as the FT size, the error computed based on the failure data set and the Minimal Cut Sets. Our results show that, for six case studies from the literature, our approach successfully achieved automatic, efficient, and consistent inference of the associated FT models. We also present the results of a parametric analysis that tests our algorithm for different relevant conditions that influence its performance, as well as an overview of the data-driven methods used to automatically infer FT models.

A Framework to Design Approximation Algorithms for Finding Diverse Solutions in Combinatorial Problems

Finding a \emph{single} best solution is the most common objective in combinatorial optimization problems. However, such a single solution may not be applicable to real-world problems as objective functions and constraints are only ``approximately'' formulated for original real-world problems. To solve this issue, finding \emph{multiple} solutions is a natural direction, and diversity of solutions is an important concept in this context. Unfortunately, finding diverse solutions is much harder than finding a single solution. To cope with the difficulty, we investigate the approximability of finding diverse solutions. As a main result, we propose a framework to design approximation algorithms for finding diverse solutions, which yields several outcomes including constant-factor approximation algorithms for finding diverse matchings in graphs and diverse common bases in two matroids and PTASes for finding diverse minimum cuts and interval schedulings.

Edge connectivity of simplicial polytopes

We show that the graph of a simplicial polytope of dimension d≥3 has no nontrivial minimum edge cut with fewer than d(d+1)/2 edges, hence the graph is min{δ,d(d+1)/2}-edge-connected where δ denotes the minimum degree. When d=3, this implies that every minimum edge cut in a plane triangulation is trivial. When d≥4, we construct a simplicial d-polytope whose graph has a nontrivial minimum edge cut of cardinality d(d+1)/2, proving that the aforementioned result is best possible.

The usefulness of (d,b)-MCs and (d,b)-MPs in network reliability evaluation under delivery or maintenance cost constraints

Reliability and delivery cost or maintenance cost are important indicators showing the stability and the quality of operation of a multi-state flow network. Many authors developed (d,b)-minimal cut based or (d,b)-minimal path based algorithms to compute the network reliability, i.e., the probability that a demand d can be delivered from the source node to the sink node subject to the constraint that the total delivery cost (maintenance cost) is not greater than b. It is proven that contrary to prevalent methods in existing literature, both (d,b)-minimal cuts and (d,b)-minimal paths are insufficient to compute the exact value of the reliability of a multi-state flow network under budget or maintenance budget constraints. Moreover, it is also shown how this value can be determined.

Quantum bit threads

We give a bit thread prescription that is equivalent to the quantum extremal surface prescription for holographic entanglement entropy. Our proposal is inspired by considerations of bit threads in doubly holographic models, and equivalence is established by proving a generalisation of the Riemannian max-flow min-cut theorem. We explore our proposal’s properties and discuss ways in which islands and spacetime are emergent phenomena from the quantum bit thread perspective.

Extreme Min – Cut Max – Flow Algorithm

In this article, the authors propose a maximum flow algorithm based on flow matrix. The algorithm only requires the effort to reduce the capacity of the underutilized arcs to that of the respective flow. The optimality of the algorithm is proved by the max-flow min-cut theorem. The algorithm is table-based, thus avoiding augmenting path and residual network concepts. The authors used numerical examples and computational comparisons to demonstrate the efficiency of the algorithm. These examples and comparisons revealed that the proposed algorithm is capable of computing exact solutions while using few iterations as compared to some existing algorithms.

Speeding Up the Structural Analysis of Metabolic Network Models Using the Fredman-Khachiyan Algorithm B.

The problem of computing the Elementary Flux Modes (EFMs) and Minimal Cut Sets (MCSs) of metabolic network is a fundamental one in metabolic networks. A key insight is that they can be understood as a dual pair of monotone Boolean functions (MBFs). Using this insight, this computation reduces to the question of generating from an oracle a dual pair of MBFs. If one of the two sets (functions) is known, then the other can be computed through a process known as dualization. Fredman and Khachiyan provided two algorithms, which they called simply A and B that can serve as an engine for oracle-based generation or dualization of MBFs. We look at efficiencies available in implementing their algorithm B, which we will refer to as FK-B. Like their algorithm A, FK-B certifies whether two given MBFs in the form of Conjunctive Normal Form and Disjunctive Normal Form are dual or not, and in case of not being dual it returns a conflicting assignment (CA), that is, an assignment that makes one of the given Boolean functions True and the other one False. The FK-B algorithm is a recursive algorithm that searches through the tree of assignments to find a CA. If it does not find any CA, it means that the given Boolean functions are dual. In this article, we propose six techniques applicable to the FK-B and hence to the dualization process. Although these techniques do not reduce the time complexity, they considerably reduce the running time in practice. We evaluate the proposed improvements by applying them to compute the MCSs from the EFMs in the 19 small- and medium-sized models from the BioModels database along with 4 models of biomass synthesis in Escherichia coli that were used in an earlier computational survey Haus et al. (2008).

Risk-informed collision avoidance system design for maritime autonomous surface ships

The maritime industry is paving the way towards developing Maritime Autonomous Surface Ships (MASSs) through the adoption of key enabling technologies for safety-critical operations, which are associated with new challenges, especially at their early design phase. This study aims to develop a methodology to conduct the risk-informed design for the Collision Avoidance System (CAS) of MASSs. Pertinent regulatory instruments are reviewed to identify functional and system requirements and develop a baseline CAS configuration at the component level. Quantitative Fault Tree Analysis is performed to derive risk metrics, such as probability of failure, Importance measures, and Minimal Cut Sets, whereas criticality analysis is conducted to recommend risk-reducing measures. A Short Sea Shipping case study is investigated considering four operating modes based on various weather and illumination conditions. Results demonstrate that the developed Fault Tree diagram provides a robust representation of the CAS failure. The most critical components are found to be related to the Intention Communication and Situation Awareness Systems, the redundancy of which leads to 91% reduction of the CAS probability of failure. This study contributes towards the risk-informed design of safety-critical systems required for the development of MASSs.

Minimum Cut and Minimum k -Cut in Hypergraphs via Branching Contractions

On hypergraphs with m hyperedges and n vertices, where p denotes the total size of the hyperedges, we provide the following results: We give an algorithm that runs in \(\widetilde{O}(mn^{2k-2})\) time for finding a minimum k -cut in hypergraphs of arbitrary rank. This algorithm betters the previous best running time for the minimum k -cut problem, for k > 2. We give an algorithm that runs in \(\widetilde{O}(n^{\max \lbrace r,2k-2\rbrace })\) time for finding a minimum k -cut in hypergraphs of constant rank r . This algorithm betters the previous best running times for both the minimum cut and minimum k -cut problems for dense hypergraphs. Both of our algorithms are Monte Carlo, i.e., they return a minimum k -cut (or minimum cut) with high probability. These algorithms are obtained as instantiations of a generic branching randomized contraction technique on hypergraphs, which extends the celebrated work of Karger and Stein on recursive contractions in graphs. Our techniques and results also extend to the problems of minimum hedge-cut and minimum hedge- k -cut on hedgegraphs, which generalize hypergraphs.

A new Coplanar Dual Core Buckling-Restrained Brace

This study presents a new all-steel Buckling-Restrained Brace (BRB) named Coplanar Dual Core (CDC) BRB, that consists of a rectangular cross-sectional core divided into two coplanar yielding lateral bands, whose movements are constrained by an external restrainer (ER) and by an internal restrainer (IR). In contrast with the traditional BRB with dog-bone shaped core, the CDC-BRB has the advantage that the core can be easily extracted and inspected. The ER is composed of two rectangular tubes welded to a pair of rectangular solid calibrated bars. The core and IR are simultaneously laser cut from the same sheet of steel, which provides additional advantages: minimum material waste, minimum laser cut length, regular and well controlled gap, reduction of weight and manufacturing cost. Formulae to design the CDC-BRB, to determine the appropriate values of the gaps, and to predict its response and ultimate energy dissipation capacity are proposed. Four specimens representing the new CDC-BRBs with different gaps are tested under cyclic loadings to failure. The results of the tests show that the cyclic behavior of the new CDC-BRB is stable, highly dissipative, and its response and ultimate energy dissipation capacity can be well predicted insofar the gaps fit the proposed values.

On the foundations and extremal structure of the holographic entropy cone

The holographic entropy cone (HEC) is a polyhedral cone first introduced in the study of a class of quantum entropy inequalities. It admits a graph-theoretic description in terms of minimum cuts in weighted graphs, a characterization which naturally generalizes the cut function for complete graphs. Unfortunately, no complete facet or extreme-ray representation of the HEC is known. In this work, starting from a purely graph-theoretic perspective, we develop a theoretical and computational foundation for the HEC. The paper is self-contained, giving new proofs of known results and proving several new results as well. These are also used to develop two systematic approaches for finding the facets and extreme rays of the HEC, which we illustrate by recomputing the HEC on $5$ terminals and improving its graph description. We also report on some partial results for $6$ terminals. Some interesting open problems are stated throughout.