Almost-Linear-Time Algorithms for Maximum Flow and Minimum-Cost Flow

Augmenting path algorithms, first introduced by Ford and Fulkerson (Ford, L. R., D. R. Fulkerson. 1962. Flows in Networks. Princeton University Press, Princeton, N.J.), are widely used in optimization. Examples include Schönsleben's polymatroid intersection algorithm (Schönsleben, P. 1980. Ganzzahlige polymatroid-intersection-algorithmen. Ph.D. thesis, Eidgenössischen Technischen Hochschule, Zürich.), the maximum polymatroidal network flow algorithm of Lawler and Martel (Lawler, E. L., C. U. Martel. 1982. Computing maximal polymatroidal network flows. Math. Oper. Res. 7 334–347.), Frank's algorithm for the Edmonds–Giles polyhedron (Frank, A. 1984. Finding feasible vectors of Edmonds-Giles polyhedra. J. Combin. Theory Ser. B 36 221-239.) and Cunningham's algorithm for testing membership in matroid polyhedra (Cunningham, H. W. 1984. Testing membership in matroid polyhedra. J. Combin. Theory Ser. B 36 161–188.). Here we give an order of magnitude improvement for the above algorithms by using an approach analogous to that of Dinits’ maximum flow algorithm (Dinits, E. A. 1970. Algorithm for solution of a problem of maximum flow in a network with power estimation. Soviet Math. Dokl. 11 1277–1280.).

With an increasing number of processing elements (PEs) integrated on a single chip, fault-tolerant techniques are critical to ensure the reliability of such complex systems. In current reconfigurable architectures, redundant PEs are utilized for fault tolerance. In the presence of faulty PEs, the physical topologies of various chips may be different, so the concept of virtual topology from network embedding problem has been used to alleviate the burden for the operating systems. With limited hardware resources, how to reconfigure a system into the most effective virtual topology such that the maximum repair rate can be reached presents a significant challenge. In this article, a new approach using a maximum flow (MF) algorithm is proposed for an efficient topology reconfiguration in reconfigurable architectures. In this approach, topology reconfiguration is converted into a network flow problem by constructing a directed graph; the solution is then found by using the MF algorithm. This approach optimizes the use of spare PEs with minimal impacts on area, throughput, and delay, and thus it significantly improves the repair rate of faulty PEs. In addition, it achieves a polynomial reconfiguration time. Experimental results show that compared to previous methods, the MF approach increases the probability to repair faulty PEs by up to 50% using the same redundant resources. Compared to a fault-free system, the throughput only decreases by less than 2.5% and latency increases by less than 4%. To consider various types of PEs in a practical application, a cost factor is introduced into the MF algorithm. An enhanced approach using a minimum-cost MF algorithm is further shown to be efficient in the fault-tolerant reconfiguration of heterogeneous reconfigurable architectures.

Predictive business process monitoring aims at providing the predictions about running instances by analyzing logs of completed cases of a business process. Recently, a lot of research focuses on increasing productivity and efficiency in a business process by forecasting potential problems during its executions. However, most of the studies lack suggesting concrete actions to improve the process. They leave it up to the subjective judgment of a user. In this paper, we propose a novel method to connect the results from predictive business process monitoring to actual business process improvements. More in detail, we optimize the resource allocation in a non-clairvoyant online environment, where we have limited information required for scheduling, by exploiting the predictions. The proposed method integrates offline prediction model construction that predicts the processing time and the next activity of an ongoing instance using LSTM with online resource allocation that is extended from the minimum cost and maximum flow algorithm. To validate the proposed method, we performed experiments using an artificial event log and a real-life event log from a global financial organization.

The bulk power system is one of the largest man-made networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (max-flow) algorithm. The graph-theory-based max-flow is applied to the power system for UPFC placement and scheduling. A distributed version of max-flow is described to coordinate the actions of the UPFCs distributed in a power network. Two sample power systems were tested using max-flow for UPFC placement and settings. The resulting system characteristics are examined over all single-line contingencies and the appropriateness of the maximum flow algorithm for power flow control is discussed.

A new maximal flow algorithm for solving optimization problems with linguistic capacities and flows

Dynamic network models are widely employed for system analysis and performance evaluation of computer and communication systems. In dynamic networks each arc is weighted with a capacity and a transit time. The former is the maximum amount of flow that can travel through the arc per unit time, the later is the time period needed for flow to travel through the arc. In this article, an additional weight operation probability, is assigned to each arc to note the capability of the arc's successful transition. The performance of dynamic networks is defined by the probability that the maximum dynamic flow from source to sink within required time horizon is no less than specified threshold. To the best of our knowledge, no existing article had considered dynamic flow and network reliability simultaneously. Based on facts that there is always a maximum dynamic flow among the temporally repeated flows and the maximum temporally repeated flow can be obtained via an equivalent minimum cost maximum flow problem, we present a pioneering algorithm to compute performance of dynamic networks by solving minimum cost maximum flow and two-terminal capacity reliability at the same time.

Histograms are a concise and flexible way to construct summary structures for large data sets. They have attracted a lot of attention in database research due to their utility in many areas, including query optimization, and approximate query answering. They are also a basic tool for data visualization and analysis.In this paper, we present a formal study of dynamic multidimensional histogram structures over continuous data streams. At the heart of our proposal is the use of a dynamic summary data structure (vastly different from a histogram) maintaining a succinct approximation of the data distribution of the underlying continuous stream. On demand, an accurate histogram is derived from this dynamic data structure. We propose algorithms for extracting such an accurate histogram and we analyze their behavior and tradeoffs. The proposed algorithms are able to provide approximate guarantees about the quality of the estimation of the histograms they extract.We complement our analytical results with a thorough experimental evaluation using real data sets.

In this paper, we focus on the dynamic traffic assignment (DTA) problem on large-scale expressway networks especially under the condition of traffic events (such as severe weather, large traffic accidents etc.). We formulate the expressway network DTA problem as a nonconvex optimization problem. Considering the difficulty of solving the problem, we put forward an approximate solution algorithm based on the network flow theory with high computational efficiency and strong applicability. Specifically, a machine learning approach and a series of customization strategies of network flow model (including multi-origin and multidestination split, flow discretization and parallel edges addition methods) are proposed to transform the DTA problem into a resoluble network flow problem. After that, in order to improve the operational efficiency of an expressway network after traffic events occurring, the proposed network flow based two-layer DTA model (NTDM) is separated into two parts-traffic flow limitation model based on maximum flow algorithm and traffic assignment model based on minimum cost flow algorithm. Experiments on real expressway network data in some provinces of China show that our method can make a significant 26.03% reduction in saturation rate in a particular scenario. In efficiency test, the NTDM is at least 600 times faster than the traditional analytic method in a network with thousands of nodes.

In rider demand responsive systems riders submit requests to demand transit services and the incoming requests and vehicles are matched by the system. This demand-responsive transport problem is viewed by existing research as a kind of spatial matching problem between vehicles and riders. However, existing schemes mainly focus on maximizing the number of matching pairs. It neglects other factors like the length of pickup trajectories and riders’ waiting time. And the matching is not revocable, which loses the chance for further optimizations. Moreover, there is still not much work on handling and matching the appointment-based requests that play a key role in the demand-responsive transport market. In this article, we propose an algorithm called BMCF ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bipartite Minimal-Cost Flow</i> ) to solve the taxi-rider matching problem with appointment-based rider requests on a time-dependent road network. Unlike existing solutions that map the problem into the max flow problem, we transform the optimal matching to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">minimal cost flow problem</i> which aims to maximize a utility value and could be solved efficiently. Riders and vehicles are modeled as vertices in a bipartite graph, and factors like the length of pickup trajectories, riders’ waiting time, etc. are abstracted as utilities and denoted as weights of the edges which are taken into account during the matching process. To the best of our knowledge, the proposed scheme is the first to efficiently integrate and process both the real-time and appointment-based requests within the same framework, and the assignments between vehicles and requests could be dynamically adjusted and revoked to maximise the overall utility. Experimental results show that the proposed algorithm can effectively increase the appointment-based matching ratio and decrease the riders’ waiting time (> 10.4%) and vacant vehicles’ picking up time (> 60.9%) compared with existing schemes, while at the cost of acceptable increase on the running time.

In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the Edmonds-Karp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n(m + n log n)) time. Using a standard transformation, this approach yields an O(m log n(m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n2/m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say m′, is much less than m. In this case, the running time of the algorithm is O((m′ + n) log n(m + n log n)).

Forwarding packets based on networking names is essential for network protocols on different layers, where the `names' could be addresses, packet/flow IDs, and content IDs. For long there have been efforts using dynamic and compact data structures for fast and memory-efficient forwarding. In this work, we identify that the recently developed programmable network paradigm has the potential to further reduce the time/memory complexity of forwarding structures by separating the data plane and control plane. This work presents the new designs of network forwarding structures under the programmable network paradigm, applying three typical dynamic and compact data structures: Bloom filters, Cuckoo hashing, and Othello hashing. We conduct careful analyses and experiments in real networks of these forwarding methods for multiple performance metrics, including lookup throughput, memory footprint, construction time, dynamic updates, and lookup errors. The results give rich insights on designing forwarding algorithms with dynamic and compact data structures. In particular, the new designs based on Cuckoo hashing and Othello hashing show significant advantages over the extensively studied Bloom filter based methods, in all situations discussed in this paper.

We present a new approach to the minimum-cost integral flow problem for small values of the flow. It reduces the problem to the tests of simple multi-variable polynomials over a finite field of characteristic two for non-identity with zero. In effect, we show that a minimum-cost flow of value k in a network with n vertices, a sink and a source, integral edge capacities and positive integral edge costs polynomially bounded in n can be found by a randomized PRAM, with errors of exponentially small probability in n, running in O(klog(kn)+log2 (kn)) time and using 2k(kn)O(1) processors. Thus, in particular, for the minimum-cost flow of value O(logn), we obtain an RNC2 algorithm.

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2m lognC, n 2m2 logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm 2 logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).

A theoretical and experimental study of a new algorithm for minimum cost flow in dynamic graphs

Chapter 1 Applications of network optimization