A flow model based on polylinking system

We present a new class of polynomial-time algorithms for submodular function minimization (SFM) as well as a unified framework to obtain strongly polynomial SFM algorithms. Our algorithms are based on simple iterative methods for the minimum-norm problem, such as the conditional gradient and Fujishige–Wolfe algorithms. We exhibit two techniques to turn simple iterative methods into polynomial-time algorithms. First, we adapt the geometric rescaling technique, which has recently gained attention in linear programming, to SFM and obtain a weakly polynomial bound [Formula: see text]. Second, we exhibit a general combinatorial black box approach to turn [Formula: see text]-approximate SFM oracles into strongly polynomial exact SFM algorithms. This framework can be applied to a wide range of combinatorial and continuous algorithms, including pseudo-polynomial ones. In particular, we can obtain strongly polynomial algorithms by a repeated application of the conditional gradient or of the Fujishige–Wolfe algorithm. Combined with the geometric rescaling technique, the black box approach provides an [Formula: see text] algorithm. Finally, we show that one of the techniques we develop in the paper can also be combined with the cutting-plane method of Lee et al., yielding a simplified variant of their [Formula: see text] algorithm.

We consider a ground set E and a submodular function acting on it. We first propose a “set multicovering” problem in which the value (price) of any is . We show that the linear program (LP) of this problem can be directly solved by applying a submodular function minimization (SFM) algorithm on the dual LP. However, the main results of this study concern prize‐collecting multicovering with submodular pricing, that is, a more general and more difficult “multicovering” problem in which elements can be left uncovered by paying a penalty. We formulate it as a large‐scale LP (with variables representing all subsets of E) that could be tackled by column generation (CG; for a CG algorithm for “set‐covering” problems with submodular pricing). However, we do not solve this large‐scale LP by CG, but we solve it in polynomial time by exploiting its integrality properties. More exactly, after appropriate restructuring, the dual LP can be transformed into an LP that has been thoroughly studied (as a primal) in the SFM theory. Solving this LP reduces to optimizing a strong map of submodular functions. For this, we use the Fleischer–Iwata framework that optimizes all these functions within the same asymptotic running time as solving a single SFM, that is, in , where and γ is the complexity of one submodular evaluation. Besides solving the problem, the proposed approach can be useful to (a) simultaneously find the best solution of up to functions under strong map relations in time, (b) perform sensitivity analysis in very short time (even at no extra cost), and (c) reveal combinatorial insight into the primal–dual optimal solutions. We present several potential applications throughout the paper, from production planning to combinatorial auctions.

Traffic researchers have developed a huge number of traffic theories for optimization and control of traffic and transportation networks. In particular, to generally accepted fundamentals and methodologies of traffic and transportation theory belong to the Lighthill-Whitham-Richards (LWR) model and the General Motors (GM) model class. These classical theories explain many real traffic flow phenomena. In this Chapter, we show that although these classical traffic flow models exhibit many important achievements, the classical models are inconsistent with the empirical nucleation nature of traffic breakdown at highway bottlenecks. Because the classical models and theories have failed in the explanation of the empirical fundamental of transportation science, the use of the classical approaches for a study of the reliability of applications of intelligent transportation systems (ITS) leads usually to invalid conclusions for real traffic. Moreover, ITS based on these classical traffic flow theories and models cannot also be used for reliable traffic control and dynamic optimization of traffic and transportation networks.

Bisubmodular functions are a natural "directed", or "signed", extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata's fully combinatorial version of IFF to BSFM.

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)).

Network optimization is being an increasingly important and fundamental issue in the fields such as engineering, computer science, operations research, transportation, telecommunication, decision support systems, manufacturing, and airline scheduling. In many applications, however, there are several criteria associated with traversing each edge of a network. For example, cost and flow measures are both important in the networks. As a result, there has been recent interest in solving Bicriteria Network Optimization Problem. The Bicriteria Network Optimization Problem is known a NP-hard. The efficient set of paths may be very large, possibly exponential in size. Thus the computational effort required to solve it can increase exponentially with the problem size in the worst case. In this paper, we propose a genetic algorithm (GA) approach used a priority-based chromosome for solving the bicriteria network optimization problem including maximum flow (MXF) model and minimum cost flow (MCF) model. The objective is to find the set of Pareto optimal solutions that give possible maximum flow with minimum cost. This paper also combines Adaptive Weight Approach (AWA) that utilizes some useful information from the current population to readjust weights for obtaining a search pressure toward a positive ideal point. Computer simulations show the several numerical experiments by using some difficult-to-solve network design problems, and show the effectiveness of the proposed method.

Recently, research on logistics caught more and more attention. One of the important issues on logistics system is to find optimal delivery routes with the least cost for products delivery. Numerous models have been developed for that reason. However, due to the diversity and complexity of practical problem, the existing models are usually not very satisfying to find the solution efficiently and convinently.In this paper, we treat a real-world logistics case with a company named ABC Co. ltd., in Kitakyusyu Japan. Firstly, based on the natures of this conveyance routing problem, as an extension of transportation problem (TP) and fixed charge transportation problem (fcTP) we formulate the problem as a minimum cost flow (MCF) model. Due to the complexity of fcTP, we proposed a priority-based genetic algorithm (pGA) approach to find the most acceptable solution to this problem. In this pGA approach, a two-stage path decoding method is adopted to develop delivery paths from a chromosome. We also apply the pGA approach to this problem, and compare our results with the current logistics network situation, and calculate the improvement of logistics cost to help the management to make decisions. Finally, in order to check the effectiveness of the proposed method, the results acquired are compared with those come from the two methods/ software, such as LINDO and CPLEX.

Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

Ford and Fulkerson around 50 years ago introduced flows over time by adding time dimension to the traditional network flow model. Road and air traffic control, production systems, communication networks (e.g., the internet) and financial flows are examples of this subject. What distinguishes flows over time from the traditional model is transit time on every arc which specifies the amount of time, flow units need to traverse the arc. In this model, flow rate entering an arc may change over time. One of the problems arising in this issue is the minimum cost flow over time problem which aims to determine an s-t flow over time f that satisfies demand d within given time horizon T at minimum cost. It is already shown that this problem is NP-hard, thus as usual a fair amount of study devoted to finding an efficient approximation algorithm for this issue. In this paper, we introduce an approximation algorithm for the T-length bounded discrete minimum cost flows over time problem.

Introduction to distribution logistics

In this work, we contribute a parallel implementation of the network simplex algorithm that is used for the solution of minimum cost flow problem. In the network simplex algorithm, finding an entering arc requires searching through many arcs to decide which one should be included in the spanning tree solution on the next iteration. We propose finding the entering arc in parallel as it often takes the majority of the execution time. A usual strategy is to pick the arc violating the optimality the most out of all possible candidates. Scanning all arcs can take quite some time, so it is common to consider only a fixed number of arcs which is referred as the block search pivoting rule. Arc scans can easily be done in parallel to find the best candidate as the calculations are independent of each other. We used shared memory parallelism using OpenMP along with vectorization using AVX instructions. We also tried adjusting block sizes to increase the parallel portion of the algorithm. Our dataset consists of various natural and synthetic graphs with sizes up to a billion arc. Our experiments show speedups up to four are possible, though they are typically lower.

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).

Decision support for truckload carriers in one-shot combinatorial auctions

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

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.