Some biological experiments show that the tubular structures of Physarum polycephalum are often analogous to those of Steiner trees. Therefore, the emerging Physarum-inspired Algorithms (PAs) have the potential of computing Steiner trees. In this paper, we propose two PAs to solve the Steiner Tree Problem in Graphs (STPG). We apply some widely-used artificial and real-world VLSI design instances to evaluate the performance of our PAs. The experimental results show that: 1) for instances with hundreds of vertices, our first PA can find feasible solutions with an average error of 0.19 , while the Genetic Algorithm (GA), the Discrete Particle Swarm Optimization (DPSO) algorithm and a widely-used Steiner tree approximation algorithm: the Shortest Path Heuristic (SPH) algorithm can only find feasible solutions with an average error above 4.96 ; and 2) for larger instances with up to tens of thousands of vertices, where our first PA, GA and DPSO are too slow to be used, our second PA can find feasible solutions with an average error of 3.69 , while SPH can only find feasible solutions with an average error of 6.42 . These experimental results indicate that PAs can compute Steiner trees, and it may be preferable to apply our PAs to solve STPG in some cases.
Use @math to signify the threshold value of edge conductivity. Edges with conductivities smaller than @math will be cut from the network. Ultimately, if there is a unique shortest path between the source node and the sink node, then this path can be found by iteratively updating edge conductivities and cutting edges @cite_26 . However, even though Physarum Solver can solve the shortest path problem, it cannot solve STPG directly because there are only two terminals in its model, which are the source node and the sink node, while there are multiple terminals in STPG. Thus, new PAs are required to solve STPG.
@cite_26
2095284441 2951159514 2123562143 2121638441
We consider the problem of finding the minimum capacity cut in a network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut separating a designated source node s from a designated sink node t, and by varying the sink node one can find a minimum cut in G as a sequence of at most 2n - 2 maximum flow problems. We then show how to reduce the running time of these 2n - 2 maximum flow algorithms to the running time for solving a single maximum flow problem. The resulting running time is O(nm log n 2 m) for finding the minimum cut in either a directed or undirected network. The algorithm also determines the arc connectivity of either a directed or undirected network in O(nm) steps. Given a large edge-weighted network @math with @math terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of @math is to construct a : a small network @math with the same @math terminals, in which the minimum cut value between every bipartition of terminals is the same as in @math . This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS '98], who proved that such @math of size at most @math always exists. Obviously, by having access to the smaller network @math , certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doubly-exponential to only singly-exponential, both for planar and for general graphs. Our first and main result is that every @math -terminal planar network admits a mimicking network @math of size @math , which is moreover a minor of @math . On the other hand, some planar networks @math require @math . For general networks, we show that certain bipartite graphs only admit mimicking networks of size @math , and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use @math machine words. The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures. We study the Steiner tree problem and the single-cable single-sink network design problem under a two-stage stochastic model with recourse and finitely many scenarios. In these models, some edges are purchased in a first stage when only probabilistic information about the second stage is available. In the second stage, one of a finite number of specified scenarios is realized, which results in the set of terminals becoming known and the opportunity to purchase additional edges (under an inflated cost function) to augment the first-stage solution. We provide constant factor approximation algorithms for these problems by rounding the linear relaxation of IP formulations of the problems. Our algorithms involve solving the linear relaxation first, followed by a primal-dual routine that is guided by the LP solution. We also show that because our bounds are local (the cost of each component is bounded by its cost in the LP solution), we are able to obtain bounds that guard against a form of downside risk.