A near-optimum parallel algorithm for bipartite subgraph problem using gradient ascent learning algorithm of the Hopfield neural networks is presented. This parallel algorithm, uses the Hopfield neural network updating to get a near-maximum bipartite subgraph and then performs gradient ascent learning on the Hopfield network to help the network escape from the state of the near-maximum bipartite subgraph until the state of the maximum bipartite subgraph or better one is obtained. A large number of instances have been simulated to verify the proposed algorithm, with the simulation result showing that our algorithm finds the solution quality is superior to that of best existing parallel algorithm. We also test the proposed algorithm on maximum cut problem. The simulation results also show the effectiveness of this algorithm.

In this paper we addresses the problem of finding feasible solutions for the Group Multicast Routing Problem (GMRP). This problem is a generalization of the multicast routing problem whereby every member of the group is allowed to multicast messages to other members from the same group. The routing problem involves the construction of a set of low cost multicast trees with bandwidth requirements for all the group members in the network. We first prove that the problem of finding feasible solutions to GMRP is NP-complete. Following that we propose a new heuristic algorithm for constructing feasible solutions for GMRP. Simulation results show that our proposed algorithm is able to achieve good performance in terms of its ability of finding feasible solutions whenever one exist.

A gradient ascent learning algorithm of the Hopfield neural networks for graph planarization is presented. This learning algorithm uses the Hopfield neural network to get a near-maximal planar subgraph, and increases the energy by modifying parameters in a gradient ascent direction to help the network escape from the state of the near-maximal planar subgraph to the state of the maximal planar subgraph or better one. The proposed algorithm is applied to several graphs up to 150 vertices and 1064 edges. The performance of our algorithm is compared with that of Takefuji/Lee's method. Simulation results show that the proposed algorithm is much better than Takefuji/Lee's method in terms of the solution quality for every tested graph.

Tree task structures occur frequently in many applications where parallelization may be desirable. We present a formal treatment of non-preemptively scheduling task trees on distributed memory multiprocessors and show that the fundamental problems of scheduling (i) a task tree in absence of any inter-task communication on a fixed number of processors and (ii) a task tree with inter-task communication on an unbounded number of processors are NP-complete. For task trees that satisfy certain constraints, we present an optimal scheduling algorithm. The algorithm is shown optimal over a wider set of task trees than previous works.

This paper deals with a study of a problem for finding the minimum-cost spanning tree with a response-time bound. The relation of cost and response-time is given as a monotonous decreasing and convex function. Regarding communication bandwidth as cost in an information network, this problem means a minimum-cost tree shaped routing for response-time constrained broadcasting, where any response-time from a root vertex to other vertex is less than a given time bound. This problem is proven to be NP-hard and consists of the minimum-cost assignment to a rooted tree and the minimum-cost tree finding. A nonlinear programming algorithm solves the former problem for the globally optimal solution. For the latter problem, different types of heuristic algorithms evaluate to find a near optimal solution experimentally.

In a multihop network, radio packets are often relayed through inter-mediate stations (repeaters) in order to transfer a radio packet from a source to its destination. We consider a scheduling problem in a multihop network using a graphtheoretical model. Let D=(V,A) be the digraph with a vertex set V and an arc set A. Let f be a labeling of positive integers on the arcs of A. The value of f(u,v) means a frequency band assigned on the link from u to v. We call f antitransitive if f(u,v)f(v,w) for any adjacent arcs (u,v) and (v,w) of D. The minimum antitransitive-labeling problem is the problem of finding a minimum antitransitive-labeling such that the number of integers assigned in an antitransitive labeling is minimum. In this paper, we prove that this problem is NP-hard, and we propose a simple distributed approximation algorithm for it.

A reallocation problem is defined as determining whether blocks can be moved from their current boxes to their destination boxes in the number of moves less than or equal to a given positive integer. This problem is defined simply and has many practical applications. We previously proved the following results: The reallocation problem such that the block volume is restricted to one volume unit for all blocks can be solved in linear time. But the reallocation problem such that the block volume is not restricted is NP-complete, even if no blocks have bypass boxes. But the computational complexity of the reallocation problems such that the range of the block volume is restricted has not yet been known. We prove that the reallocation problem is NP-complete even if the block volume is restricted to one or two volume units for all blocks and no blocks have bypass boxes. Furthermore, we prove that the reallocation problem is NP-complete, even if there are only two blocks such that each block has the volume units of fixed positive integer greater or equal than two, the volume of the other blocks is restricted to one volume unit, and no blocks have bypass boxes. Next, we consider such a block that must stays in a same box both in the initial state and in the final state but can be moved to another box for making room for other blocks. We prove that the reallocation problem such that an instance has such blocks is also NP-complete even if the block volume is restricted to one volume unit for all blocks.

Given a Petri net N=(P, T, E), a siphon is a set S of places such that the set of input transitions to S is included in the set of output transitions from S. Concerning extraction of one or more minimal siphons containing a given specified set Q of places, the paper shows several results on polynomial time solvability and NP-completeness, mainly for the case |Q| 1.

A novel combinatorial optimization algorithm called 2-stage discrete optimization method (2DOM) is proposed for the largest common subgraph problem (LCSP) in this paper. Given two graphs G=(V1, E1) and H=(V2, E2), the goal of LCSP is to find a subgraph G'=(V1', E1') of G and a subgraph H'=(V2', E2') of H such that G' and H' are not only isomorphic to each other but also their number of edges is maximized. The two graphs G' and H' are isomorphic when |V1'|=|V2'| and |E1'|=|E2'|, and there exists one-to-one vertex correspondence f: V1' V2' such that {u, v} E1' if and only if{f(u), f(v)} E2'. LCSP is known to be NP-complete in general. The 2DOM consists of a construction stage and a refinement stage to achieve the high solution quality and the short computation time for large size difficult combinatorial optimization problems. The construction stage creates a feasible initial solution with considerable quality, based on a greedy heuristic method. The refinement stage improves it keeping the feasibility, based on a random discrete descent method. The performance is evaluated by solving two types of randomly generated 1200 LCSP instances with a maximum of 500 vertices for G and 1000 vertices for H. The simulation result shows the superiority of 2DOM to the simulated annealing in terms of the solution quality and the computation time.

A problem of obtaining an optimal file transfer on a file transmission net N is to consider how to distribute, with a minimum total cost, copies of a certain file of information from some vertices to others on N by the respective vertices' copy demand numbers. This paper proves such a problem to be NP-hard in general.

An approximation algorithm composed of a digital neural network (DNN) and a modified greedy algorithm (MGA) is presented for the board-level routing problem (BLRP) in a logic emulation system based on field-programmable gate arrays (FPGA's) in this paper. For a rapid prototyping of large scale digital systems, multiple FPGA's provide an efficient logic emulation system, where signals or nets between design partitions embedded on different FPGA's are connected through crossbars. The goal of BLRP, known to be NP-complete in general, is to find a net assignment to crossbars subject to the constraint that all the terminals of any net must be connected through a single crossbar while the number of I/O pins designated for each crossbar m is limited in an FPGA. In the proposed combination algorithm, DNN is applied for m = 1 and MGA is for m 2 in order to achieve the high solution quality. The DNN for the N-net-M-crossbar BLRP consists of N M digital neurons of binary outputs and range-limited non-negative integer inputs with integer parameters. The MGA is modified from the algorithm by Lin et al. The performance is verified through massive simulations, where our algorithm drastically improves the routing capability over the latest greedy algorithms.

This paper investigates the relations between the computational complexity and the restrictions for several problems that determine whether a given graph with edge costs and edge lengths has a spanning subgraph with such restrictions as the diameter, the connectivity, and the NA-distance and the NA-(edge)-connectivity proposed and investigated in [1]-[5]. The NA-distance and the NA-(edge)-connectivity are the measures for the distance and the connectivity between a vertex and a vertex subset (area). In this paper we prove that the minimum diameter spanning subgraph problem considering the restrictions of the diameter and the sum of edge costs is NP-complete even if the following restrictions are satisfied: all edge costs and all edge lengths are equal to one, and the upper bound of the diameter is restricted to four. Next, we prove that the minimum NA-distance spanning subgraph problem considering the restrictions of the NA-distances and the sum of edge costs is NP-complete even if the following conditions are satisfied: all edge costs and all edge lengths are equal to one, the upper bound of the NA-distance is restricted to four, each area is composed of a vertex, and the number of areas is restricted to two. Finally, we investigate the preserving NA-distance and NA-edge-connectivity spanning subgraph problem considering the preservations of the NA-distances and the NA-edge-connectivity and the restrictions of the sum of edge costs, and prove that a sparse spanning subgraph can be constructed in polynomial time if all edge costs are equal to one.

The lower-bounded p-collection problem is the problem where to locate p sinks in a flow network with lower bounds such that the value of a maximum flow is maximum. This paper discusses the cover problems corresponding to the lower bounded p-collection problem. We consider the complexity of the cover problem, and we show polynomial time algorithms for its subproblems in a network with tree structure.

A neural network of massively interconnected digital neurons is presented for the total coloring problem in this paper. Given a graph G (V, E), the goal of this NP-complete problem is to find a color assignment on the vertices in V and the edges in E with the minimum number of colors such that no adjacent or incident pair of elements in V and E receives the same color. A graph coloring is a basic combinatorial optimization problem for a variety of practical applications. The neural network consists of (N+M) L neurons for the N-vertex-M-edge-L-color problem. Using digital neurons of binary outputs and range-limited non-negative integer inputs with a set of integer parameters, our digital neural network is greatly suitable for the implementation on digital circuits. The performance is evaluated through simulations in random graphs with the lower bounds on the number of colors. With a help of heuristic methods, the digital neural network of up to 530, 656 neurons always finds a solution in the NP-complete problem within a constant number of iteration steps on the synchronous parallel computation.

A digital neural network approach is presented for the multilayer channel routing problem with the objective of crosstalk minimization in this paper. As VLSI fabrication technology advances, the reduction of crosstalk between interconnection wires on a chip has gained important consideration in VLSI design, because of the closer interwire spacing and the circuit operation at higher frequencies. Our neural network is composed of N M L digital neurons with one-bit output and seven-bit input for the N-net-M-track-2L-layer problem using a set of integer parameters, which is greatly suitable for the implementaion on digital technology. The digital neural network directly seeks a routing solution of satisfying the routing constraint and the crosstalk constraint simultaneously. The heuristic methods are effectively introduced to improve the convergence property. The performance is evaluated through solving 10 benchmark problems including Deutsch difficult example in 2-10 layers. Among the existing neural networks, the digital neural network first achieves the lower bound solution in terms of the number of tracks in any instance. Through extensive simulation runs, it provides the best maximum crosstalks of nets for valid routing solutions of the benchmark problems in multilayer channels.

In this paper we extend the p-collection problem to a flow network with lower bounds, and call the extended problem the lower-bounded p-collection problem. First we discuss the complexity of this problem to show NP-hardness for a network with path structure. Next we present a linear time algorithm for the lower-bounded 1-collection problem in a network with tree structure, and a pseudo-polynomial time algorithm with dynamic programming type for the lower-bounded p-collection problem in a network with tree structure. Using the pseudo-polynomial time algorithm, we show an exponential algorithm, which is efficient in a connected network with few cycles, for the lower-bounded p-collection problem.

The reallocation problem is defined as determining whether products can be moved from their current storehouses to their target storehouses in a number of moves that is less than or equal to a given number. This problem is defined simply and has many practical applications. We previously presented necessary and sufficient conditions whether an instance of the reallocation problem is feasible, as well as a linear-time algorithm that determines whether aall products can be moved, when the volume of the products is restricted to one. However, a linear-time algorithm that generates the order of moving the products has not been reported yet. Such an algorithm is proposed in this paper. We have also previously proved that the reallocation problem is NP-complete in the strong sense when the volume of the products is not restricted and the products have evacuation storehouses show that the reallocation problem is NP-complete in the strong sense even when none of the products has evacuation storehouses.

The Hopfield neural network for optimization problems often falls into local minima. To escape from the local minima, the neuron unit in the neural network is modified to become an oscillatory unit by adding a simple self-feedback circuit. By combining the oscillatory unit with an energy-value extraction circuit, an oscillatory neural network is constructed. The network can repeatedly extract solutions, and can simultaneously evaluate them. In this paper, the network is applied to four NP-complete problems to demonstrate its generality and efficiency. The network can solve each problem and can obtain better solutions than the original Hopfield neural network and simple algorithms.

A siphon (or alternatively a structutal deadlock) of a Petri net is defined as a set S of places such that existence of any edge from a transition t to a place of S implies that there is an edge from some place of S to t. A minimal siphon is a siphon such that any proper subset is not a siphon. The results of the paper are as follows. (1) The problem of deciding whether or not a given Petri net has a minimum siphon (i.e., a minimum-cardinality minimal siphon) is NP-complete. (2) A polynomial-time algorithm to find, if any, a minimal siphon or even a maximal calss of mutually disjoint minimal siphons of a general Petri net is proposed.

A minimal siphon (or alternatively a structural deadlock) of a Petri net is defined as a minimal set S of places such that existence of any edge from a transition t to a place of S implies that there is an edge from some place of S to t. The subject of the paper is to find a minimal siphon containing a given set of specified places of a general Petri net.