Graph-theoretic Problem Research Articles

Formal timing analysis for distributed real-time programs

A static analysis method for verifying timing properties of real-time distributed programs is presented. The goal is to calculate the worst-case response time of concurrent tasks which run mainly independently but share, and may have to wait for, logical or physical devices. For such tasks, the determination of the worst-case waiting time is a crucial problem because of the unpredictable order of synchronization events. We investigate the class of distributed Client-Server programs in which independent, time-critical tasks (clients) are synchronized only through additional server tasks, playing the role of monitors or resource managers. This model follows well-known real-time design guidelines for distributed ADA programs proposed to enhance schedulability and synchronization analysis. Our formal analysis approach is flow graph oriented. It leads to generating reduced program paths each of which represents a sequence of ordered local and global operations, thus transforming and reducing the original problem of computing the worst-case waiting time of a concurrent task into a graph-theoretic problem of calculating the maximal blocking time for one of its corresponding program paths. While local operations are completely independent global operations require mutually exclusive access to shared resources. We prove that computing the worst-case blocking time for a program path is NP-complete. Even for a reduced problem solution—which would yield a good upper bound for the worst-case blocking time—there was a conjecture maintained over many years that this problem was NP-complete. A major result of this paper is to show that this is wrong. Instead, we construct a polynomial solution algorithm, and we prove its correctness. The effectiveness and complexity of our method are discussed, with particular emphasis on distributed real-time debugging.

A SELF-STABILIZING DISTRIBUTED ALGORITHM FOR ALL-PAIRS SHORTEST PATH PROBLEM

In this paper we propose a simple and elegant distributed self-stabilizing algorithm to solve the all-pairs shortest path problem in graph theory. We have proved the correctness of our algorithm without using a bounded function (which is customary for proving correctness of self-stabilizing algorithms); the new approach is simple and seems to have potential in developing self-stabilizing algorithms for other graph-theoretical problems.

Scheduling of incompatible jobs on unrelated machines

We study a general scheduling problem where each job j∈J has an execution time e(j, Mi)∈ℝ+ on a machine of type Mi∈M and where we have an incompatibility relation between the jobs. Given a price p(Mi)∈ℝ+ for each machine of type Mi and a deadline d∈IR+, we consider the problem to select a minimum cost set of machines for which there exists a schedule with make-span at most d. We propose an approximation method where the solution is at most [Formula: see text] away from the optimum. The method is based on a graph theoretical problem which we have studied for several graph classes.

Genetic algorithm for node partitioning problem and applications in VLSI design

The group of classical graph-theoretic problems, including graph colouring, clique cover, and maximal clique, may be viewed as instances of a general node partitioning problem (NPP). A wide variety of real life problems can be modelled as instances of NPP. Finding an optimal partition for the NPP is said to be NP-complete. In this work a stochastic search by a genetic algorithm (GA) is employed to find a near optimal solution for the NPP. The critical aspects of the GA for NPP, such as the solution representation by elegant data structure, together with genetic operations and selection policies employed in the GA procedure, are also presented. The proposed GA does not require the number of disjunct node sets to be given a priori. Three application problems is VLSI design are solved as instances of NPP. The experimental results presented in each case of these application problems bring out the efficacy of genetic algorithms.

On minimizing the number of L-shaped channels in building-block layout (VLSI)

Given a building-block layout of rectangular modules, the routing area is to be decomposed into straight and L-shaped channels and routed in a certain order. Since straight channels are easier to route and channel routers usually produce near-optimal results, it is desirable to minimize the number of L-shaped channels used in such a decomposition. An algorithm for minimizing the number of L-shaped channels used in the routing area decomposition is presented. The algorithm is based on a careful study of the structure of layouts of rectangular modules and a transformation of the original problem to a graph theoretical problem. For examples of up to 136 channels, the algorithm took less than one tenth of a second on a SUN SPARC station 1 to finish the computation and obtained up to 29% reduction in the mumber of L-shaped channels over the results produced by the algorithm of W.M. Dai et al. (1985). >

A self-stabilizing algorithm for coloring bipartite graphs

We propose a new simple self-stabilizing distributed algorithm for coloring a bipartite graph. The algorithm has been proved to be correct, and the simple reasoning-based proof technique seems to have potential to be useful in proving correctness of other self-stabilizing algorithms for graph theoretical problems.

Checking robust nonsingularity is NP-hard

We consider the following problem: givenk+1 square matrices with rational entries,A0,A1,...,Ak, decide ifA0+r1A1+···+rkAk is nonsingular for all possible choices of real numbersr1, ...,rk in the interval [0, 1]. We show that this question, which is closely related to the robust stability problem, is NP-hard. The proof relies on the new concept ofradius of nonsingularity of a square matrix and on the relationship between computing this radius and a graph-theoretic problem.

Configurations and graphs

The main aim of this paper is not to present new results but to give a short survey on some relations between graphs and configurations. In the first part the analogy between configurations and regular bipartite graphs is described. Some instances of parallel research in graph theory and configuration theory are mentioned. In the second part certain graph-theoretic problems related to configurations are discussed.

Multicast communication in multicomputer networks

Efficient routing of messages is a key to the performance of multicomputers. Multicast communication refers to the delivery of the same message from a source node to an arbitrary number of destination nodes. While multicast communication is highly demanded in many applications, most of the existing multicomputers do not directly support this service; rather it is indirectly supported by multiple one-to-one or broadcast communications, which result in more network traffic and a waste of system resources. The authors study routing evaluation criteria for multicast communication under different switching technologies. Multicast communication in multicomputers is formulated as a graph theoretical problem. Depending on the evaluation criteria and switching technologies, they study three optimal multicast communication problems, which are equivalent to the finding of the following three subgraphs: optimal multicast path, optimal multicast cycle, and minimal Steiner tree, where the interconnection of a multicomputer defines a host graph. They show that all these optimization problems are NP-complete for the popular 2D-mesh and hypercube host graphs. Heuristic multicast algorithms for these routing problems are proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Maximumk-covering of weighted transitive graphs with applications

A weighted transitive graph is considered where each vertex is assigned a positive weight. Given a positive integer k, the maximum k-covering problem is to find k disjoint cliques covering a set of vertices with maximum total weight. An O(kn/sup 2/) time algorithm to solve the problem in a transitive graph is proposed, where n is the number of vertices. Based on this algorithm the weighted version of a number of VLSI problems can be solved-for example, k-layer topological via minimization, assignment of intervals into k tracks (with applications to channel routing), k-layer subset of multiterminal nets, and k-layer subset of nets in a leveled routing region (e.g. standard cells). The proposed technique solves several fundamental geometric and graph-theoretic problems. >

A SELF-STABILIZING DISTRIBUTED ALGORITHM TO CONSTRUCT BFS SPANNING TREES OF A SYMMETRIC GRAPH

We propose a simple and efficient self-stabilizing distributed algorithm to construct the breadth first search (BFS) spanning tree of an arbitrary connected symmetric graph. We develop a completely new direct approach of graph theoretical reasoning to prove the correctness of our algorithm. The approach seems to have potential to have applications in proving correctness of other self-stabilizing algorithms for graph theoretical problems.

Routing with Polynomial Communication-Space Trade-Off

This paper presents a family of memory-balanced routing schemes that use relatively short paths while storing relatively little routing information. The quality of the routes provided by a scheme is measured in terms of their stretch, namely, the maximum ratio between the length of a route connecting some pair of processors and their distance. The hierarchical schemes$\mathcal{H}_k $ (for every integer $k\geqq 1$) presented in this paper guarantee a stretch factor of $O( k^2 )$ on the length of the routes and require storing at most $O( k \cdot n^{1/k} \cdot \log n\log D )$ bits of routing information per vertex in an n-processor network with diameter D. The schemes are name independent and applicable to general networks with arbitrary edge weights. This improves on previous designs whose stretch bound was exponential in k. The proposed schemes are based on a new efficient solution to a certain graph-theoretic problem concerning sparse graph covers. The new cover technique has already found several other applications in the area of distributed computing.

An Optimal Algorithm for the Maximum Two-Chain Problem

Given a point set $\rho $, a chain is a subset $\mathcal{C} \subseteq \rho $ of points in which, for any two points, one is dominated by the other. A two-chain is a subset of $\rho $ that can be partitioned into two chains. A two-chain with maximum cardinality among all possible two-chains is called a maximum two-chain. This paper presents a $\Theta ( n \log n )$ time and $\Theta ( n )$ space algorithm for finding a maximum two-chain in a point set $\rho $, where $n = | \rho |$. Maximum two-chain has applications in, for example, graph-theoretic problems, VLSI layout, and sequence manipulation.

M-Median and m-Center Problems with Mutual Communication: Solvable Special Cases

In this paper, we consider the network version of the m-median problem with mutual communication (MMMC). We reformulate this problem as a graph theoretic node selection problem defined on a special graph. We give a polynomial time algorithm to solve the node selection problem when the flow graph (graph that denotes the interaction between pairs of new facilities in MMMC) has a special structure. We also show that with some modification in the algorithm for MMMC, the m-center problem with mutual communication can also be solved when the flow graph has a special structure.

Distributed subcube identification algorithms for reliable hypercubes

The subcube identification problem is addressed. The number of faulty nodes is assumed to be bounded above by the dimension of the hypercube. The problem is modeled as a graph theoretical problem and shown to be NP complete. Based on this result, two heuristic identification algorithms are presented.

Weighted arcs, the finite radon transform and a Ramsey problem

We establish a link between the theory of (k, v)-arcs in affine planes and a graph theoretic Ramsey problem: A (n, k)-coloring of the complete graphK u is a coloring of the edges ofK u withk colours such that monochromatic connected subgraphs have at mostn vertices. The Ramsey numberf(n, k) is the smallestu such thatK u does not admit a (n, k)-coloring. Let[Figure not available: see fulltext.] be an affine plane of orderq. Aweighted v-arc is a function w:[Figure not available: see fulltext.]? ? such that "line integrals" are at mostv. Themass ofw is the sum of all the weights. In certain cases the maximum massM q (v) can be determined (maximum over all weightedv-arcs and all affine planes of orderq). This yields the exact value off(n, q + 1) for all large enoughn.

Unit disk graphs

Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in computational geometry. We show that many standard graph theoretic problems remain NP-complete on unit disk graphs, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs. In contrast, we give a polynomial time algorithm for finding cliques when the geometric representation (circles in the plane) is provided.

The discrete p-dispersion problem

Considered is the problem of selecting p out of n given points in some space, such that the minimum distance between pairs of selected points is maximized. This objective may be appropriate if the selected points correspond to facility sites and the objective is to have as ‘dispersed’ a set as possible. This problem is NP-complete. Related graph theoretical problems are discussed, integer programming models are proposed, and an outline is given on a line search procedure to solve this problem optimally. Using a branch-and-bound procedure, problems with n = 40 and p = 16 can be solved on a microcomputer. The heuristic, developed to get an initial lower bound, finds an optimal solution for most of our random test problems. Also described is an extension to the basic problem that allows for preselected points, which may correspond to existing facility locations. This more general version can be solved by slight modifications of the algorithms.

Yield enhancement of field programmable logic arrays by inherent component redundancy

A complete technique that does not use any additional components for enhancing the yield of field-programmable logic arrays (FPLAs) is presented. In this approach, the inherent sparsity (absence of devices at crosspoints) of programmable logic arrays (PLAs) is utilized to mask certain types of manufacturing defects within the unprogrammed FPLAs, thus reclaiming chips which are otherwise discarded. Two categories of faults (called type 1 and type 2) are considered. Type-1 faults, which can be diagnosed a priori, are considered first. After diagnosing type 1 faults, the mask can be reconfigured around the faulty crosspoints. A streamlined bipartite matching algorithm is presented to enhance the speed of this reconfiguration. The uniqueness of the approach is that the programming of an FPLA is formulated as a graph theoretic problem for which a polynomial time solution exists. Type-2 faults in general cannot be diagnosed a priori. Therefore, a dynamic technique is presented for the repair of type-2 faults. Unused product lines of the FPLA are utilized for the repair. With a sufficient number of excess product lines, it is shown that a defective FPLA is guaranteed to be rendered usable. A probability measure for the usability of defective FPLA is obtained both with and without the implementation of this technique. Computer studies have shown that FPLAs with even a large number of defects can be successfully repaired, thereby increasing the yield.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ordered vertex removal and subgraph problems

Several new graph theoretic problems, which arise naturally from existing coloring algorithms, are defined. The complementary High Degree Vertex Removal Problem and Low Degree Vertex Removal Problem are both shown to be NP-complete. The Low Degree Subgraph Problem is defined and shown to be NP-complete, whereas, the “complementary” problem for high degree subgraphs was previously shown to be P-complete. Since the obvious sequential algorithms for computing the low degree subgraph and high degree subgraph are based on high degree vertex removal and low degree vertex removal, respectively, we find this is an interesting result. The “greedy” versions of the vertex removal and subgraph are shown to be P-complete. In addition, a natural lexicographic version of the Low Degree Subgraph Problem is shown to be NP-complete.