A novel approach for solving all-pairs shortest path problem in an interval-valued fuzzy network: Suggested modifications

Enayattabr et al. (Journal of Intelligent and Fuzzy Systems 37 (2019) 6865– 6877) claimed that till now no one has proposed an approach to solve interval-valued trapezoidal fuzzy all-pairs shortest path problems (all-pairs shortest path problems in which distance between every two nodes is represented by an interval-valued trapezoidal fuzzy number). Also, to fill this gap, Enayattabr et al. proposed an approach to solve interval-valued trapezoidal fuzzy all-pairs shortest path problems. In this paper, an interval-valued trapezoidal fuzzy shortest path problem is considered to point out that Enayattabr et al.’s approach fails to find correct shortest distance between two fixed nodes. Hence, it is inappropriate to use Enayattabr et al.’s approach in its present from. Also, the required modifications are suggested to resolve this inappropriateness of Enayattabr et al.’s approach.

Fine-Grained Reductions from Approximate Counting to Decision

In this article, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus, we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead and can therefore be applied to subpolynomial improvements such as the n 3 / exp(Θ (√ log n ))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF( m ) d for constant m can be solved in time n · poly ( d ) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time. We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O ( c n )-time algorithm for k -SAT. Then we prove that for all k , there is an O (( c + o (1)) n )-time algorithm for approximate # k -SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1+ɛ in time 2 o ( n )/ ɛ 2 (taking ɛ > 0 as part of the input).

Non-crossing Rectilinear Shortest Minimum Bend Paths in the Presence of Rectilinear Obstacles

The paper presents a new algorithm to determine the shortest, non-crossing, rectilinear paths in a twodimensional grid graph. The shortest paths are determined in a manner ensuring that they do not cross each other and bypass any obstacles present. Such shortest paths are applied in robotic chip design, suburban railway track layouts, routing traffic in wireless sensor networks, printed circuit board design routing, etc. When more than one equal length noncrossing path is present between the source and the destination, the proposed algorithm selects the path which has the least number of corners (bends) along the path. This feature makes the path more suitable for moving objects, such as unmanned vehicles. In the author’s scheme presented herein, the grid points are the vertices of the graph and the lines joining the grid points are the edges of the graph. The obstacles are represented by their boundary grid points. Once the graph is ready, an adjacency matrix is generated and the Floyd-Warshall all-pairs shortest path algorithm is used iteratively to identify the non-crossing shortest paths. To get the minimum number of bends in a path, we make a modification to the Floyd-Warshall algorithm, which is constitutes the main contribution of the author presented herein.