One-to-one disjoint path covers on multi-dimensional tori

The multi-dimensional torus is one of the most popular underlying topologies for massively parallel systems. In this study, we consider a non-bipartite n-dimensional torus where n≥2 and prove that for 1≤m≤2n, m vertex disjoint paths exist that cover all vertices between any two distinct vertices. In other words, we construct the one-to-one m-disjoint path cover of a non-bipartite torus for any m where 1≤m≤2n.     

Many-to-many disjoint path covers in hypercube-like interconnection networks with faulty elements

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G/sub 0/ /spl oplus/ G/sub 1/ and (G/sub 0/ /spl oplus/ G/sub 1/) /spl oplus/ (G/sub 2/ /spl oplus/ G/sub 3/ ), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G/sub i/ are satisfied, 0 /spl les/ i /spl les/ 3. We apply our main results to recursive circulant G(2/sup m/, 4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclasses includes twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k-DPC joining any k distinct source-sink pairs for any k /spl ges/ 1 and f /spl ges/ 0 such that f+2k /spl les/ m - 1.     

边故障K元3立方体的二不交路覆盖

研究具有故障边的[k]元3立方体的非指定二不交路覆盖问题。证明了在具有至多3条故障边的[k]元3立方体[Qk3]中,任意给定两个源点和两个汇点,则存在两条顶点不交的路[P1]和[P2],分别连接一个源点和汇点,且[V(P1)?V(P2)=V(Qk3)]。     

On a relaxed maximally disjoint path pair problem: a bicriteria approach

In some application areas in telecommunication and transportation networks, there are problems requiring the determination of pairs of paths, aiming at minimizing the number of links, or link groups that they share, and their total cost. In this paper, a new bicriteria algorithm is proposed to deal with this problem. The algorithm is based on ranking pairs of paths by order of the total cost, using an adaptation of a path‐ranking algorithm, after a suitable modification of the network topology. Nondominated solutions are then filtered by means of a dominance test. First, computational experiments are reported in order to assess the efficiency of the algorithm to calculate the whole set of nondominated pairs of paths. Second, we present computational results focused on the nondominated solutions close to the maximal disjoint pair (i.e., quasi‐disjoint pairs only, for a predefined admissible relaxation value) because in some application problems, such as shared risk link group pairs of paths, only those solutions have practical relevance.     

On approximating the longest path in a graph

We consider the problem of approximating the longest path in undirected graphs. In an attempt to pin down the best achievable performance ratio of an approximation algorithm for this problem, we present both positive and negative results. First, a simple greedy algorithm is shown to find long paths in dense graphs. We then consider the problem of finding paths in graphs that are guaranteed to have extremely long paths. We devise an algorithm that finds paths of a logarithmic length in Hamiltonian graphs. This algorithm works for a much larger class of graphs (weakly Hamiltonian), where the result is the best possible. Since the hard case appears to be that of sparse graphs, we also consider sparse random graphs. Here we show that a relatively long path can be obtained, thereby partially answering an open problem of Broderet al. To explain the difficulty of obtaining better approximations, we also prove hardness results. We show that, for any ε<1, the problem of finding a path of lengthn-n ^ε in ann-vertex Hamiltonian graph isNP-hard. We then show that no polynomial-time algorithm can find a constant factor approximation to the longest-path problem unlessP=NP. We conjecture that the result can be strengthened to say that, for some constant δ>0, finding an approximation of ration ^δ is alsoNP-hard. As evidence toward this conjecture, we show that if any polynomial-time algorithm can approximate the longest path to a ratio of , for any ε>0, thenNP has a quasi-polynomial deterministic time simulation. The hardness results apply even to the special case where the input consists of bounded degree graphs. D. Karger was supported by an NSF Graduate Fellowship, NSF Grant CCR-9010517, and grants from the Mitsubishi Corporation and OTL. R. Motwani was supported by an Alfred P. Sloan Research Fellowship, an IBM Faculty Development Award, grants from Mitsubishi and OTL, NSF Grant CCR-9010517, and NSF Young Investigator Award CCR-9357849, with matching funds from IBM, the Schlumberger Foundation, the Shell Foundation, and the Xerox Corporation, G. D. S. Ramkumar was supported by a grant from the Toshiba Corporation. Communicated by M. X. Goemans.     

Improved algorithms for finding length-bounded two vertex-disjoint paths in a planar graph and minmax k vertex-disjoint paths in a directed acyclic graph

This paper is composed of two parts. In the first part, an improved algorithm is presented for the problem of finding length-bounded two vertex-disjoint paths in an undirected planar graph. The presented algorithm requires O(n³bmin) time and O(n²bmin) space, where bmin is the smaller of the two given length bounds. In the second part of this paper, we consider the minmax k vertex-disjoint paths problem on a directed acyclic graph, where k?2 is a constant. An improved algorithm and a faster approximation scheme are presented. The presented algorithm requires O(nk+1Mk−1) time and O(nkMk−1) space, and the presented approximation scheme requires O((1/?)^k−1n2klogk−1M) time and O((1/?)^k−1n2k−1logk−1M) space, where ? is the given approximation parameter and M is the length of the longest path in an optimal solution.     

Cycles passing through a prescribed path in a hypercube with faulty edges

Let Qn denote an n-dimensional hypercube with n?2, P be a path of length h in Qn and F⊂E(Qn)\E(P). Recently, Tsai proved that if 1?h?n−1 and |F|?n−1−h, then in the graph Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if |F|?h−2. In this paper, we show that if 1?h?2n−3 and |F|?n−2−⌊h/2⌋, then in Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if P contains two edges of the same dimension or P is a shortest path and |F∩E(Qh)|?h−2, where Qh is the h-dimensional subcube containing the path P. Moreover, the upper bound 2n−3 of h is sharp and the upper bound n−2−⌊h/2⌋ of |F| is sharp for any given h with 1?h?2n−3.     

Linear time algorithms for counting the number of minimal vertex covers with minimum/maximum size in an interval graph

This paper presents efficient algorithms for an interval graph. These are (1) an algorithm for counting the number of minimum vertex covers, and (2) an algorithm for counting the number of maximum minimal vertex covers.     

Global exponential stability analysis for a class of neural networks with time delays

Employing Brouwer's fixed point theorem, matrix theory, we made a further investigation of a class of neural networks with delays in this paper. A family of sufficient conditions were given for checking global exponential stability. These results have important leading significance in the design and applications of globally stable neural networks with delays. Our results extended and improved some earlier publications. Copyright © 2011 John Wiley & Sons, Ltd.     

Tight upper bound on the number of edges in a bipartite K3,3-free or K5-free graph with an application

We show that an n-vertex bipartite K_3,3-free graph with n?3 has at most 2n−4 edges and that an n-vertex bipartite K₅-free graph with n?5 has at most 3n−9 edges. These bounds are also tight. We then use the bound on the number of edges in a K_3,3-free graph to extend two known NC algorithms for planar graphs to K_3,3-free graphs.     

Synthesis of a New Robust Exponential Sliding Mode Differentiator with its Observer Applications

This paper proposes a new robust exponential sliding mode differentiator for estimating the (n ? 1) derivatives of a non‐linear function. Finite time convergence, robustness, and exactness are also ensured analytically with the proposed methodology. In addition, the results of the proposed sliding mode differentiator are extended to derive theorems for the novel state (SO) and extended state observers (ESO), which would estimate the system states as well as uncertainties recursively in finite time. Finally, three examples are implemented to validate the proposed methodologies and the obtained simulations are compared with the previously developed methods in literature.     

Accentuating the rank positions in an agreement index with reference to a consensus order

In this paper, we propose an index that measures the agreement level between an individual opinion and a collective opinion when both are expressed by rankings of a set of alternatives. This index constitutes an interesting weighted version of the well‐known Kendall's ranks correlation index. The originality of the proposed index arises from the fact that it accounts for the relevance of the specific position of the alternatives in an individual order to quantify the agreement level of the individual order with respect to a collective temporary order. The paper also introduces a new consensus measure model. The core of the consensus model is the proposed agreement index. We present an illustrative example to describe the consensus process. We can obtain a faster convergence to a consensus solution using this new index compared to Kendall's index.