On input read-modes of alternating Turing machines

AbstractA number of input read-modes of Turing machines have appeared in the literature. To investigate the differences among these input read-modes, we study log-time alternating Turing machines of constant alternations. For each fixed integer k ⩾ 1 and for each read-mode, a precise circuit characterization is established for log-time alternating Turing machines of k alternations, which is a nontrivial refinement of Ruzzo's circuit characterization of alternating Turing machines. These circuit characterizations indicate clearly the differences among the input read-modes. Complete languages in strong sense for each level of the log-time hierarchy are presented, refining a result by Buss. An application of these results to computational optimization problems is described

On strong Menger-connectivity of star graphs

AbstractMotivated by parallel routing in networks with faults, we study the following graph theoretical problem. Let G be a graph of minimum vertex degree d. We say that G is strongly Menger-connected if for any copy Gf of G with at most d−2 nodes removed, every pair of nodes u and v in Gf are connected by min{degf(u),degf(v)} node-disjoint paths in Gf, where degf(u) and degf(v) are the degrees of the nodes u and v in Gf, respectively. We show that the star graphs, which are a recently proposed attractive alternative to the widely used hypercubes as network models, are strongly Menger-connected. An algorithm of optimal running time is developed that constructs the maximum number of node-disjoint paths of nearly optimal length in star graphs with faults

On PTAS for planar graph problems

Approximation algorithms for a class of planar graph problems, including planar independent set, planar vertex cover and planar dominating set, were intensively studied. The current upper bound on the running time of the polynomial time approximation schemes (PTAS) for these planar graph problems is of 2O(1/∈ )nO(1). Here we study the lower bound on the running time of the PTAS for these planar graph problems. We prove that there is no PTAS of time 2=(√(1/∈ )nO(1) for planar independent set, planar vertex cover and planar dominating set unless an unlikely collapse occurs in parameterized complexity theory. For the gap between our lower bound and the current known upper bound, we speci cally show that to further improve the upper bound on the running time of the PTAS for planar vertex cover, we can concentrate on planar vertex cover on pla- nar graphs of degree bounded by three.4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI

Graph Imbeddings and Overlap Matrices (Preliminary Report)

Mohar has shown an interesting relationship between graph imbeddings and certain boolean matrices. In this paper, we show some interesting properties of this kind of matrices. Using these properties, we give the distributions of nonorietable imbeddings of several interesting infinite families of graphs, including cobblestone paths, closed-end ladders for which the distributions of orientable imbeddings are known