73 research outputs found
Subgraphs, Closures and Hamiltonicity
Closure theorems in hamiltonian graph theory are of the following type: Let G be a 2- connected graph and let u, v be two distinct nonadjacent vertices of G. If condition c(u,v) holds, then G is hamiltonian if and only if G + uv is hamiltonian. We discuss several results of this type in which u and v are vertices of a subgraph H of G on four vertices and c(u, v) is a condition on the neighborhoods of the vertices of H (in G). We also discuss corresponding sufficient conditions for hamiltonicity of G
A closure concept based on neighborhood unions of independent triples
The well-known closure concept of Bondy and Chvatal is based on degree-sums of pairs of nonadjacent (independent) vertices. We show that a more general concept due to Ainouche and Christofides can be restated in terms of degree-sums of independent triples. We introduce a closure concept which is based on neighborhood unions of independent triples and which also generalizes the closure concept of Bondy and Chvatal
Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing
We study the Parallel Task Scheduling problem with a
constant number of machines. This problem is known to be strongly NP-complete
for each , while it is solvable in pseudo-polynomial time for each . We give a positive answer to the long-standing open question whether
this problem is strongly -complete for . As a second result, we
improve the lower bound of for approximating pseudo-polynomial
Strip Packing to . Since the best known approximation algorithm
for this problem has a ratio of , this result
narrows the gap between approximation ratio and inapproximability result by a
significant step. Both results are proven by a reduction from the strongly
-complete problem 3-Partition
Applications of Two-Body Dirac Equations to the Meson Spectrum with Three versus Two Covariant Interactions, SU(3) Mixing, and Comparison to a Quasipotential Approach
In a previous paper Crater and Van Alstine applied the Two Body Dirac
equations of constraint dynamics to the meson quark-antiquark bound states
using a relativistic extention of the Adler-Piran potential and compared their
spectral results to those from other approaches, ones which also considered
meson spectroscopy as a whole and not in parts. In this paper we explore in
more detail the differences and similarities in an important subset of those
approaches, the quasipotential approach. In the earlier paper, the
transformation properties of the quark-antiquark potentials were limited to a
scalar and an electromagnetic-like four vector, with the former accounting for
the confining aspects of the overall potential, and the latter the short range
portion. A part of that work consisted of developing a way in which the static
Adler-Piran potential was apportioned between those two different types of
potentials in addition to covariantization. Here we make a change in this
apportionment that leads to a substantial improvement in the resultant
spectroscopy by including a time-like confining vector potential over and above
the scalar confining one and the electromagnetic-like vector potential. Our fit
includes 19 more mesons than the earlier results and we modify the scalar
portion of the potential in such a way that allows this formalism to account
for the isoscalar mesons {\eta} and {\eta}' not included in the previous work.
Continuing the comparisons made in the previous paper with other approaches to
meson spectroscopy we examine in this paper the quasipotential approach of
Ebert, Faustov, and Galkin for a comparison with our formalism and spectral
results.Comment: Revisions of earlier versio
Approximation Algorithms for Scheduling Parallel Jobs: Breaking the Approximation Ratio of 2
In this paper we study variants of the non-preemptive parallel job scheduling problem in which the number of machines is polynomially bounded in the number of jobs. For this problem we show that a schedule with length at most (1 + ε)OPT can be calculated in polynomial time. Unless P = NP, this is the best possible result (in the sense of approximation ratio), since the problem is strongly NP-hard. For the case, where all jobs must be allotted to a subset of consecutive machines, a schedule with length at most (1.5 + ε)OPT can be calculated in polynomial time. The previously best known results are algorithms with absolute approximation ratio 2. Furthermore, we extend both algorithms to the case of malleable jobs with the same approximation ratios
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