Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.Comment: 18 pages, 1 figur

Cosmic Magnetic Fields from Particle Physics

I review a number of particle-physics models that lead to the creation of magnetic fields in the early universe and address the complex problem of evolving such primordial magnetic fields into the fields observed today. Implications for future observations of the Cosmic Microwave Background (CMB) are briefly discussed.Comment: 8 pages, 1 figure, talk presented at the 7th International Symposium on Particles, Strings and Cosmology (PASCOS-99) at Granlibakken, Lake Tahoe, 10-16 Dec 1999, to appear in the proceeding

Magnetic Fields from the Electroweak Phase Transition

I review some of the mechanisms through which primordial magnetic fields may be created in the electroweak phase transition. I show that no magnetic fields are produced initially from two-bubble collisions in a first-order transition. The initial field produced in a three-bubble collision is computed. The evolution of fields at later times is discussed.Comment: Talk presented at the International Workshop on Particle Physics and the Early Universe (COSMO--97) in Ambleside, England, 15-19 Sept 1997. To appear in the proceedings (World Scientific). LaTeX, 5 page

The Origin of Cosmic Magnetic Fields

In this talk, I review a number of particle-physics models that lead to the creation of magnetic fields in the early universe and address the complex problem of evolving such primordial magnetic fields into the fields observed today. Implications for future observations of the Cosmic Microwave Background (CMB) are discussed. Focussing on first-order phase transitions in the early universe, I describe how magnetic fields arise in the collision of expanding true-vacuum bubbles both in Abelian and non-Abelian gauge theories.Comment: 9 pages, 1 figure, talk presented at the 3rd International Conference on Particle Physics and the Early Universe (COSMO-99), Trieste, Italy, 27 Sept - 3 Oct, 1999, to be published in the proceedings. Added reference

Approximating ATSP by Relaxing Connectivity

The standard LP relaxation of the asymmetric traveling salesman problem has been conjectured to have a constant integrality gap in the metric case. We prove this conjecture when restricted to shortest path metrics of node-weighted digraphs. Our arguments are constructive and give a constant factor approximation algorithm for these metrics. We remark that the considered case is more general than the directed analog of the special case of the symmetric traveling salesman problem for which there were recent improvements on Christofides' algorithm. The main idea of our approach is to first consider an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. For this relaxed problem, it is quite easy to give an algorithm with a guarantee of 3 on node-weighted shortest path metrics. More surprisingly, we then show that any algorithm (irrespective of the metric) for the relaxed problem can be turned into an algorithm for the asymmetric traveling salesman problem by only losing a small constant factor in the performance guarantee. This leaves open the intriguing task of designing a "good" algorithm for the relaxed problem on general metrics.Comment: 25 pages, 2 figures, fixed some typos in previous versio

Massive Gravity with N=1 local Supersymmetry

A consistent theory of massive gravity, where the graviton acquires mass by spontaneously breaking diffeomorphism invariance, is now well established. We supersymmetrize this construction using N =1 fields. Coupling to N = 1 supergravity is done by applying the rules of tensor calculus to construct an action invariant under local N = 1 supersymmetry. The supersymmetric action is shown, at the quadratic level, to be free of ghosts and have as its spectrum a massive graviton, two gravitinos with different masses, and a massive vector.Comment: change in wording, references adde