Directed Steiner Tree and the Lasserre Hierarchy

The goal for the Directed Steiner Tree problem is to find a minimum cost tree in a directed graph G=(V,E) that connects all terminals X to a given root r. It is well known that modulo a logarithmic factor it suffices to consider acyclic graphs where the nodes are arranged in L <= log |X| levels. Unfortunately the natural LP formulation has a |X|^(1/2) integrality gap already for 5 levels. We show that for every L, the O(L)-round Lasserre Strengthening of this LP has integrality gap O(L log |X|). This provides a polynomial time |X|^{epsilon}-approximation and a O(log^3 |X|) approximation in O(n^{log |X|) time, matching the best known approximation guarantee obtained by a greedy algorithm of Charikar et al.Comment: 23 pages, 1 figur

EDF-schedulability of synchronous periodic task systems is coNP-hard

In the synchronous periodic task model, a set \tau_1,...,\tau_n of tasks is given, each releasing jobs of running time c_i with relative deadline d_i, at each integer multiple of the period p_i. It is a classical result that Earliest Deadline First (EDF) is an optimal preemptive uni-processor scheduling policy. For constrained deadlines, i.e. d_i = 0: \sum_{i=1}^n (floor(Q-d_i)/p_i) + 1) * c_i <= Q. Though an enormous amount of literature deals with this topic, the complexity status of this test has remained unknown. We prove that testing EDF-schedulability of such a task system is (weakly) coNP-hard. This solves Problem 2 from the survey "Open Problems in Real-time Scheduling" by Baruah & Pruhs. The hardness result is achieved by applying recent results on inapproximability of Diophantine approximation

Diameter of Polyhedra: Limits of Abstraction

We investigate the diameter of a natural abstraction of the 1-skeleton of polyhedra. Although this abstraction is simpler than other abstractions that were previously studied in the literature, the best upper bounds on the diameter of polyhedra continue to hold here. On the other hand, we show that this abstraction has its limits by providing a superlinear lower bound

Exact quantification of the sub-optimality of uniprocessor fixed-priority pre- emptive scheduling

This paper examines the relative effectiveness of fixed priority pre-emptive scheduling in a uniprocessor system, compared to an optimal algorithm. The quantitative metric used in this comparison is the processor speedup factor, equivalent to the factor by which processor speed needs to increase to ensure that any taskset that is schedulable according to an optimal scheduling algorithm can be scheduled using fixed priority pre-emptive scheduling. The maximum value for the processor speedup factor is shown to be 1/Omega = 1.76322 for tasksets where all task deadlines are less than or equal to their periods, and 1/ln(2) = 1.44270 for tasksets where all task deadlines are equal to their periods. The derivation of this latter result provides an alternative proof of the well-know Liu and Layland result.We refer to this factor as the processor speedup factor for fixed priority pre- emptive scheduling. Liu and Layland (1973) considered the pre-emptive scheduling of synchronous1 tasksets comprising independent periodic tasks, with bounded execution times, and deadlines equal to their periods. We refer to such tasksets as implicit-deadline tasksets. Liu and Layland showed that Earliest Deadline First (EDF) can schedule any implicit-deadline taskset with a total utilisation2 less than or equal to 100% ( 1 ≤ U ). Liu and Layland also showed that rate monotonic3 priority assignment is the optimal fixed priority assignment policy for implicit-deadline tasksets, and that using this priority assignment policy, fixed priority pre- emptive scheduling can schedule any implicit-deadlin

Approximating Connected Facility Location Problems via Random Facility Sampling and Core Detouring

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00, and for the single-sink rent-or-buy problem from 3.55 to 2.92. We show that our connected facility location algorithms can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems