Santa Claus Schedules Jobs on Unrelated Machines

One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines, i.e., job j requires time p_{ij} if processed on machine i. More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form p_{ij} in {p_j, \infty}. This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than 1.5 which is also the best known lower bound for the general version. Our main result is a polynomial time algorithm that estimates the optimal makespan of the restricted assignment problem within a factor 33/17 + \epsilon \approx 1.9412 + \epsilon, where \epsilon > 0 is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as configuration LP, that was previously successfully used for the related Santa Claus problem. Similar to the strongest analysis for that problem our proof is based on a local search algorithm that will eventually find a schedule of the mentioned approximation guarantee, but is not known to converge in polynomial time.Comment: 22 pages, 1 figure; corrected typos and changed some notatio

Young researchers of the Barents Region

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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

Who must pay bribes and how much? Evidence from a cross-section of firms

The author exploits a unique data set on corruption containing information about estimated bribe payments by Ugandan firms. To guide the empirical analysis, he develops a simple rent-extortion model, which yields predictions on both the incidence of bribery, and the amount paid. Both predictions are consistent with the data. Firms typically have to pay bribes when dealing with public officials whose actions directly affect the firms'business operations. And the amount paid in bribes is not a fixed sum for a set of public services, but depends on the firm's ability to pay. Controlling for other potential explanations of the relationship between"ability to pay"and equilibrium graft, the author shows that the more a firm can pay, the more it has to pay.Economic Theory&Research,Corruption&Anitcorruption Law,Public Sector Corruption&Anticorruption Measures,Environmental Economics&Policies,Social Policy

When is foreign aid policy credible : aid dependence and conditionality

The author studies foreign aid policy within a principal-agent framework. He shows that one reason for foreign aid's poor overall record may be a moral hazard problem that shapes the aid recipient s incentive to undertake structural reform. The model's basic prediction is a two-way relationship: Disbursements of foreign aid are guided (in part) by the needs of the poor. Anticipating this, recipients have little incentive to improve the welfare of the poor. Preliminary econometric work shows that the data support this hypothesis. In principle, conditionality could partly solve this problem, but only if the donor can make a binding commitment to increase disbursements in good relative to bad states. Without such a commitment technology, aid disbursements remain guided by the needs of the poor and recipient countries maintain a low effort to reduce poverty. Contrary to the conventional wisdom found in the aid literature, the author shows that the welfare of all parties might be improved by using tied project aid or by delegating part of the aid budget to an (international) agency with less aversion to poverty.Development Economics&Aid Effectiveness,School Health,Economic Adjustment and Lending,Environmental Economics&Policies,Economic Theory&Research