Move-optimal schedules for parallel machines to minimize total weighted completion time

We study the minimum total weighted completion time problem on identical machines, which is known to be strongly $\mathcal{NP}$-hard. We analyze a simple local search heuristic, moving jobs from one machine to another. The local optima can be shown to be approximately optimal with approximation ratio $1.5$. In case all jobs have equal Smith ratios, the approximation ratio is at most $1.092$

Quality of Move-Optimal Schedules for Minimizing the Vector Norm of the Workloads

We study the problem of minimizing the vector norm $||\cdot||_p$ of the workloads. We examine move-optimal assignments and prove a performance guarantee of $\frac{2^p-1}{p} \cdot \left(\frac{p-1}{2^p-2}\right)^{\frac{p-1}{p}},$ for any integer $p>1$ and moreover, we show that this guarantee is tight. Additionally, we consider assignments obtained by applying the LPT-heuristic of Graham (1969). We prove that an LPT-assignment has a performance guarantee of $\frac{3^p-2^p}{p} \cdot \left(\frac{p-1}{2 \cdot 3^p - 3 \cdot 2^p}\right)^{\frac{p-1}{p}},$ which reproves a result of Chandra and Wong (1975)

A new optimal packing algorithm for telecommunications networks planning

AbstractIn telecommunications network planning, the problem of multiplexing low speed lines onto high speed transmission channels provides opportunities for attractive economic solutions. In many cases, the overall costs for transmission channels runs into hundreds of thousands of dollars per year. Thus, the problem of finding the minimum number of transmission channels to support low speed lines might be very important.This paper describes an exact packing algorithm (OPTPACK) for determining the packing of low speed lines onto high speed channels in such a way that the number of required channels is minimized

A new model for selfish routing

AbstractIn this work, we introduce and study a new, potentially rich model for selfish routing over non-cooperative networks, as an interesting hybridization of the two prevailing such models, namely the KPmodel [E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413] and the Wmodel [J.G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the of the Institute of Civil Engineers 1 (Pt. II) (1952) 325–378].In the hybrid model, each of n users is using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links. In a Nash equilibrium, no user can unilaterally improve its Expected Individual Cost. To evaluate Nash equilibria, we introduce Quadratic Social Cost as the sum of the expectations of the latencies, incurred by the squares of the accumulated traffic. This modeling is unlike the KP model, where Social Cost [E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413] is the expectation of the maximum latency incurred by the accumulated traffic; but it is like the W model since the Quadratic Social Cost can be expressed as a weighted sum of Expected Individual Costs. We use the Quadratic Social Cost to define Quadratic Coordination Ratio. Here are our main findings: •Quadratic Social Cost can be computed in polynomial time. This is unlike the #P-completeness [D. Fotakis, S. Kontogiannis, E. Koutsoupias, M. Mavronicolas, P. Spirakis, The structure and complexity of Nash equilibria for a selfish routing game, in: P. Widmayer, F. Triguero, R. Morales, M. Hennessy, S. Eidenbenz, R. Conejo (Eds.), Proceedings of the 29th International Colloquium on Automata, Languages and Programming, in: Lecture Notes in Computer Science, vol. 2380, Springer-Verlag, 2002, pp. 123–134] of computing Social Cost for the KP model.•For the case of identical users and identical links, the fully mixed Nash equilibrium [M. Mavronicolas, P. Spirakis, The price of selfish routing, Algorithmica 48 (1) (2007) 91–126], where each user assigns positive probability to every link, maximizes Quadratic Social Cost.•As our main result, we present a comprehensive collection of tight, constant (that is, independent of m and n), strictly less than 2, lower and upper bounds on the Quadratic Coordination Ratio for several, interesting special cases. Some of the bounds stand in contrast to corresponding super-constant bounds on the Coordination Ratio previously shown in [A. Czumaj, B. Vöcking, Tight bounds for worst-case equilibria, ACM Transactions on Algorithms 3 (1) (2007); E. Koutsoupias, M. Mavronicolas, P. Spirakis, Approximate equilibria and ball fusion, Theory of Computing Systems 36 (6) (2003) 683–693; E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413; M. Mavronicolas, P. Spirakis, The price of selfish routing, Algorithmica 48 (1) (2007) 91–126] for the KP model

A unified approach to truthful scheduling on related machines

We present a unified framework for designing deterministic monotone polynomial time approximation schemes (PTAS's) for a wide class of scheduling problems on uniformly related machines. This class includes (among others) minimizing the makespan, maximizing the minimum load, and minimizing the l_p norm of the machine loads vector. Previously, this kind of result was only known for the makespan objective. Monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it. The key idea of our novel method is to show that for goal functions that are sufficiently well-behaved functions of the machine loads, it is possible to compute in polynomial time a highly structured nearly optimal schedule. Monotone approximation schemes have an important role in the emerging area of algorithmic mechanism design. In the game-theoretical setting of these scheduling problems there is a social goal, which is one of the objective functions that we study. Each machine is controlled by a selfish single-parameter agent, where its private information is its cost of processing a unit sized job, which is also the inverse of the speed of its machine. Each agent wishes to maximize its own profit, defined as the payment it receives from the mechanism minus its cost for processing all jobs assigned to it, and places a bid which corresponds to its private information. For each one of the problems, we show that we can calculate payments that guarantee truthfulness in an efficient manner. Thus, there exists a dominant strategy where agents report their true speeds, and we show the existence of a truthful mechanism which can be implemented in polynomial time, where the social goal is approximated within a factor of 1+epsilon for every epsilon>0