arXiv subjects

José Verschae

Publications and source records attributed to José Verschae.

24 records · Page 2Linked to original sources

arXiv2017

The Primal-Dual Greedy Algorithm for Weighted Covering Problems

We present a general approximation framework for weighted integer covering problems. In a weighted integer covering problem, the goal is to determine a non-negative integer solution $x$ to system $\{ Ax \geq r \}$ minimizing a non-negative cost function $c^T x$ (of appropriate dimensions). All coefficients in matrix $A$ are assumed to be non-negative. We analyze the performance of a very simple primal-dual greedy algorithm and discuss conditions of system $(A,r)$ that guarantee feasibility of the constructed solutions, and a bounded approximation factor. We call system $(A,r)$ a \emph{greedy system} if it satisfies certain properties introduced in this work. These properties highly rely on monotonicity and supermodularity conditions on $A$ and $r$, and can thus be seen as a far reaching generalization of contra-polymatroids. Given a greedy system $(A,r)$, we carefully construct a truncated system $(A',r)$ containing the same integer feasible points. We show that our primal-dual greedy algorithm when applied to the truncated system $(A',r)$ obtains a feasible solution to $(A,r)$ with approximation factor at most $2δ+ 1$, or $2δ$ if $r$ is non-negative. Here, $δ$ is some characteristic of the truncated matrix $A'$ which is small in many applications. The analysis is shown to be tight up to constant factors. We also provide an approximation factor of $k (δ+ 1)$ if the greedy algorithm is applied to the intersection of multiple greedy systems. The parameter $k$ is always bounded by the number of greedy systems but may be much smaller. Again, we show that the dependency on $k$ is tight. We conclude this paper with an exposition of classical approximation results based on primal-dual algorithms that are covered by our framework. We match all of the known results. Additionally, we provide some new insight in a generalization of the flow cover on a line problem.

cs.DM↗

arXiv2016

A 4-approximation for scheduling on a single machine with general cost function

We consider a single machine scheduling problem that seeks to minimize a generalized cost function: given a subset of jobs we must order them so as to minimize $\sum f_j(C_j)$, where $C_j$ is the completion time of job $j$ and $f_j$ is a job-dependent cost function. This problem has received a considerably amount of attention lately, partly because it generalizes a large number of sequencing problems while still allowing constant approximation guarantees. In a recent paper, Cheung and Shmoys provided a primal-dual algorithm for the problem and claimed that is a 2-approximation. In this paper we show that their analysis cannot yield an approximation guarantee better than $4$. We then cast their algorithm as a local ratio algorithm and show that in fact it has an approximation ratio of $4$. Additionally, we consider a more general problem where jobs has release dates and can be preempted. For this version we give a $4κ$-approximation algorithm where $κ$ is the number of distinct release dates.

cs.DS↗

arXiv2016

A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

We consider the following single-machine scheduling problem, which is often denoted $1||\sum f_{j}$: we are given $n$ jobs to be scheduled on a single machine, where each job $j$ has an integral processing time $p_j$, and there is a nondecreasing, nonnegative cost function $f_j(C_{j})$ that specifies the cost of finishing $j$ at time $C_{j}$; the objective is to minimize $\sum_{j=1}^n f_j(C_j)$. Bansal \& Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the LP is less than 4. Finally, we show how the technique can be adapted to yield, for any $ε>0$, a $(4+ε)$-approximation algorithm for this problem.

cs.DS↗

arXiv2016

Closing the Gap for Makespan Scheduling via Sparsification Techniques

Makespan scheduling on identical machines is one of the most basic and fundamental packing problems studied in the discrete optimization literature. It asks for an assignment of $n$ jobs to a set of $m$ identical machines that minimizes the makespan. The problem is strongly NP-hard, and thus we do not expect a $(1+ε)$-approximation algorithm with a running time that depends polynomially on $1/ε$. Furthermore, Chen et al. [3] recently showed that a running time of $2^{(1/ε)^{1-δ}}+\text{poly}(n)$ for any $δ>0$ would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on $1/ε$, the better of which achieves a running time of $2^{\tilde{O}(1/ε^2)}+O(n\log n)$ [11]. In this paper we obtain an algorithm with a running time of $2^{\tilde{O}(1/ε)}+O(n\log n)$, which is tight under ETH up to logarithmic factors on the exponent. Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we obtain an efficient PTAS with running time $2^{\tilde{O}(1/ε)} + \text{poly}(n)$.

cs.DS↗

arXiv2014

Dual techniques for scheduling on a machine with varying speed

We study scheduling problems on a machine with varying speed. Assuming a known speed function we ask for a cost-efficient scheduling solution. Our main result is a PTAS for minimizing the total weighted completion time in this setting. This also implies a PTAS for the closely related problem of scheduling to minimize generalized global cost functions. The key to our results is a re-interpretation of the problem within the well-known two-dimensional Gantt chart: instead of the standard approach of scheduling in the {\em time-dimension}, we construct scheduling solutions in the weight-dimension. We also consider a dynamic problem variant in which deciding upon the speed is part of the scheduling problem and we are interested in the tradeoff between scheduling cost and speed-scaling cost, which is typically the energy consumption. We observe that the optimal order is independent of the energy consumption and that the problem can be reduced to the setting where the speed of the machine is fixed, and thus admits a PTAS. Furthermore, we provide an FPTAS for the NP-hard problem variant in which the machine can run only on a fixed number of discrete speeds. Finally, we show how our results can be used to obtain a~$(2+\eps)$-approximation for scheduling preemptive jobs with release dates on multiple identical parallel machines.

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arXiv2010

On the Configuration-LP for Scheduling on Unrelated Machines

One of the most important open problems in machine scheduling is the problem of scheduling a set of jobs on unrelated machines to minimize the makespan. The best known approximation algorithm for this problem guarantees an approximation factor of 2. It is known to be NP-hard to approximate with a better ratio than 3/2. Closing this gap has been open for over 20 years. The best known approximation factors are achieved by LP-based algorithms. The strongest known linear program formulation for the problem is the configuration-LP. We show that the configuration-LP has an integrality gap of 2 even for the special case of unrelated graph balancing, where each job can be assigned to at most two machines. In particular, our result implies that a large family of cuts does not help to diminish the integrality gap of the canonical assignment-LP. Also, we present cases of the problem which can be approximated with a better factor than 2. They constitute valuable insights for constructing an NP-hardness reduction which improves the known lower bound. Very recently Svensson studied the restricted assignment case, where each job can only be assigned to a given set of machines on which it has the same processing time. He shows that in this setting the configuration-LP has an integrality gap of 33/17<2. Hence, our result imply that the unrelated graph balancing case is significantly more complex than the restricted assignment case. Then we turn to another objective function: maximizing the minimum machine load. For the case that every job can be assigned to at most two machines we give a purely combinatorial 2-approximation algorithm which is best possible, unless P=NP. This improves on the computationally costly LP-based (2+eps)-approximation algorithm by Chakrabarty et al.

cs.DM↗