Restricted Assignment Problem Research Articles

Makespan minimization on unrelated parallel machines with a few bags

Let there be a set M of m parallel machines and a set J of n jobs, where each job j takes pi,j time units on machine Mi. In makespan minimization the goal is to schedule each job non-preemptively on a machine such that the length of the schedule, the makespan, is minimum. We investigate a generalization of makespan minimization on unrelated parallel machines (R||Cmax) where J is partitioned into b bags B=(B1,…,Bb), and no two jobs belonging to the same bag can be scheduled on the same machine. First we present a simple b-approximation algorithm for R||Cmax with bags (R|bag|Cmax). We also give a polynomial-time approximation scheme (PTAS) for R|bag|Cmax with machine types where both the number of machine types and bags are constant; two machines Mi and Mi′ have the same machine type if pi,j=pi′,j for all j∈J. This result infers the existence of a PTAS for uniform parallel machines (Q|bag|Cmax) when the number of machine speeds and number of bags are both constant. Then, we present a b/2-approximation algorithm for the graph balancing problem with b≥2 bags; the approximation ratio is tight for b=3, unless P=NP, and this algorithm solves the graph balancing problem with b=2 bags in polynomial time. In addition, we present a polynomial-time algorithm for the restricted assignment problem on uniform parallel machines with bags when all the jobs have unit length. To complement our algorithmic results, we show that when the jobs have lengths 1 or 2 it is NP-hard to approximate the optimum makespan within a factor smaller than 3/2 for both the restricted assignment and graph balancing problems with b=2 bags and b=3 bags, respectively. We also prove that Q|bag|Cmax with b=2 bags is strongly NP-hard.

A Quasi-Polynomial Approximation for the Restricted Assignment Problem

The Restricted Assignment problem is a prominent special case of Scheduling on Unrelated Parallel Machines. For the strongest known linear programming relaxation, the configuration LP, we improve the nonconstructive bound on its integrality gap from 1.9412 to 1.8334 and significantly simplify the proof. Then we give a constructive variant, yielding a 1.8334-approximation in quasi-polynomial time. This is the first quasi-polynomial algorithm for this problem improving on the long-standing approximation rate of 2.

Makespan minimization on unrelated parallel machines with simple job-intersection structure and bounded job assignments

Let there be a set J of n jobs and a set M of m parallel machines, where each job Jj takes pi,j∈Z+ time units on machine Mi and assume pi,j=∞ implies job Jj cannot be scheduled on machine Mi. In makespan minimization on unrelated parallel machines (R||Cmax), the goal is to schedule each job non-preemptively on a machine so as to minimize the makespan. A job-intersection graph GJ=(J,EJ) is an unweighted undirected graph where there is an edge {Jj,Jj′}∈EJ if there is a machine Mi such that both pi,j≠∞ and pi,j′≠∞. In this paper we consider two variants of R||Cmax where there are a small number of eligible jobs per machine. First, we prove that there are no approximation algorithms with approximation ratios less than 3/2 for R||Cmax when restricted to instances with job-intersection graphs belonging to some graph classes such as diamondless graphs and planar graphs, unless P=NP. Second, we match this lower bound by presenting a 3/2-approximation algorithm for R||Cmax restricted to instances with diamondless job-intersection graphs, and furthermore show that when GJ is triangle free R||Cmax is solvable in polynomial time. For R||Cmax restricted to instances when every machine can process at most ℓ jobs, we give an approximation algorithm with approximation ratios 3/2 and 5/3 for ℓ=3 and ℓ=4 respectively, a polynomial-time algorithm when ℓ=2, and prove that it is NP-hard to approximate the optimum solution within a factor less than 3/2 when ℓ≥3. In the special case where every pi,j∈{pj,∞}, called the restricted assignment problem, and there are only two job lengths pj∈{α,β} we present a (2−1/(ℓ−1))-approximation algorithm when ℓ≥3. In addition, we give a 2-approximation algorithm for the so-called unrelated graph balancing problem for ℓ=3 in the case where J is partitioned into b sets called bags, and no two jobs from the same bag can be scheduled on the same machine.

Approximation algorithms for the graph balancing problem with two speeds and two job lengths

Consider a set of n jobs and m uniform parallel machines, where each job has a length \(p_j \in {\mathbb {Q}}^+\) and each machine has a speed \(s_i \in {\mathbb {Q}}^+\). The goal of the graph balancing problem with speeds is to schedule each job j non-preemptively on one of a subset \({\mathcal {M}}_j\) of at most 2 machines so that the makespan is minimized. This is a \(\textsf {NP}\)-hard special case of the makespan minimization problem on unrelated parallel machines, where for the latter a longstanding open problem is to find an approximation algorithm with approximation ratio better than 2. Our main contribution is an approximation algorithm for the graph balancing problem with two speeds and two job lengths with approximation ratio \((\sqrt{65}+7)/8 \approx 1.88278\). In addition, we consider when every \({\mathcal {M}}_j\) has no cardinality constraints, this is the restricted assignment problem in the uniform parallel machine setting. We present a simple \((2-\alpha /\beta )\)-approximation algorithm in this case when every job has one of two job lengths \(p_j \in \{\alpha , \beta \}\) where \(\alpha < \beta \).

Estimating the Makespan of the Two-Valued Restricted Assignment Problem

We consider a special case of the scheduling problem on unrelated machines, namely the restricted assignment problem with two different processing times. We show that the configuration LP has an integrality gap of at most $$\frac{5}{3} \approx 1.667$$ for this problem. This allows us to estimate the optimal makespan within a factor of $$\frac{5}{3}$$ , improving upon the previously best known estimation algorithm with ratio $$\frac{11}{6} \approx 1.833$$ due to Chakrabarty et al. (in: Proceedings of the twenty-sixth annual ACM-SIAM symposium on discrete algorithms (SODA 2015), pp 1087–1101, 2015).

On some special cases of the restricted assignment problem

We consider some special cases of the restricted assignment problem. In this scheduling problem on parallel machines, any job j can only be assigned to one of the machines in its given subset Mj of machines. We give an LP-formulation for the problem with two job sizes and show that it has an integral optimal solution. We also present a PTAS for the case that the Mj's are intervals of the same length. Further, we give a new and very simple algorithm for the case that |Mj|=2 (known as the graph balancing problem) with ratio 11/6.

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 the configuration LP, that was previously successfu...

All-norm approximation algorithms

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ p norms. We address this problem by introducing the concept of an all-norm ρ-approximation algorithm , which supplies one solution that guarantees ρ-approximation to all ℓ p norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each job is associated with a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al. [Math. Program. 46 (1990) 259–271] showed a 2-approximation algorithm for the problem with respect to the ℓ ∞ norm. For any fixed ℓ p norm the previously known approximation algorithm has a performance of θ( p). We provide an all-norm 2-approximation polynomial algorithm for the restricted assignment problem. On the other hand, we show that for any given ℓ p norm ( p>1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given ℓ p norm a FPTAS for any fixed number of machines.

RESTRICTED TRACK ASSIGNMENT WITH APPLICATIONS

Consider a set of intervals S = {I1, I2, …, In}, where Ii = (li, ri), li, and ri are real numbers, and li &lt; ri. We study a restricted track assignment problem (RTAP): if an interval Ia contains another interval Ib, then Ia must be assigned to a higher track than Ib, and the goal is to minimize the number of tracks used. The problem RTAP is shown to be NP-hard. An approximation algorithm that produces a solution within twice of the optimal is also presented and the bound is shown to be tight. The algorithm runs in O(n log n) time and requires linear space. The proposed approximation algorithm is employed to solve the problem of finding a maximum-weighted independent set in a circle graph, and related problems.