Restricted Assignment Research Articles

EmbAssi: embedding assignment costs for similarity search in large graph databases

The graph edit distance is an intuitive measure to quantify the dissimilarity of graphs, but its computation is mathsf {NP}-hard and challenging in practice. We introduce methods for answering nearest neighbor and range queries regarding this distance efficiently for large databases with up to millions of graphs. We build on the filter-verification paradigm, where lower and upper bounds are used to reduce the number of exact computations of the graph edit distance. Highly effective bounds for this involve solving a linear assignment problem for each graph in the database, which is prohibitive in massive datasets. Index-based approaches typically provide only weak bounds leading to high computational costs verification. In this work, we derive novel lower bounds for efficient filtering from restricted assignment problems, where the cost function is a tree metric. This special case allows embedding the costs of optimal assignments isometrically into ell _1 space, rendering efficient indexing possible. We propose several lower bounds of the graph edit distance obtained from tree metrics reflecting the edit costs, which are combined for effective filtering. Our method termed EmbAssi can be integrated into existing filter-verification pipelines as a fast and effective pre-filtering step. Empirically we show that for many real-world graphs our lower bounds are already close to the exact graph edit distance, while our index construction and search scales to very large databases.

Malleable scheduling beyond identical machines

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. In this setting, jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S with respect to j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than \(\frac{e}{e-1}\), unless \(P = NP\). On the positive side, we present polynomial-time algorithms with approximation ratios \(\frac{2e}{e-1}\) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding. They result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of \(1+\varphi \) for unrelated speeds (\(\varphi \) is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms for a variant where we determine the effective speed of a set of allocated machines based on the \(L_p\) norm of their speeds.

Online load balancing with general reassignment cost

We investigate a semi-online variant of load balancing with restricted assignment. In this problem, we are given n jobs, which need to be processed by m machines with the goal to minimize the maximum machine load. Since strong lower bounds rule out any competitive ratio of o(log⁡n), we may reassign jobs at a certain job-individual cost. We generalize a result by Gupta, Kumar, and Stein (SODA 2014) by giving a O(log⁡log⁡mn)-competitive algorithm with constant amortized reassignment cost.

Scheduling uniform machines with restricted assignment.

The problem of minimizing makespan (maximum completion time) on uniform machines with restricted assignment is considered. The machines differ in their speeds and functionalities. Each job has a set of machines to which it can be assigned, called its processing set. The goal is to finish the jobs as soon as possible. There exist 4/3-approximation algorithms for the cases of inclusive and tree-hierarchical assignment restrictions, under an assumption that machines with higher capabilities also run at higher speeds. We eliminate the assumption and present algorithms with approximation ratios 2 and 4/3 for both cases.

Greedy is Optimal for Online Restricted Assignment and Smart Grid Scheduling for Unit Size Jobs

Greedy is Optimal for Online Restricted Assignment and Smart Grid Scheduling for Unit Size Jobs

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.

Restricted assignment scheduling with resource constraints

We consider parallel machine scheduling with job assignment restrictions, i.e., each job can only be processed on a certain subset of the machines. Moreover, each job requires a set of renewable resources. Any resource can be used by only one job at any time. The objective is to minimize the makespan. We present approximation algorithms with constant worst-case bound in the case that each job requires only a fixed number of resources. For some special cases optimal algorithms with polynomial running time are given. If any job requires at most one resource and the number of machines is fixed, we give a PTAS. On the other hand we prove that the problem is APX-hard, even when there are just three machines and the input is restricted to unit-time jobs.

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 \(\alpha < \beta \).

Rejecting jobs to minimize load and maximum flow-time

The notion of competitive ratio often turns out to be too pessimistic for the analysis of online algorithms. Although the approach of resource augmentation (introduced by Kalyanasundaram and Pruhs) has been very successful in dealing with a variety of objective functions, there are problems for which even a (arbitrary) constant speedup cannot lead to a constant competitive algorithm. Here we propose a rejection model which permits the online algorithm to not serve epsilon-fraction of requests. We present O(log2⁡1/ε) and O(1/ε4)-competitive algorithms for the problems of load balancing and minimizing maximum flow time in the restricted assignment setting.

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).

Interference Minimization in D2D Communication Underlaying Cellular Networks

Interference minimization while maintaining a target system sum rate by sharing radio resources among cellular user equipments (UEs) and device-to-device (D2D) pairs is an important research question in long term evolution (LTE) and beyond (4G and 5G). Total system sum rate of a cellular network can be improved if cellular UEs and D2D pairs share resource blocks. However, some sharing can also decrease the sum rate and increase the system interference. Considering this observation, we address two types of assignments (fair and restricted) in resource allocation for the interference minimization resource allocation problem. We propose a two-phase resource allocation algorithm for both fair and restricted assignments, where our objective is to minimize the system interference and at the same time, maintaining a target system sum rate. In the phase-I of our proposed algorithm, a weighted bipartite matching algorithm is used to minimize the interference and get a feasible initial solution. In some cases, we can decrease the interference introduced in phase-I of our algorithm. Therefore, in the phase-II, local search techniques are used to improve the solution. We compare the fair assignment of our proposed algorithms with a two-phase auction-based fair and interference aware resource allocation algorithm (TAFIRA), which addresses the same research problem. As well as, we compare the restricted assignment of our proposed algorithm with a minimum knapsack-based interference resource allocation algorithm (MIKIRA). We prove that the MIKIRA fails to provide feasible solutions in most of the cases. We also show that the performance ratio of the TAFIRA can be unbounded in the worst case. Moreover, in some cases, TAFIRA cannot provide any solution to the problem though the solutions exist, whereas our proposed algorithms always provide a solution whenever the solution exists. We perform extensive simulations of the algorithms and find that in all the cases, our proposed algorithm outperforms a number of state-of-the-art algorithms.

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.

The price of envy-freeness in machine scheduling

We consider k-envy-free assignments for scheduling problems in which the completion time of each machine is not k times larger than the one she could achieve by getting the jobs of another machine, for a given factor k≥1. We introduce and investigate the notion of price of k-envy-freeness, defined as the ratio between the makespan of the best k-envy-free assignment and that of an optimal allocation achievable without envy-freeness constraints. We provide exact or asymptotically tight bounds on the price of k-envy-freeness for all the basic scheduling models, that is unrelated, related and identical machines. Moreover, we show how to efficiently compute such allocations with a worsening multiplicative factor being at most the best approximation ratio for the minimum makespan problem guaranteed by a polynomial time algorithm for each specific model. Finally, we extend our results to the case of restricted assignments and to the objective of minimizing the sum of the completion times of all the machines.

Minimizing average flow-time under knapsack constraint

We give the first logarithmic approximation for minimizing average flow-time of jobs in the subset parallel machine setting (also called the restricted assignment setting) under a single knapsack constraint. In a knapsack constraint setting, each job has a profit, and the set of jobs which get scheduled must have a total profit of at least a quantity Π. Our result extends the work of Gupta et al. (2009) [8] who considered the special case where the profit of each job is unity. Our algorithm is based on rounding a natural LP relaxation for this problem. In fact, we show that one can use techniques based on iterative rounding.

The Strong Price of Anarchy of Linear Bottleneck Congestion Games

We study the inefficiency of equilibrium outcomes in Bottleneck Congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We analyze the (strong) Price of Anarchy of these games for a natural load balancing social cost objective, i.e., minimize the maximum latency of a facility. In our studies, we focus on Bottleneck Congestion games with linear latency functions. These games still constitute a rich class of games and generalize, for example, Load Balancing games with identical or uniformly related machines (with or without restricted assignments). We derive upper and (asymptotically) matching lower bounds on the (strong) Price of Anarchy of these games. We also derive more refined bounds for several special cases of these games, including the cases of identical player weights, identical latency functions and symmetric strategy sets. Further, we provide lower bounds on the Price of Anarchy for k-strong equilibria.

SU-E-T-227: Least Restrictive Assignment of Dose-Distance Differences (LRAD) in IMRT On a Conventional Linac

Purpose: To develop a method to compare dose distributions based on the least restrictive of dose and distance differences, and to evaluate the method for IMRT QA on a conventional linac. Methods: Lrad is defined as the least restrictive of the dose difference at each point in the distributions being compared (%D), with dose normalized by the same factor, and the distance to agreement (DTA in millimeters), limited to directions parallel and perpendicular to MLC motion. If %D<%Do, Lrad=%D. Otherwise, if L=%D/DTA>Lo, Lrad=DTA, else Lrad=%D. Lrad fails at any point when Lrad>DTAo or Lrad>%Do. We compared LRAD (Lo=1%/mm, DTAo=2mm, %Do=2%) with Gamma index tool (2%/2mm pass/fail criteria) for IMRT QA with 2D detector arrays. Dose distribution measured for 6 MV and 18 MV x-rays for various field sizes and shapes, including 6 IMRT plans, were compared to measurements made with output, beam flatness, beam symmetry, MLC leaf positions, and detector array translation and rotation adjusted at up to twice published acceptable tolerance. Results: LRAD was more effective than Gamma in detecting a 2% drift in output (dose per monitor unit), 1–2 mm drifts in MLC leaf position for 1–8 leaves in the field, and beam flatness and symmetry drifts that resulted in a 3% change in the off-axis ratio. LRAD and Gamma were equally effective for larger output changes and for detector translations and rotations. For all ∼400 measurements performed, LRAD detected 78% of the error and Gamma detected 52% of the errors. By using a color scale to display dose differences and grey scale for spatial errors, the reason for failure was self evident, whether it be beam or mechanical changes. Conclusion: LRAD proved to be advantageous in its ability to detect and distinguish between realistic changes in beam and mechanical settings in comparison to the Gamma Index.

Robust Algorithms for Preemptive Scheduling

Preemptive scheduling problems on parallel machines are classic problems. Given the goal of minimizing the makespan, they are polynomially solvable even for the most general model of unrelated machines. In these problems, a set of jobs is to be assigned to run on a set of m machines. A job can be split into parts arbitrarily and these parts are to be assigned to time slots on the machines without parallelism, that is, for every job, at most one of its parts can be processed at each time. Motivated by sensitivity analysis and online algorithms, we investigate the problem of designing robust algorithms for constructing preemptive schedules. Robust algorithms receive one piece of input at a time. They may change a small portion of the solution as an additional part of the input is revealed. The capacity of change is based on the size of the new piece of input. For scheduling problems, the supremum ratio between the total size of the jobs (or parts of jobs) which may be re-scheduled upon the arrival of a new job j, and the size of j, is called migration factor. We design a strongly optimal algorithm with the migration factor $1-\frac{1}{m}$ for identical machines. Strongly optimal algorithms avoid idle time and create solutions where the (non-increasingly) sorted vector of completion times of the machines is lexicographically minimal. In the case of identical machines this results not only in makespan minimization, but the created solution is also optimal with respect to any l p norm (for p>1). We show that an algorithm of a smaller migration factor cannot be optimal with respect to makespan or any other l p norm, thus the result is best possible in this sense as well. We further show that neither uniformly related machines nor identical machines with restricted assignment admit an optimal algorithm with a constant migration factor. This lower bound holds both for makespan minimization and for any l p norm. Finally, we analyze the case of two machines and show that in this case it is still possible to maintain an optimal schedule with a small migration factor in the cases of two uniformly related machines and two identical machines with restricted assignment.

Santa claus meets hypergraph matchings

We consider the restricted assignment version of the problem of max-min fair allocation of indivisible goods, also known as the Santa Claus problem . There are m items and n players. Every item has some nonnegative value, and every player is interested in only some of the items. The goal is to distribute the items to the players in a way that maximizes the minimum of the sum of the values of the items given to any player. It was previously shown via a nonconstructive proof that uses the Lovász local lemma that the integrality gap of a certain configuration LP for the problem is no worse than some (unspecified) constant. This gives a polynomial-time algorithm to estimate the optimum value of the problem within a constant factor, but does not provide a polynomial-time algorithm for finding a corresponding allocation. We use a different approach to analyze the integrality gap. Our approach is based upon local search techniques for finding perfect matchings in certain classes of hypergraphs. As a result, we prove that the integrality gap of the configuration LP is no worse than 1/4. Our proof provides a local search algorithm which finds the corresponding allocation, but is nonconstructive in the sense that this algorithm is not known to converge to a local optimum in a polynomial number of steps.