PTAS for Ordered Instances of Resource Allocation Problems

We consider the problem of fair allocation of indivisible goods where we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource j in I has a same value vj > 0 for a subset of customers interested in j and it has no value for other customers. The goal is to find a feasible allocation of the resources to the interested customers such that in the Max-Min scenario (also known as Santa Claus problem) the minimum utility (sum of the resources) received by each of the customers is as high as possible and in the Min-Max case (also known as R||C_max problem), the maximum utility is as low as possible. In this paper we are interested in instances of the problem that admit a PTAS. These instances are not only of theoretical interest but also have practical applications. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; there exists an ordering of the resources such that each customer is interested (has positive evaluation) in a set of consecutive resources and we demonstrate a PTAS. For the Min-Max allocation problem, we obtain a PTAS for instances in which there is an ordering of the customers (machines) and each resource (job) is adjacent to a consecutive set of customers (machines). Next we show that our method for the Max-Min scenario, can be extended to a broader class of bipartite graphs where the resources can be viewed as a tree and each customer is interested in a sub-tree of a bounded number of leaves of this tree (e.g. a sub-path)

Inapproximability Results for Scheduling with Interval and Resource Restrictions

In the restricted assignment problem, the input consists of a set of machines and a set of jobs each with a processing time and a subset of eligible machines. The goal is to find an assignment of the jobs to the machines minimizing the makespan, that is, the maximum summed up processing time any machine receives. Herein, jobs should only be assigned to those machines on which they are eligible. It is well-known that there is no polynomial time approximation algorithm with an approximation guarantee of less than 1.5 for the restricted assignment problem unless P=NP. In this work, we show hardness results for variants of the restricted assignment problem with particular types of restrictions. For the case of interval restrictions - where the machines can be totally ordered such that jobs are eligible on consecutive machines - we show that there is no polynomial time approximation scheme (PTAS) unless P=NP. The question of whether a PTAS for this variant exists was stated as an open problem before, and PTAS results for special cases of this variant are known. Furthermore, we consider a variant with resource restriction where the sets of eligible machines are of the following form: There is a fixed number of (renewable) resources, each machine has a capacity, and each job a demand for each resource. A job is eligible on a machine if its demand is at most as big as the capacity of the machine for each resource. For one resource, this problem has been intensively studied under several different names and is known to admit a PTAS, and for two resources the variant with interval restrictions is contained as a special case. Moreover, the version with multiple resources is closely related to makespan minimization on parallel machines with a low rank processing time matrix. We show that there is no polynomial time approximation algorithm with a rate smaller than 48/47 ? 1.02 or 1.5 for scheduling with resource restrictions with 2 or 4 resources, respectively, unless P=NP. All our results can be extended to the so called Santa Claus variants of the problems where the goal is to maximize the minimal processing time any machine receives

Designing the Game to Play: Optimizing Payoff Structure in Security Games

Effective game-theoretic modeling of defender-attacker behavior is becoming increasingly important. In many domains, the defender functions not only as a player but also the designer of the game's payoff structure. We study Stackelberg Security Games where the defender, in addition to allocating defensive resources to protect targets from the attacker, can strategically manipulate the attacker's payoff under budget constraints in weighted L^p-norm form regarding the amount of change. Focusing on problems with weighted L^1-norm form constraint, we present (i) a mixed integer linear program-based algorithm with approximation guarantee; (ii) a branch-and-bound based algorithm with improved efficiency achieved by effective pruning; (iii) a polynomial time approximation scheme for a special but practical class of problems. In addition, we show that problems under budget constraints in L^0-norm form and weighted L^\infty-norm form can be solved in polynomial time. We provide an extensive experimental evaluation of our proposed algorithms

A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

(In-)Approximability Results for Interval, Resource Restricted, and Low Rank Scheduling

We consider variants of the restricted assignment problem where a set of jobs has to be assigned to a set of machines, for each job a size and a set of eligible machines is given, and the jobs may only be assigned to eligible machines with the goal of makespan minimization. For the variant with interval restrictions, where the machines can be arranged on a path such that each job is eligible on a subpath, we present the first better than 2-approximation and an improved inapproximability result. In particular, we give a (2-1/24)-approximation and show that no better than 9/8-approximation is possible, unless P=NP. Furthermore, we consider restricted assignment with R resource restrictions and rank D unrelated scheduling. In the former problem, a machine may process a job if it can meet its resource requirements regarding R (renewable) resources. In the latter, the size of a job is dependent on the machine it is assigned to and the corresponding processing time matrix has rank at most D. The problem with interval restrictions includes the 1 resource variant, is encompassed by the 2 resource variant, and regarding approximation the R resource variant is essentially a special case of the rank R+1 problem. We show that no better than 3/2, 8/7, and 3/2-approximation is possible (unless P=NP) for the 3 resource, 2 resource, and rank 3 variant, respectively. Both the approximation result for the interval case and the inapproximability result for the rank 3 variant are solutions to open challenges stated in previous works. Lastly, we also consider the reverse objective, that is, maximizing the minimal load any machine receives, and achieve similar results