Bin packing approximation algorithms: Survey and classification

Following recent interest in the study of computer science problems in a game theoretic setting, we consider the well known bin packing problem where the items are controlled by selfish agents. Each agent is charged with a cost according to the fraction of the used bin space its item requires. That is, the cost of the bin is split among the agents, proportionally to their sizes. Thus, the selfish agents prefer their items to be packed in a bin that is as full as possible. The social goal is to minimize the number of the bins used. The social cost in this case is therefore the number of bins used in the packing.

A pure Nash equilibrium is a packing where no agent can obtain a smaller cost by unilaterally moving his item to a different bin, while other items remain in their original positions. A Strong Nash equilibrium is a packing where there exists no subset of agents, all agents in which can profit from jointly moving their items to different bins. We say that all agents in a subset profit from moving their items to different bins if all of them have a strictly smaller cost as a result of moving, while the other items remain in their positions.

We measure the quality of the equilibria using the standard measures PoA and PoS that are defined as the worst case worst/best asymptotic ratio between the social cost of a (pure) Nash equilibrium and the cost of an optimal packing, respectively. We also consider the recently introduced measures SPoA and SPoS, that are defined similarly to the PoA and the PoS, but consider only Strong Nash equilibria.

We give nearly tight lower and upper bounds of 1.6416 and 1.6428, respectively, on the PoA of the bin packing game, improving upon previous result by Bilo. We study the Strong Nash equilibria of the bin packing game, and show that a packing is a Strong Nash equilibrium iff it is produced by the Subset Sum algorithm for bin packing. This characterization implies that the SPoA of the bin packing game equals the approximation ratio of the Subset Sum algorithm, for which an almost tight bound is known. Moreover, the fact that any lower bound instance for the Subset Sum algorithm can be converted by a small modification of the item sizes to a lower bound instance on the SPoS, implies that in the bin packing game SPoA=SPoS. Finally, we address the issue of complexity of computing a Strong Nash packing and show that no polynomial time algorithm exists for finding Strong Nash equilibria, unless P=NP.

Selfish Bin Packing

Non-cooperative games on multidimensional resource allocation

This article formulates two variants of packing problems in which the set of items is known, but the arrival order is unknown. The goal is to certify that the items can be packed in a given container and/or to optimize the size or cost of a container so that that the items are guaranteed to be packable, regardless of arrival order. The nondeterministically ordered packing (NDOP) variant asks to generate a certificate that a packing plan exists for every ordering of items. Quasi-online packing (QOP) asks to generate a partially observable packing policy that chooses the location of each item as their arrival order is revealed one-by-one, with all of the remaining items certified to be packable regardless of their future arrival order. Theoretical analysis demonstrates that even the simple subproblem of verifying the feasibility of a packing policy is NP-complete. Despite this worst case complexity, practical solvers for both NDOP and QOP are developed. Multiple extensions to the basic nondeterministic problem are presented, including packing with a fixed-capacity buffer and packing with equivalent objects. Experiments demonstrate that these algorithms can be applied to packing irregular 3-D shapes with stability and manipulator loading constraints. Note to Practitioners —Automatic packing of objects into containers, such as a shipping box, has applications in warehouse automation for e-commerce applications and less-than-truckload shipping. In many scenarios, a container needs to be preselected for a set of objects before they arrive, and the order of the objects cannot be controlled. This article studies the theoretical foundations of packing problems, in which the set of items is known, but the arrival order is unknown. The algorithms presented in this article can certify whether a container can hold a set of objects in the case where the arrival order is revealed at once, one-by-one, and where a limited set of objects can be put aside in a buffer before packing. These algorithms can be used to choose optimal containers and packing strategies, both for robot and human packers.

Robot Packing with Known Items and Nondeterministic Arrival Order.

In a system in which noncooperative agents share a common resource, we propose the ratio between the worst possible Nash equilibrium and the social optimum as a measure of the effectiveness of the system. Deriving upper and lower bounds for this ratio in a model in which several agents share a very simple network leads to some interesting mathematics, results, and open problems.

Worst-case equilibria

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008, pp. 368---380) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the Bin Packing problem is precisely the approximation ratio of this heuristic. In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the Bin Packing problem where item sizes lie on the interval $$(0, \alpha ]$$(0,?] for some $$\alpha \le 1$$?≤1, yielding tight bounds for the Strong Price of Anarchy for all $$\alpha \le 1$$?≤1. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all $$\alpha \le 1$$?≤1.

Parametric Packing of Selfish Items and the Subset Sum Algorithm

The subset sum algorithm is a natural heuristic for the classical Bin Packing problem: In each iteration, the algorithm finds among the unpacked items, a maximum size set of items that fits into a new bin. More than 35 years after its first mention in the literature, establishing the worst-case performance of this heuristic remains, surprisingly, an open problem. 
Due to their simplicity and intuitive appeal, greedy algorithms are the heuristics of choice of many practitioners. Therefore, better understanding simple greedy heuristics is, in general, an interesting topic in its own right. Very recently, Epstein and Kleiman (Proc. ESA 2008) provided another incentive to study the subset sum algorithm by showing that the Strong Price of Anarchy of the game theoretic version of the bin-packing problem is precisely the approximation ratio of this heuristic. 
In this paper we establish the exact approximation ratio of the subset sum algorithm, thus settling a long standing open problem. We generalize this result to the parametric variant of the bin packing problem where item sizes lie on the interval (0,\alpha] for some \alpha \leq 1, yielding tight bounds for the Strong Price of Anarchy for all \alpha \leq 1. Finally, we study the pure Price of Anarchy of the parametric Bin Packing game for which we show nearly tight upper and lower bounds for all \alpha \leq 1.

Parametric packing of selfish items and the subset sum algorithm

We consider a scheduling problem where each job is controlled by a selfish agent, who is only interested in minimizing its own cost, which is defined as the total load on the machine that its job is assigned to. We consider the objective of maximizing the minimum load (cover) over the machines. Unlike the regular makespan minimization problem, which was extensively studied in a game theoretic context, this problem has not been considered in this setting before.

We study the price of anarchy (poa) and the price of stability (pos). We show that on related machines, both these values are unbounded. We then focus on identical machines. We show that the $\textsc{pos}$ is 1, and we derive tight bounds on the $\textsc{poa}$ for m ≤ 6 and nearly tight bounds for general m. In particular, we show that the $\textsc{poa}$ is at least 1.691 for larger m and at most 1.7. Hence, surprisingly, the $\textsc{poa}$ is less than the $\textsc{poa}$ for the makespan problem, which is 2. To achieve the upper bound of 1.7, we make an unusual use of weighting functions. Finally, in contrast we show that the mixed $\textsc{poa}$ grows exponentially with m for this problem, although it is only ?(logm/loglogm) for the makespan.

Maximizing the Minimum Load: The Cost of Selfishness

In the well-known scheduling game, a set of jobs controlled by selfish players wishes each to minimize the load of the machine on which it is executed, while the social goal is to minimize the makespan, that is, the maximum load of any machine. We consider this problem on the three most common machines models, identical machines, uniformly related machines and unrelated machines, with respect to both weak and strict Pareto optimal Nash equilibria. These are kinds of equilibria which are stable not only in the sense that no player can improve its cost by changing its strategy unilaterally, but in addition, there is no alternative choice of strategies for the entire set of players where no player increases its cost, and at least one player reduces its cost (in the case of strict Pareto optimality), or where all players reduce their costs (in the case of weak Pareto optimality).We give a complete classification of the social quality of such solutions with respect to an optimal solution, that is, we find the Price of Anarchy of such schedules as a function of the number of machines, m. In addition, we give a full classification of the recognition complexity of such schedules.

/pdf/on-the-quality-and-complexity-of-pareto-equilibria-in-the-1iv771px7m.pdf

On the quality and complexity of pareto equilibria in the job scheduling game

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