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We study the multidimensional vector packing problem with selfish items. An item is a d-dimensional non-zero vector, whose rational components are in [0, 1]. A set of items can be packed into a bin if for every $$1 \le i \le d$$ , the sum of the ith components of all items of this set does not exceed 1. Items share costs of bins proportionally to the $$\ell _1$$ -norms of items, and each item corresponds to a selfish player in the sense that it prefers to be packed into a bin minimizing its resulting cost. This defines a class of games called vector packing games. We show that any game in this class has a packing that is a strong equilibrium, and that both the strong price of anarchy and the strong price of stability are logarithmic in d. We also provide an algorithm that constructs a packing that is a strong equilibrium. Furthermore, we show improved and nearly tight lower and upper bounds of $$d+0.657067$$ and $$d+0.657143$$ , respectively, for any $$d\ge 2$$ , on the price of anarchy. This exhibits a difference between the multidimensional problem and the one-dimensional problem, for which that price of anarchy is at most 1.6428.

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.

Pyroclastic and lava vent-facies, from the Late Jurassic El Quemado Complex, are described at the southern Lake Ghío, in the Cordillera Patagónica Austral. Based on the comprehensive study of lithology and structures, the reconstruction of the volcanic architecture has been carried out. Four ignimbrites and one rhyolitic lava unit, affected by oblique-slip normal faults have been recognized. The evolution of La Peligrosa Caldera has been modeled in three different stages:1) initial collapse, consisting of a precursory downsag subsidence, related to a dilatational zone, which controlled the location of the caldera, 2) main collapse, with the emplacement of large volume crystal-rich ignimbrites and megabreccias, under a progressive subsidence controlled by a pull-apart structure related to a transtensional regime and 3) post-collapse, in which lava flows and associated domes were emplaced under an oblique-extensional regime. The caldera records a remarkable change from transtension to oblique extension, which may represent an important variation in regional deformation conditions during Jurassic times. La Peligrosa Caldera may be considered a key event to understand the eruptive mechanisms of the flare-up volcanism in the Chon Aike Silicic Province. Fil: Sruoga, Patricia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Secretaría de Industria y Minería. Servicio Geológico Minero Argentino; Argentina Fil: Japas, Maria Silvia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires; Argentina Fil: Salani, Flavia Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires; Argentina Fil: Kleiman, Laura Elena. Comisión Nacional de Energía Atómica; Argentina

Infiernillo and Las Picazas are two small hydrothermal ore deposits in the northern San Rafael Massif (Argentina). They are genetically linked to the Permian Choiyoi volcanic province which reflects the transition from a convergent plate margin to an extensional regime: Calc-alkaline Early Permian Lower Choiyoi magmatism was syntectonic with transpressional deformation of the San Rafael Orogeny whereas transitional Late Permian Upper Choiyoi sequences were emplaced under a transtensional regime during the Post-San Rafael extension. Infiernillo is a Cu–(Mo) porphyry-type deposit hosted by pyroclastic rocks of the Lower Choiyoi. It consists of a central quartz plug surrounded by a potassic and a phyllic halo, and a set of peripheral polymetallic (Cu–Pb–Zn) veins cropping out close to the alteration zone. The Las Picazas deposit consists of a group of low-sulfidation epithermal galena-bearing veins hosted by Early Palaeozoic meta-sediments. Strain fabric and available 2-D and 3-D kinematic analyses were performed in order to define the relationships between both deposits and the geodynamic scenario. These data reveal that a) porphyry emplacement at Infiernillo occurred during the declination of the San Rafael Orogeny, shortly before the change in stress regime, and b) Las Picazas low sulfidation epithermal veins were emplaced during the subsequent transtensional Post-San Rafael regime. These results prove that each ore deposit is linked to a different stress regime reflecting the change in geodynamical conditions that prevailed during the emplacement of the Choiyoi volcanism. Thus, strain fabric analysis at both ore deposits allows us to confirm that the tectonic regime is a major factor in controlling the mineralization style of hydrothermal ore deposits. Fil: Japas, Maria Silvia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Geociencias Basicas, Aplicadas y Ambientales de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Geociencias Basicas, Aplicadas y Ambientales de Buenos Aires; Argentina Fil: Rubinstein, Nora Alicia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Geociencias Basicas, Aplicadas y Ambientales de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Geociencias Basicas, Aplicadas y Ambientales de Buenos Aires; Argentina Fil: Kleiman, Laura Elena. Comision Nacional de Energia Atomica. Regional Noroeste. Gerencia de Exploración de Materias Primas; Argentina

Consider the following scheduling game. A set of jobs, each controlled by a selfish agent, are to be assigned to m uniformly related machines. The cost of a job is defined as the total load of the machine that its job is assigned to. A job is interested in minimizing its cost, while the social objective is maximizing the minimum load (the value of the cover) over the machines. This goal is different from the regular makespan minimization goal, which was extensively studied in a game theoretic context. We study the price of anarchy (poa) and the price of stability (pos) for uniformly related machines. The results are expressed in terms of s, which is the maximum speed ratio between any two machines. For uniformly related machines, we prove that the pos is unbounded for s>2, and the poa is unbounded for s?2. For the remaining cases we show that while the poa grows to infinity as s tends to 2, the pos is at most 2 for any s≤2.

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.

The Choiyoi rhyolitic province of Chile and Argentina (23°S-42°S) was emplaced at the SW margin of Gondwana during the Permian. The San Rafael Massif (Mendoza, Argentina, 34°-36°S), is a key area to analyse the relative timing of Choyoi magmatism and related deformation as it bears one of the most complete and well exposed succession. Stratigraphic, structural and magmatic studies indicate that major changes of geodynamic conditions occurred during the Permian since arc-related sequences syntectonic with transpression (lower Choiyoi) were followed by transitional to intraplate, postorogenic suites coeval with transtension (upper Choiyoi). During the Early Permian, a major event of N-NNW dextral transpressional motions deformed the Carboniferous foreland basin in the San Rafael Massif. This event is attributed to the first episode of the San Rafael orogeny and can be related to oblique subduction (Az. 30°) of the Palaeo-Pacific plate. Ca. 280 Ma the inception of voluminous calc-alkaline volcanism (lower Choiyoi) syntectonic with WNW sinistral transpression of the second episode of the San Rafael orogeny, is associated with an eastward migration of the magmatic arc at this latitude. To the southeast of San Rafael, magmatism and transpression continued to migrate inland suggesting that a progressively younger, WNW, sinistral, thick skinned deformation belt broadens into the foreland and can be traced from San Rafael to Sierra de la Ventana, linking the San Rafael orogeny with the Gondwanide orogeny of the Cape Fold Belt in South Africa. This distribution of magmatism and deformation is interpreted as being the consequence of a progressive shallowing of the Palaeo-Pacific plate starting to the north of San Rafael, and culminating with a flat-slab region south of 36°S. Ca. 265 Ma the onset of predominantly felsic volcanism (upper Choiyoi) in San Rafael occurred in a Post-San Rafael extensional setting. Kinematic indicators and strain fabric analyses of San Rafael orogeny transpression and Post-San Rafael extension show a tectonic reversion. The Post-San Rafael event could be the result of the extensional collapse of the San Rafael orogen, triggered by continental-scale clockwise rotations. These rotations would account for subduction ceasing earlier in the north (31°S-36°S) than in the south, thus explaining the coexistence, after ∼ 265 Ma, of extension in San Rafael with compression in the Sierra de la Ventana-Cape Fold Belt area. Fil: Kleiman, Laura Elena. Comisión Nacional de Energía Atómica; Argentina Fil: Japas, Maria Silvia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Geociencias Básicas, Aplicadas y Ambientales de Buenos Aires; Argentina

We study the multidimensional vector packing problem with selfish items. An item is d-dimensional non-zero vector, whose rational components are in [0,1], and a set of items can be packed into a bin if for any 1 ≤ i ≤ d, the sum of the ith components of all items of this set does not exceed 1. Items share costs of bins proportionally to their l1-norms, and each item corresponds to a selfish player in the sense that it prefers to be packed into a bin minimizing its resulting cost. This defines a class of games called vector packing games. We show that any game in this class has a packing that is a strong equilibrium, and that the strong price of anarchy (and the strong price of stability) is logarithmic in d, and provide an algorithm that constructs such a packing. We also show improved and nearly tight lower and upper bounds of d + 0.657067 and d + 0.657143 respectively, on the price of anarchy, exhibiting a difference between the multidimensional problem and the one dimensional problem, for which that price of anarchy is at most 1.6428.

We study the multidimensional vector scheduling problem with selfish jobs, both in non-cooperative and in cooperative versions, in two variants. These variants differ in the private costs of the jobs: in the first variant, the cost is the l ∞ norm (maximum over components) while in the second variant it is the l 1 norm (sum of components) of the load. We show existence of assignments that are Nash, strong Nash, weakly and strictly Pareto optimal Nash equilibria in these settings. For the first variant, we improve upon the previous bounds on the price of anarchy for the non-cooperative case, and find tight bounds for every number of machines and dimension. For the cooperative case we provide tight bounds on the strong prices of anarchy and stability, as well as tight bounds on weakly and strictly Pareto optimal prices of anarchy and stability, for every number of machines and dimension. For the second variant, which was not considered before, we again consider all the aforementioned measures, and find tight bounds, each of which being a function of the number of machines and the dimension, showing cardinal differences in the behavior of these measures both between the two variants, and in comparison to the one-dimensional case.

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, pages 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, ?] for some ? ≤ 1, yielding tight bounds for the Strong Price of Anarchy for all ? ≤ 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 ? ≤ 1.

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.

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 PoAand PoSthat 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 SPoAand SPoS, that are defined similarly to the PoAand 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 PoAof the bin packing game, improving upon previous result by Bilo, and establish the fact that PoS= 1. We show that the bin packing game admits a Strong Nash equilibrium, and that SPoA=SPoS. We prove that this value is equal to the approximation ratio of a natural greedy algorithm for bin packing.