We study fair distribution of a collection of m indivisible goods among a group of n agents, using the widely recognized fairness principles of Maximin Share (MMS) and Any Price Share (APS). These principles have undergone thorough investigation within the context of additive valuations. We explore these notions for valuations that extend beyond additivity. First, we study approximate MMS under the separable (piecewise-linear) concave (SPLC) valuations, an important class generalizing additive, where the best known factor was 1/3-MMS. We show that 1/2-MMS allocation exists and can be computed in polynomial time, significantly improving the state-of-the-art. We note that SPLC valuations introduce an elevated level of intricacy in contrast to additive. For instance, the MMS value of an agent can be as high as her value for the entire set of items. We use a relax-and-round paradigm that goes through competitive equilibrium and LP relaxation. Our result extends to give (symmetric) 1/2-APS, a stronger guarantee than MMS. APS is a stronger notion that generalizes MMS by allowing agents with arbitrary entitlements. We study the approximation of APS under submodular valuation functions. We design and analyze a simple greedy algorithm using concave extensions of submodular functions. We prove that the algorithm gives a 1/3-APS allocation which matches the best-known factor. Concave extensions are hard to compute in polynomial time and are, therefore, generally not used in approximation algorithms. Our approach shows a way to utilize it within analysis (while bypassing its computation), and hence might be of independent interest.

We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents’ valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with a factor independent of m was known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations before this work. In this article, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high-valued items are done separately by un-matching certain items and re-matching them by different processes in both algorithms. We show that these approaches achieve approximation factors of O ( n ) and O ( n log n ) for additive and submodular cases, independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item ( EF1 ). Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an \(\frac{\mathrm{e}}{\mathrm{e}-1}\) factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of \(\frac{\mathrm{e}}{\mathrm{e}-1}\) , hence resolving it completely.

We study fair allocation of indivisible items, both goods and chores, under the popular fairness notion of maximin share (MMS). The problem is well-studied when there are only goods (or chores), where a PTAS to compute the MMS values of agents is well-known. In contrast, for the mixed manna, a recent result showed that finding even an approximate MMS value of an agent up to any approximation factor in (0,1] is NP-hard for general instances. In this paper, we complement the hardness result by obtaining a PTAS to compute the MMS value when its absolute value is at least 1/p times either the total value of all the goods or total cost of all the chores, for some constant p valued at least 1.

Data races are among the most common bugs in concurrency. The standard approach to data-race detection is via dynamic analyses, which work over executions of concurrent programs, instead of the program source code. The rich literature on the topic has created various notions of dynamic data races, which are known to be detected efficiently when certain parameters (e.g., number of threads) are small. However, the fine-grained complexity of all these notions of races has remained elusive, making it impossible to characterize their trade-offs between precision and efficiency. In this work we establish several fine-grained separations between many popular notions of dynamic data races. The input is an execution trace σ with 𝒩 events, 𝒯 threads and ℒ locks. Our main results are as follows. First, we show that happens-before HB races can be detected in O(𝒩⋅ min(𝒯, ℒ)) time, improving over the standard O(𝒩⋅ 𝒯) bound when ℒ = o(𝒯). Moreover, we show that even reporting an HB race that involves a read access is hard for 2-orthogonal vectors (2-OV). This is the first rigorous proof of the conjectured quadratic lower-bound in detecting HB races. Second, we show that the recently introduced synchronization-preserving races are hard to detect for 3-OV and thus have a cubic lower bound, when 𝒯 = Ω(𝒩). This establishes a complexity separation from HB races which are known to be strictly less expressive. Third, we show that lock-cover races are hard for 2-OV, and thus have a quadratic lower-bound, even when 𝒯 = 2 and ℒ = ω(log 𝒩). The similar notion of lock-set races is known to be detectable in O(𝒩⋅ ℒ) time, and thus we achieve a complexity separation between the two. Moreover, we show that lock-set races become hitting-set (HS)-hard when ℒ = Θ(𝒩), and thus also have a quadratic lower bound, when the input is sufficiently complex. To our knowledge, this is the first work that characterizes the complexity of well-established dynamic race-detection techniques, allowing for a rigorous comparison between them.

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We study the smoothed complexity of finding pure Nash equilibria in Network Coordination Games, a PLS-complete problem in the worst case, even when each player has two strategies. This is a potential game where the sequential-better-response algorithm is known to converge to a pure NE, albeit in exponential time. First, we prove polynomial (respectively, quasi-polynomial) smoothed complexity when the underlying game graph is complete (resp. arbitrary), and every player has constantly many strategies. The complete graph assumption is reminiscent of perturbing all parameters, a common assumption in most known polynomial smoothed complexity results. We develop techniques to bound the probability that an (adversarial) better-response sequence makes slow improvements to the potential. Our approach combines and generalizes the local-max-cut approaches of Etscheid and Röglin (SODA `14; ACM TALG, `17) and Angel, Bubeck, Peres, and Wei (STOC `17), to handle the multi-strategy case. We believe that the approach and notions developed herein could be of interest in addressing the smoothed complexity of other potential games. Further, we define a notion of a smoothness-preserving reduction among search problems, and obtain reductions from 2-strategy network coordination games to local-max-cut, and from k-strategy games (k arbitrary) to local-max-bisection. The former, with the recent result of Bibak, Chandrasekaran, and Carlson (SODA `18) gives an alternate O(n^8)-time smoothed algorithm when k=2. These reductions extend smoothed efficient algorithms from one problem to another.

We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents' valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with factor independent of m is known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations. In this paper, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations respectively. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high valued items are done separately by un-matching certain items and re-matching them, by processes that are different in both algorithms. We show that these approaches achieve approximation factors of O(n) and O(n log n) for additive and submodular case respectively, which is independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1). Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of , hence resolving it completely.