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We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an Ω(log m / log log m)-approximation to the optimal makespan, where m is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a 3/2-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations. 

We study two recent combinatorial contract design models, which highlight different sources of complexity that may arise in contract design, where a principal delegates the execution of a costly project to others. In both settings, the principal cannot observe the choices of the agent(s), only the project’s outcome (success or failure), and incentivizes the agent(s) using a contract, a payment scheme that specifies the payment to the agent(s) upon a project’s success. We present results that resolve open problems and advance our understanding of the computational complexity of both settings. In the multi-agent setting, the project is delegated to a team of agents, where each agent chooses whether or not to exert effort. A success probability function maps any subset of agents who exert effort to a probability of the project’s success. For the family of submodular success probability functions, Dütting et al. [2023] established a poly-time constant factor approximation to the optimal contract, and left open whether this problem admits a PTAS. We answer this question on the negative, by showing that no poly-time algorithm guarantees a better than 0.7-approximation to the optimal contract. For XOS functions, they give a poly-time constant approximation with value and demand queries. We show that with value queries only, one cannot get any constant approximation. In the multi-action setting, the project is delegated to a single agent, who can take any subset of a given set of actions. Here, a success probability function maps any subset of actions to a probability of the project’s success. Dütting et al. [2021a] showed a poly-time algorithm for computing an optimal contract for gross substitutes success probability functions, and showed that the problem is NP-hard for submodular functions. We further strengthen this hardness result by showing that this problem does not admit any constant factor approximation. Furthermore, for the broader class of XOS functions, we establish the hardness of obtaining a n^{-1/2+ε}-approximation for any ε > 0. 

The celebrated model of auctions with interdependent valuations, introduced by Milgrom and Weber in 1982, has been studied almost exclusively under private signals $$s_1, \ldots, s_n$$ of the $$n$$ bidders and public valuation functions $$v_i(s_1, \ldots, s_n)$$. Recent work in TCS has shown that this setting admits a constant approximation to the optimal social welfare if the valuations satisfy a natural property called submodularity over signals (SOS). More recently, Eden et al. (2022) have extended the analysis of interdependent valuations to include settings with private signals and \emph{private valuations}, and established $$O(\log^2 n)$$-approximation for SOS valuations. In this paper we show that this setting admits a {\em constant} factor approximation, settling the open question raised by Eden et al. (2022). 

We study combinatorial auctions with interdependent valuations, where each agent i has a private signal s_ithat captures her private information and the valuation function of every agent depends on the entire signal profile, [Formula: see text]. The literature in economics shows that the interdependent model gives rise to strong impossibility results and identifies assumptions under which optimal solutions can be attained. The computer science literature provides approximation results for simple single-parameter settings (mostly single-item auctions or matroid feasibility constraints). Both bodies of literature focus largely on valuations satisfying a technical condition termed single crossing (or variants thereof). We consider the class of submodular over signals (SOS) valuations (without imposing any single crossing-type assumption) and provide the first welfare approximation guarantees for multidimensional combinatorial auctions achieved by universally ex post incentive-compatible, individually rational mechanisms. Our main results are (i) four approximation for any single-parameter downward-closed setting with single-dimensional signals and SOS valuations; (ii) four approximation for any combinatorial auction with multidimensional signals and separable-SOS valuations; and (iii) (k + 3) and (2 log(k) + 4) approximation for any combinatorial auction with single-dimensional signals, with k-sized signal space, for SOS and strong-SOS valuations, respectively. All of our results extend to a parameterized version of SOS, d-approximate SOS, while losing a factor that depends on d. Funding: A. Eden was partially supported by NSF Award IIS-2007887, the European Research Council (ERC) under the European Union's Seventh Framework Programme [FP7/2007-2013]/ERC Grant Agreement 337122, by the Israel Science Foundation [Grant 317/17], and by an Amazon research award. M. Feldman received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program [Grant Agreement 866132], by the Israel Science Foundation [Grant 317/17], by an Amazon research award, and by the NSF-BSF [Grant 2020788]. The work of K. Goldner was supported partially by NSF awards DMS-1903037 and CNS-2228610 and a Shibulal Family Career Development Professorship. A. R. Karlin was supported by the NSF-CCF [Grant 1813135]. 

Abstract Gliomas are one of the most common and lethal brain tumors among adults. One process that contributes to glioma progression and recurrence is the epithelial to mesenchymal transition (EMT). EMT is regulated by a set of defined transcription factors which tightly regulate this process, among them is the basic helix-loop-helix family member, TWIST1. Here we show that TWIST1 is methylated on lysine-33 at chromatin by SETD6, a methyltransferase with expression levels correlating with poor survival in glioma patients. RNA-seq analysis in U251 glioma cells suggested that both SETD6 and TWIST1 regulate cell adhesion and migration processes. We further show that TWIST1 methylation attenuates the expression of the long-non-coding RNA, LINC-PINT, thereby promoting EMT in glioma. Mechanistically, TWIST1 methylation represses the transcription of LINC-PINT by increasing the occupancy of EZH2 and the catalysis of the repressive H3K27me3 mark at the LINC-PINT locus. Under un-methylated conditions, TWIST1 dissociates from the LINC-PINT locus, allowing the expression of LINC-PINT which leads to increased cell adhesion and decreased cell migration. Together, our findings unravel a new mechanistic dimension for selective expression of LINC-PINT mediated by TWIST1 methylation. 

We consider a revenue-maximizing seller with m heterogeneous items and a single buyer whose valuation for the items may exhibit both substitutes and complements. We show that the better of selling the items separately and bundling them together— guarantees a [Formula: see text]-fraction of the optimal revenue, where d is a measure of the degree of complementarity; it extends prior work showing that the same simple mechanism achieves a constant-factor approximation when buyer valuations are subadditive (the most general class of complement-free valuations). Our proof is enabled by a recent duality framework, which we use to obtain a bound on the optimal revenue in the generalized setting. Our technical contributions are domain specific to handle the intricacies of settings with complements. One key modeling contribution is a tractable notion of “degree of complementarity” that admits meaningful results and insights—we demonstrate that previous definitions fall short in this regard.