The problem of identifying the satisfiability threshold of random 3-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The classical assumption in this line of research is that, for a given set of n Boolean variables, each clause is drawn uniformly at random among all sets of three literals from these variables, independently from other clauses. Here, we keep the uniform distribution of each clause, but deviate significantly from the independence assumption and consider richer families of probability distributions. For integer parameters n, m, and k, we denote by ℱ_k(n,m) the family of probability distributions that produce formulas with m clauses, each selected uniformly at random from all sets of three literals from the n variables, so that the clauses are k-wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in ℱ_k(n,m) for different values of the parameters n, m, and k. Our technical results are as follows: First, all probability distributions in ℱ₂(n,m) with m ∈ Ω(n³) return unsatisfiable formulas with high probability. This result is tight. We show that there exists a probability distribution 𝒟 ∈ ℱ₃(n,m) with m ∈ O(n³) so that a random formula drawn from 𝒟 is almost always satisfiable. In contrast, for m ∈ Ω(n²), any probability distribution 𝒟 ∈ ℱ₄(n,m) returns an unsatisfiable formula with high probability. This is our most surprising and technically involved result. Finally, for any integer k ≥ 2, any probability distribution 𝒟 ∈ ℱ_k(n,m) with m ∈ O(n^{1-1/k}) returns a satisfiable formula with high probability.

The design of energy markets is a subject of ongoing debate, particularly concerning the choice between the widely adopted Pay-as-Clear (PC) pricing mechanism and the alternative Pay-as-Bid (PB). These mechanisms determine how energy producers are compensated: under PC, all selected producers are paid the market-clearing price (i.e., the highest accepted bid), while under PB, each selected producer is paid their own submitted bid. The overarching objective is to meet the total demand for energy at minimal cost in the presence of strategic behavior. We present two key theoretical results. First, no mechanism can uniformly dominate PC or PB. This means that for any mechanism $$\mathcal {M}$$ , there exists a market configuration and a mixed-strategy Nash equilibrium of PC (respectively for PB) that yields strictly lower total energy costs than under $$\mathcal {M}$$ . Second, in terms of worst-case equilibrium outcomes, PB consistently outperforms PC: across all market instances, the highest possible equilibrium price under PB is strictly lower than that under PC. This suggests a structural robustness of PB to strategic manipulation. These theoretical insights are further supported by extensive simulations based on no-regret learning dynamics, which consistently yield lower average market prices in several energy market settings.

Cut games are among the most fundamental strategic games in algorithmic game theory. It is well-known that computing an exact pure Nash equilibrium in these games is PLS-hard, so research has focused on computing approximate equilibria. We present a polynomial-time algorithm that computes 2.7371-approximate pure Nash equilibria in cut games. This is the first improvement to the previously best-known bound of 3, due to the work of Bhalgat, Chakraborty, and Khanna from EC 2010. Our algorithm is based on a general recipe proposed by Caragiannis, Fanelli, Gravin, and Skopalik from FOCS 2011 and applied on several potential games since then. The first novelty of our work is the introduction of a phase that can identify subsets of players who can simultaneously improve their utilities considerably. This is done via semidefinite programming and randomized rounding. In particular, a negative objective value to the semidefinite program guarantees that no such considerable improvement is possible for a given set of players. Otherwise, randomized rounding of the SDP solution is used to identify a set of players who can simultaneously improve their strategies considerably and allows the algorithm to make progress. The way rounding is performed is another important novelty of our work. Here, we exploit an idea that dates back to a paper by Feige and Goemans from 1995, but we take it to an extreme that has not been analyzed before.

We propose a Multi-task learning approach for Abstractive Text Summarization (MATS), motivated by the fact that humans have no difficulty performing such task because they have the capabilities of multiple domains. Specifically, MATS consists of three components: (i) a text categorization model that learns rich category-specific text representations using a bi-LSTM encoder; (ii) a syntax labeling model that learns to improve the syntax-aware LSTM decoder; and (iii) an abstractive text summarization model that shares its encoder and decoder with the text categorization and the syntax labeling tasks, respectively. In particular, the abstractive text summarization model enjoys significant benefit from the additional text categorization and syntax knowledge. Our experimental results show that MATS outperforms the competitors.1