Ioannis Panageas, Avi Shein, Shayan Taherijam, Vijay V. Vazirani
The Hylland-Zeckhauser (HZ) mechanism provides a market-based approach to one-sided matching, making efficient equilibrium computation central to its implementation. Bi-valued utilities admit a strongly polynomial-time algorithm [VY], whereas with four values per buyer, the problem is PPAD-hard under inverse polynomial approximation [CCPY]. To understand the tractability boundary, we study two extensions of the bi-valued case: a third linear utility value and separable concave utilities with two common marginal slopes. For every fixed rational $a\in(0,1)$, we prove that computing a normalized additive $\varepsilon_{*}(a)/N$-equilibrium in an $N$-buyer HZ market is PPAD-hard for some constant $\varepsilon_{*}(a)>0$, even when all utilities belong to the same prescribed set $\{0,a,1\}$. This resolves the main question left open by [VY, CCPY]: whether three utility values already suffice for PPAD-hardness. Using the same market gadgets, we show that hardness at a specified constant accuracy on the recognizable family generated by our construction is equivalent to the PCP-for-PPAD conjecture. Consequently, assuming that conjecture, general markets over $\{0,a,1\}$ admit no PTAS unless $\mathrm{PPAD}\subseteq\mathrm{P}$. For the concave extension, we compute an exact equilibrium in polynomial time with two marginal slopes common across goods for each buyer and arbitrary rational buyer--good breakpoints. The algorithm selects a cheapest utility-maximizing bundle for each buyer. With three common marginal slopes, suitable breakpoints encode the three valued linear case, yielding PPAD-hardness. Thus, our paper identifies two thresholds separating tractable and intractable cases of HZ models with utility functions specified by utility values or marginal slopes. In both cases, the tractability threshold lies between two and three.
David P. Woodruff, Vincent Cohen-Addad, Lalit Jain, Jieming Mao, Song Zuo, MohammadHossein Bateni, Simina Branzei, Michael P. Brenner 等 36 位作者
Recent advances in large language models (LLMs) have opened new avenues for accelerating scientific research. While models are increasingly capable of assisting with routine tasks, their ability to contribute to novel, expert-level mathematical discovery is less understood. We present a collection of case studies demonstrating how researchers have successfully collaborated with advanced AI models, specifically Google's Gemini-based models (in particular Gemini Deep Think and its advanced variants), to solve open problems, refute conjectures, and generate new proofs across diverse areas in theoretical computer science, as well as other areas such as economics, optimization, and physics. Based on these experiences, we extract common techniques for effective human-AI collaboration in theoretical research, such as iterative refinement, problem decomposition, and cross-disciplinary knowledge transfer. While the majority of our results stem from this interactive, conversational methodology, we also highlight specific instances that push beyond standard chat interfaces. These include deploying the model as a rigorous adversarial reviewer to detect subtle flaws in existing proofs, and embedding it within a "neuro-symbolic" loop that autonomously writes and executes code to verify complex derivations. Together, these examples highlight the potential of AI not just as a tool for automation, but as a versatile, genuine partner in the creative process of scientific discovery.