Local occupancy can be used to bound the expected size of a hard-core independent set. The framework of locally stationary distributions shows that for triangle-free graphs, despite slow mixing at the relevant fugacity parameter, standard Glauber dynamics can be used to find sets close to the local occupancy guarantee. We improve the application of this framework to the hard-core model by replacing a conditional mixing argument with a more direct approach based on a short sequence of single-site updates that, from every initial configuration, achieves the required occupancy inequality. In general graphs, naive conditional mixing estimates or certain problematic structures in the neighborhood can in principle incur running time cost exponential in the maximum degree $Δ$. We design short update sequences that, via our relaxed criterion, yield running times polynomial in $Δ$ for graphs with a hereditary bound $b$ on the density of $K_{t,t}$ subgraphs in neighborhoods. Our methods handle the case where $t$ and $b$ can grow with the maximum degree, and the resulting algorithm attains the random graph shattering threshold when they grow sufficiently slowly. That is, we give improved local occupancy parameters and a short update sequence analysis which together show that Glauber dynamics finds independent sets of size $(1-o(1))n(\logΔ)/Δ$ in time polynomial in $n$ and $Δ$ whenever $t$ and $b$ are constant or grow suitably slowly. In a similar regime, we also establish that the (correspondence) chromatic number is at most $(1+o(1))Δ/\logΔ$ which improves upon prior results by a factor of at least $4$ in the case $t\ge 4$.
Yuanchen Brian Tang, Jiaxin Song, Bhaskar Ray Chaudhury, Ruta Mehta
原文摘要
We study the stability of data exchange among competing firms. Unlike tangible goods, data is non-rival: sharing it does not deprive the giver of its use. On one hand, competing firms-- often operating as machine-learning-driven predictors that collect observations from a shared environment to solve similar tasks-- can improve their predictive accuracies from pooling data. On the other hand, they compete for market share, which may depend on their relative predictive performance. This raises the central question of this paper: can there be a stable data exchange in such competitive environments? We formalize this via a model of $n$ firms exchanging fractions of their datasets, with predictive performance feeding into market competition. We adopt core stability as our solution concept, accounting for externalities via worst-case responses by non-participating firms. Our main results reveal a sharp contrast driven by demand structure: (i) in fully rational markets, where customers flock entirely to the top-performing firm, a core-stable exchange always exists for any monotone continuous prediction functions; (ii) in partially rational markets with logit-style demand, core-stable exchanges may fail to exist-and even approximate cores can be empty. Our impossibility results suggest that internal profit-sharing within coalitions can restore stability in partially rational markets. We formalize this intuition by showing that when co-operating coalitions equally redistribute market-share revenues among members, a core-stable data exchange exists. Finally, we propose a natural exchange dynamic and demonstrate empirically that it converges to a core-stable data exchange in fully rational markets on simulated instances derived from real datasets.