We prove that the problem of computing an Arrow-Debreu market equilibrium is PPAD-complete even when all traders use additively separable, piecewise-linear and concave utility functions. In fact, our proof shows that this market-equilibrium problem does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time.

Recent attention has been devoted to the pursuit of learning semantic segmentation models exclusively from image tags, a paradigm known as image-level Weakly Supervised Semantic Segmentation (WSSS). Existing attempts adopt the Class Activation Maps (CAMs) as priors to mine object regions yet observe the imbalanced activation issue, where only the most discriminative object parts are located. In this paper, we argue that the distribution discrepancy between the discriminative and the non-discriminative parts of objects prevents the model from producing complete and precise pseudo masks as ground truths. For this purpose, we propose a Pixel-Level Domain Adaptation (PLDA) method to encourage the model in learning pixel-wise domain-invariant features. Specifically, a multi-head domain classifier trained adversarially with the feature extraction is introduced to promote the emergence of pixel features that are invariant with respect to the shift between the source (i.e., the discriminative object parts) and the target (i.e., the nondiscriminative object parts) domains. In addition, we come up with a Confident Pseudo-Supervision strategy to guarantee the discriminative ability of each pixel for the segmentation task, which serves as a complement to the intra-image domain adversarial training. Our method is conceptually simple, intuitive and can be easily integrated into existing WSSS methods. Taking several strong baseline models as instances, we experimentally demonstrate the effectiveness of our approach under a wide range of settings.

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