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Holographic algorithms introduced by Valiant have two ingredients: matchgates, which are gadgets realizing local constraint functions by weighted planar perfect matchings, and holographic reductions, which show equivalences among problems with different descriptions via basis transformations. In this paper, we replace matchgates in the paradigm above by the affine type and the product type constraint functions, which are known to be tractable in general (not necessarily planar) graphs. We present polynomial-time algorithms to decide if a given counting problem has a holographic reduction to another problem defined by the affine or product-type functions. We also give polynomial-time algorithms to the same problems for symmetric functions, where the complexity is measured in terms of the (exponentially more) succinct representations. The latter result implies that the symmetric Boolean Holant dichotomy (Cai, Guo, and Williams, SICOMP 2016) is efficiently decidable. Our proof techniques are mainly algebraic. [ABSTRACT FROM AUTHOR]

The Piedmont Region in NW Italy has recently deployed an ambitious and pioneering agricultural water pricing reform aimed at integrating and effectively enforcing EU's Water Framework Directive principles of cost recovery, polluter-pays and affordability. This paper develops a multi-model ensemble framework encompassing 5 mathematical programming models (2 Positive Mathematical Programming models, 2 Positive Multi-Attribute Utility Programming models and 1 Weighted Goal Programming model) that represent the observed behavior of socioeconomic agents to: 1) simulate the impacts of the Piedmontese water pricing reform on land use allocation and management, water conservation, profit and water tariff revenue; 2) sample modeling uncertainty through the ensemble spread; and 3) explore potential tipping points through use of scenario-discovery techniques. Our research suggests that the key challenge to the reform lies in the management of rice fields, an extensive (17% of the agricultural area), water-demanding and relatively low-added-value crop that nonetheless delivers significant ecosystem services (e.g. water retention) of historical and cultural relevance to the region. The ensemble experiment suggests that rice agriculture rapidly dwindles in the price range 0.012–0.074 EUR/m3 depending on the model. Before reaching this tipping point, agricultural water pricing can reduce withdrawals up to 1.7%–9.5%, while reducing profit between 4.9% and 5.6% and achieving a 57- to 65-fold increase in water tariff revenue. • We combine a multi-model ensemble experiment with scenario discovery techniques. • Our approach samples modeling and scenario uncertainty to identify tipping points. • Our approach is illustrated using the agricultural water pricing in Piedmont (IT). • Results show historical rice fields dwindle in the price range 0.014–0.074 EUR/m3. • Conserving rice fields considerably limits pricing water conservation potential. [ABSTRACT FROM AUTHOR]