We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker's layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemer\'edi's regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. On the other hand, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

翻译：我们识别出一个充分条件——树宽可塑性(treewidth-pliability)，该条件为在由允许的约束图类参数化（在无界字母表上具有任意约束）的大类Max-2-CSP问题中，对最优值进行任意精度的多项式时间近似提供了算法。我们的结果更为一般地适用于两个有理值结构之间的最大同态问题。该条件统一了设计多项式时间近似方案的两类主要方法：其一是Baker的分层技术，适用于平面图或排除子图等稀疏图；其二是基于Szemerédi正则性引理的方法，适用于稠密图。我们将这两种技术的适用范围扩展到了新的Max-CSP问题类别。另一方面，我们证明该条件通常无法用于求解具体解（相对于近似最优值）。树宽可塑性被证明是一个稳健的概念，可通过规模、树深度或Hadwiger数等多种等价方式定义。我们展示了该概念与结构图论中的分数树宽脆弱性、性质检验领域的超有限性以及稠密图极限理论中的正则划分之间的联系。这些关联可能具有独立的研究价值。特别地，我们证明：一个单调图类是超有限的，当且仅当其满足分数树宽脆弱性且具有有界度数。