Many classical problems in theoretical computer science involve norm, even if implicitly; for example, both XOS functions and downward-closed sets are equivalent to some norms. The last decade has seen a lot of interest in designing algorithms beyond the standard $\ell_p$ norms $\|\cdot \|_p$. Despite notable advancements, many existing methods remain tailored to specific problems, leaving a broader applicability to general norms less understood. This paper investigates the intrinsic properties of $\ell_p$ norms that facilitate their widespread use and seeks to abstract these qualities to a more general setting. We identify supermodularity -- often reserved for combinatorial set functions and characterized by monotone gradients -- as a defining feature beneficial for $ \|\cdot\|_p^p$. We introduce the notion of $p$-supermodularity for norms, asserting that a norm is $p$-supermodular if its $p^{th}$ power function exhibits supermodularity. The association of supermodularity with norms offers a new lens through which to view and construct algorithms. Our work demonstrates that for a large class of problems $p$-supermodularity is a sufficient criterion for developing good algorithms. This is either by reframing existing algorithms for problems like Online Load-Balancing and Bandits with Knapsacks through a supermodular lens, or by introducing novel analyses for problems such as Online Covering, Online Packing, and Stochastic Probing. Moreover, we prove that every symmetric norm can be approximated by a $p$-supermodular norm. Together, these recover and extend several results from the literature, and support $p$-supermodularity as a unified theoretical framework for optimization challenges centered around norm-related problems.

翻译：理论计算机科学中的许多经典问题都涉及范数，即使这种关联是隐式的；例如，XOS函数和向下封闭集都等价于某些范数。过去十年间，针对标准$\ell_p$范数$\|\cdot \|_p$之外的算法设计引起了广泛关注。尽管取得了显著进展，许多现有方法仍局限于特定问题，对于一般范数的更广泛适用性仍缺乏深入理解。本文研究了$\ell_p$范数得以广泛应用的内在特性，并尝试将这些特性抽象到更一般的设定中。我们发现超模性——这一通常用于组合集函数并由单调梯度刻画的特性——是$\|\cdot\|_p^p$所具有的一个有益特征。我们为范数引入了$p$-超模性的概念，主张一个范数是$p$-超模的，当其$p$次幂函数表现出超模性。超模性与范数的关联为理解和构建算法提供了新的视角。我们的研究表明，对于一大类问题，$p$-超模性是设计良好算法的充分准则。这既可以通过超模视角重构现有算法（如在线负载均衡和背包赌博机问题），也可以通过引入新颖分析来处理在线覆盖、在线包装和随机探测等问题。此外，我们证明了任何对称范数都可以用一个$p$-超模范数来逼近。这些结果共同恢复并扩展了文献中的若干结论，并支持将$p$-超模性作为围绕范数相关问题的优化挑战的统一理论框架。