Many classical problems in theoretical computer science involve norm, even if implicitly; for
\nexample, both XOS functions and downward-closed sets are equivalent to some norms. The last
\ndecade has seen a lot of interest in designing algorithms beyond the standard \u2113p norms.
\nDespite notable advancements, many existing methods remain tailored to specific problems,
\nleaving a broader applicability to general norms less understood. This paper investigates the
\nintrinsic properties of \u2113p norms that facilitate their widespread use and seeks to abstract these
\nqualities to a more general setting.
\nWe identify supermodularity\u2014often reserved for combinatorial set functions and characterized by monotone gradients\u2014as a defining feature beneficial for the \u2113p norm. We introduce the notion
\nof p-supermodularity for norms, asserting that a norm is p-supermodular if its pth power function exhibits supermodularity. The association of supermodularity with norms offers a new lens
\nthrough which to view and construct algorithms.
\nOur work demonstrates that for a large class of problems p-supermodularity is a sufficient
\ncriterion for developing good algorithms. This is either by reframing existing algorithms for
\nproblems like Online Load-Balancing and Bandits with Knapsacks through a supermodular
\nlens, or by introducing novel analyses for problems such as Online Covering, Online Packing,
\nand Stochastic Probing. Moreover, we prove that every symmetric norm can be approximated by
\na p-supermodular norm. Together, these recover and extend several results from the literature,
\nand support p-supermodularity as a unified theoretical framework for optimization challenges
\ncentered around norm-related problems<\/p>\n","protected":false},"excerpt":{"rendered":"
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 \u2113p norms. Despite notable advancements, many existing methods remain tailored to specific problems, leaving a 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