Inequalities on submodular functions via term rewriting

Yuval Filmus
Information Processing Letters, Volume 113, Issue 13, 2013, pp. 457–464

We devise a method for proving inequalities on submodular functions, with a term rewriting flavor. Our method comprises of the following steps:

  • Start with a linear combination $X$ of the values of the function.
  • Define a set of simplification rules.
  • Conclude that $X \ge Y$, where $Y$ is a linear combination of a small number of terms which cannot be simplified further.
  • Calculate the coefficients of $Y$ by evaluating $X$ and $Y$ on functions on which the inequality is tight.

The crucial third step is non-constructive, since it uses compactness of the dual cone of submodular functions. Its proof uses the classical uncrossing technique with a quadratic potential function.

We prove several inequalities using our method, and use them to tightly analyze the performance of two natural (but non-optimal) algorithms for submodular maximization, the random set algorithm and local search.


 author = {Yuval Filmus},
 title = {Inequalities on submodular functions via term rewriting},
 journal = {Information Processing Letters},
 volume = {113},
 issue = {13},
 year = {2013},
 pages = {457--464}
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