Oblivious algorithms for the Max-kAND Problem

by   Noah G. Singer, et al.
Carnegie Mellon University

Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-kAND problem. This generalizes the definition by Feige and Jozeph (Algorithmica '15) of oblivious algorithms for Max-DICUT, a special case of Max-2AND. Oblivious algorithms round each variable with probability depending only on a quantity called the variable's bias. For each oblivious algorithm, we design a so-called "factor-revealing linear program" (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all k, oblivious algorithms for Max-kAND provably outperform a special subclass of algorithms we call "superoblivious" algorithms. Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every k, O(log n)-space sketching algorithms for Max-kAND known to be optimal in o(√(n))-space can be beaten in (a) O(log n)-space under a random-ordering assumption, and (b) O(n^1-1/k D^1/k) space under a maximum-degree-D assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.


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