All-Norms and All-L_p-Norms Approximation Algorithms

Abstract

In many optimization problems, a solution can be viewed as ascribing
  a ``cost\'\' to each client, and the goal is to optimize some
  aggregation of the per-client costs.  We often optimize some
  $L_p$-norm (or some other symmetric convex function or norm) of the
  vector of costs---though different applications may suggest
  different norms to use.  Ideally, we could obtain a solution that
  optimizes several norms simultaneously.
  In this paper, we examine approximation algorithms that
  simultaneously perform well on all norms, or on all $L_p$ norms.  
  
  A natural problem in this framework is the $L_p$ Set Cover
  problem, which generalizes \textsc{Set Cover} and \textsc{Min-Sum Set
    Cover}.  We show that the greedy algorithm \emph{simultaneously
    gives a $(p + \ln p + O(1))$-approximation for all $p$, and show
    that this approximation ratio is optimal up to constants} under
  reasonable complexity-theoretic assumptions.

  We additionally show how to use our analysis techniques
  to give similar results for the more general \emph{submodular set
    cover}, and prove some results for the so-called \emph{pipelined set
    cover} problem.

  We then go on to examine approximation algorithms in the
  ``all-norms\'\' and the ``all-$L_p$-norms\'\' frameworks more broadly,
  and present algorithms and structural results for other problems
  such as $k$-facility-location, TSP, and average flow-time
  minimization, extending and unifying previously
  known results.