New Approximation Algorithms for Degree Lower-bounded Arborescences and Max-Min Allocation

Abstract:

We consider the problem of MaxMin allocation of indivisible goods. There are $m$ items to be distributed among $n$ players. Each player $i$ has a nonnegative valuation $p_{ij}$ for an item $j$, and the goal is to allocate items to players so as to maximize the minimum total valuation received by each player. There is a large gap in our understanding of this problem. The best known positive result is an $\tilde O(\sqrt n)$-approximation algorithm, while there is only a factor $2$ hardness known. Better algorithms are known for the restricted assignment case where each item has exactly one nonzero value for the players. We study the effect of bounded degree for items: each item has a nonzero value for at most $D$ players. We show that essentially the case $D = 3$ is equivalent to the general case, and give a $4$-approximation algorithm for $D = 2$.
The current algorithmic results for MaxMin Allocation are based on a complicated LP relaxation called the configuration LP. We present a much simpler LP which is equivalent in power to the configuration LP. We focus on a special case of MaxMin Allocation—a family of instances on which this LP has a polynomially large gap. The technical core of our result for this case comes from an algorithm for an interesting new optimization problem on directed graphs, MaxMinDegree Arborescence, where the goal is to produce an arborescence of large outdegree. We develop an $n$-approximation for this problem that runs in $n^{O(1/t)}$ time and obtain a polylogarithmic approximation that runs in quasipolynomial time, using a lift-and-project inspired LP formulation. In fact, we show that our results imply a rounding algorithm for the relaxations obtained by $t$ rounds of the Sherali-Adams hierarchy applied to a natural LP relaxation of the problem. Roughly speaking, the integrality gap of the relaxation obtained from $t$ rounds of Sherali-Adams is at most $n^{1/t}$.
We are able to extend the latter result to a more general class of instances called InfDegreeTwo: each item has infinite utility for at most two players. Along the way, we prove a result about the existence of a perfect matching in a probabilistically pruned graph which may be of independent interest.