Lift-and-Project Integrality Gaps for Santa Claus

TL;DR

An MMDA instance of depth 3 is constructed for which an integrality gap of $n^{\Omega(1/\ell)$ survives $1$ round of the Sherali-Adams hierarchy, for which it is conjecture that an integrality gap of $n^{\Omega(1/\ell)$ survives $\Omega(\ell)$ rounds of Sherali-Adams.

Abstract

This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth $\ell\le O(\log n/\log \log n)$, which is an important special case of the Santa Claus problem. Obtaining a polylogarithmic approximation for MMDA in polynomial time is of high interest as it is a necessary condition to improve upon the well-known 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS'87]. The only way we have to solve the MMDA problem within a polylogarithmic factor is via an elegant recursive rounding of the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, which needs time $n^{O(\ell)}$ to solve. However, it remains plausible that one could obtain a polylogarithmic approximation in polynomial time by using the same rounding with only $1$ round of the Sherali-Adams hierarchy. As a main result, we rule out this possibility by constructing an MMDA instance of depth $3$ for which an integrality gap of $n^{Ω(1)}$ survives $1$ round of the Sherali-Adams hierarchy. This result is tight since it is known that after only $2$ rounds the gap is at most polylogarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in Ω(1)\cap o(\log n/\log \log n)$ (the whole range of interest), for which we conjecture that an integrality gap of $n^{Ω(1/\ell)}$ survives $Ω(\ell)$ rounds of Sherali-Adams. We show a number of intermediate results towards this conjecture, which also suggest that our construction is a significant challenge to the techniques used so far for Santa Claus.

Figures (5)

• Figure 1: An MMDA instance of depth $3$ with $k_v=2$ for all non-sink vertices. The source is the blue vertex in $L_0$, and the set of sinks is $L_3$. All edges are oriented from left to right. The set of orange edges (dashed) forms an optimum feasible integral solution. The set of green edges (dashed+dotted) is feasible and gives out-degree $1$ to all its non-sink vertices, so it is a $2$-approximate solution.
• Figure 2: The construction with $\mathcal{U}=\{1,2,3,4\}$ and $\rho=1/4$.
• Figure 3: An instance similar to bansal2006santa translated as an MMDA instance. The source is connected to $k^2$ vertices in $L_1$, each vertex in $L_1$ is connected to a "private" subgraph containing $k$ vertices in $L_2$ and $k$ sinks in $L_3$. In every private subgraph, each vertex in $L_2$ is connected to all sinks in $L_3$. Finally, we set $k_u=k$ for all non-sink vertices.
• Figure 4: The Santa Claus instance corresponding to the MMDA instance in Figure \ref{['fig:bansal_instance']}. Circles represent players and squares represent resources. The valuation for edges is given on the left. If there is no edge between a player and a resource, that player values this resource to $0$.
• Figure 5: An example of depth $2$ which contains a subtree solution $x^{(e')}$ for any $e'$ and fools the naive LP, but does not fool the Sherali-Adams hierarchy. In this example, all non-sink vertices require out-degree $k$. There are $k^2$ vertices connected to the source, each of them connected to a single private sink, and an additional set of $k$ public sinks which are shared with all other vertices.

Theorems & Definitions (29)

• Theorem 1: Main result
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• proof
• Lemma 7: Stirling's approximation
• Lemma 8
• proof