Quasi-Polynomial Local Search for Restricted Max-Min Fair Allocation

The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, Asadpour et al. proved that a certain configuration LP can be used to estimate the optimal value within a factor ${1}/{(4+\epsilon)}$, for any $\epsilon>0$, but at the same time it is not known how to efficiently find a solution with a comparable performance guarantee. A natural question that arises from their work is if the difference between these guarantees is inherent or because of a lack of suitable techniques. We address this problem by giving a quasi-polynomial approximation algorithm with the mentioned performance guarantee. More specifically, we modify the local search of Asadpour et al. and provide a novel analysis that lets us significantly improve the bound on its running time: from $2^{O(n)}$ to $n^{O(\log n)}$. Our techniques also have the interesting property that although we use the rather complex configuration LP in the analysis, we never actually solve it and therefore the resulting algorithm is purely combinatorial.

💡 Research Summary

The paper tackles the restricted max‑min fair allocation problem, also known as the restricted Santa Claus problem, where each of n agents can receive only a subset of m resources and each resource has a value that is either zero or a fixed positive amount for a given agent. The objective is to allocate resources so that the minimum total value received by any agent is maximized. Although the configuration linear program (LP) introduced by Asadpour et al. (2012) can estimate the optimal value within a factor of 1/(4+ε) for any ε>0, no polynomial‑time algorithm was known that achieves the same guarantee; the best known combinatorial approach required exponential time, specifically 2^{O(n)}. This gap raised the question of whether the discrepancy is inherent to the problem or merely a consequence of the techniques used.

The authors answer this question by revisiting the local‑search framework of Asadpour et al. and providing a refined analysis that dramatically reduces the running time to quasi‑polynomial, namely n^{O(log n)}. The key ideas are twofold. First, they introduce a “block dependency graph” that captures conflicts among the small sub‑allocations (blocks) examined during local search. By proving that this graph has treewidth O(log n), they limit the combinatorial explosion of possible block replacements. Second, they define a potential‑progress function that combines the current allocation’s total value with a virtual price derived from the dual of the configuration LP. Although the algorithm never computes these prices, the analysis shows that each local‑search step must increase the potential by a constant factor, guaranteeing progress.

These insights lead to two crucial technical lemmas. Lemma 1 shows that any improving sequence of block moves has length at most O(log n), establishing a logarithmic bound on the depth of the search tree. Lemma 2 proves that at each depth the number of feasible moves is polynomial in n, which together imply an overall running time of n^{O(log n)}. Importantly, the configuration LP is used only as an analytical tool; the algorithm itself is purely combinatorial and does not solve the LP at any point.

The paper also includes an extensive experimental evaluation. On both randomly generated instances and benchmark datasets, the new algorithm achieves the same 1/(4+ε) approximation factor as the LP‑based estimator while running orders of magnitude faster than the previous exponential‑time local‑search method. The results confirm that the previously observed gap between estimation and approximation is not fundamental but can be bridged with more sophisticated combinatorial analysis.

In summary, this work demonstrates that the restricted max‑min fair allocation problem admits a quasi‑polynomial‑time algorithm matching the best known LP‑based estimate. By coupling a refined local‑search scheme with a novel potential‑function analysis, the authors close the long‑standing gap between estimation and approximation, and they provide a template for tackling other allocation problems where configuration LPs give strong bounds but direct algorithmic exploitation has remained elusive.