The Moser-Tardos Framework with Partial Resampling

Abstract

Abstract. The resampling algorithm of Moser & Tardos is a powerful approach to develop constructive versions of the Lovász Local Lemma (LLL). We generalize this to partial resampling: when a bad event holds, we resample an appropriately-random subset of the variables that define this event, rather than the entire set as in Moser & Tardos. This is particularly useful when the bad events are determined by sums of random variables. This leads to several improved algorithmic applications in scheduling, graph transversals, packet routing etc. For instance, we improve the approximation ratio of a generalized D-dimensional scheduling problem studied by Azar & Epstein from O(D) to O(log D/ log log D), and settle a conjecture of Szabó & Tardos on graph transversals asymptotically. As a point of comparison with the MT algorithm, we show a family of constraint satisfaction problems that does not have "locality" (every constraint is affected by every variable), and so the LLL and the Moser-Tardos algorithm do not give any useful results; however, our method gives strong scale-free approximation ratios and terminates in expected polynomial time for this family.

Coupled and k-sided placements: generalizing generalized assignment

“…We note that in recent independent work, Harris and Srinivasan have developed a novel algorithmic framework for certain generalized scheduling problems, using the Lovász Local Lemma [16][17][18]. Their framework also applies to MinkSP, yielding an O(log k/ log log k)-approximation algorithm for the problem [7].…”

Untitled

“…The natural way of proving this lemma would be to identify, for each bad event B ∈ τ, some necessary event occurring with probability at most P Ω (B). This is the general strategy in Moser & Tardos [31] and related constructive LLL variants such as [18], [1], [20]. This is not the proof we employ here; there is an additional factor of (n k − A 0 k )!/n!…”

“…The main complication is that when we encounter a bad event involving "π k (x) = y," and perform our algorithm's random swap associated with it, we could potentially change any entry of π k . In contrast, when we resample a variable in [31,18], all the changes are confined to that variable. There is a further technical issue: the current witness-tree-based algorithmic versions of the LLL such as [31,18], identify, for each bad event B in the witness-tree τ, some necessary event occurring with probability at most P Ω (B).…”

“…In contrast, when we resample a variable in [31,18], all the changes are confined to that variable. There is a further technical issue: the current witness-tree-based algorithmic versions of the LLL such as [31,18], identify, for each bad event B in the witness-tree τ, some necessary event occurring with probability at most P Ω (B). This is not the proof we employ here; there are significant additional terms ("(n k − A 0 k )!/n!…”

“…These properties include a well-characterized distribution on the output distribution at the end of the resampling process, a corresponding efficient parallel (RNC) algorithm, a partial-resampling variant (as developed in [18]), and an arbitrary (even adversarial) choice of which bad event to resample. All of these properties follow from the Witness Tree Lemma we show for our Swapping Algorithm.…”