Convex Program Duality, Fisher Markets, and Nash Social Welfare

Abstract

We study Fisher markets and the problem of maximizing the Nash social welfare (NSW), and show several closely related new results. In particular, we obtain:• A new integer program for the NSW maximization problem whose fractional relaxation has a bounded integrality gap. In contrast, the natural integer program has an unbounded integrality gap.• An improved, and tight, factor 2 analysis of the algorithm of [7]; in turn showing that the integrality gap of the above relaxation is at most 2. The approximation factor shown by [7] was 2e 1/e ≈ 2.89.• A lower bound of e 1/e ≈ 1.44 on the integrality gap of this relaxation.• New convex programs for natural generalizations of linear Fisher markets and proofs that these markets admit rational equilibria.Recently, Cole and Gkatzelis [7] gave the first constant factor approximation algorithm for the problem of maximizing the Nash social welfare (NSW). In this problem, a set of indivisible goods needs to be allocated to agents with additive utilities, and the goal is to compute an allocation that maximizes the geometric mean of the agents' utilities. The natural integer program for this problem is closely related to the Fisher market model: if we relax the integrality constraint of the allocation, i.e., assume that the the goods are divisible, this program reduces to the Eisenberg-Gale (EG) convex program [11], whose solutions correspond to market equilibria for the linear Fisher market. Therefore, a canonical approach for designing a NSW approximation algorithm would be to compute a fractional allocation via the EG program, and then "round" it to get an integral one. However, [7] observed that this program's integrality gap is unbounded, and they were forced to follow an unconventional approach in analyzing their algorithm. This algorithm used an alternative fractional allocation, the spending-restricted (SR) equilibrium, and they had to come up with an independent upper bound of the optimal NSW in order to prove that the approximation factor is at most 2e 1/e ≈ 2.89.The absence of a conventional analysis for this problem could be, in part, to blame for the lack of progress on important follow-up problems (e.g., see Section 7). For instance, the SR equilibrium introduces constraints that are incompatible with the EG program, so [7] had to use a complicated algorithm for computing this allocation. Generalizing such an algorithm may be non-trivial, and so would proving new upper bounds for the optimal NSW. In this paper we remove this obstacle by uncovering the underlying structure of the NSW problem and shedding new light on the results of [7]. Specifically, we propose a new integer program which, as we show, also computes the optimal NSW allocation. More importantly, we prove that the relaxation of this program computes the SR equilibrium, and, quite surprisingly, we also show that the objective of this program happens to be precisely the upper bound that was used in [7]. As a result, this new integer program yields a convex program for computing the SR equilibrium and, unl...

Nash Social Welfare for Indivisible Items under Separable, Piecewise-Linear Concave Utilities

“…The factor 2 algorithm for the linear case was shown to be tight in [9]. Hence, the bound for our algorithm stated above is also tight.…”

“…A solution under linear utilities obtained by the rounding procedure in Figure 5 can then be viewed as a solution under SPLC utilities. Since the rounding procedure gives a solution under linear utilities that is at least a factor 2 of the upper bound in 2.4 (as shown in [9]), the same solution is also a factor 2 approximation under SPLC utilities.…”

“…Our rounding algorithm is a generalization of the ones in [10,9]. We first allocate to agents the integral part of their allocation.…”

“…To get around this, we use information in the obtained fractional allocation to create a new instance under linear utilities. We apply the rounding procedure of [9] to the latter instance and it yields an allocation for our SPLC problem. A critical step in this proof is showing that the upper bound which [9] compare their solution to also applies to our SLPC instance.…”

“…We apply the rounding procedure of [9] to the latter instance and it yields an allocation for our SPLC problem. A critical step in this proof is showing that the upper bound which [9] compare their solution to also applies to our SLPC instance. Hence we get a factor 2 algorithm for the NSW problem with SPLC utilities.…”