Consensus Halving for Sets of Items

Abstract

Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work shows that, when the resource is represented by an interval, a consensus halving with at most n cuts always exists but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting in which the resource consists of a set of items without a linear ordering. For agents with linear and additively separable utilities, we present a polynomialtime algorithm that computes a consensus halving with at most n cuts and show that n cuts are almost surely necessary when the agents' utilities are randomly generated. On the other hand, we show that, for a simple class of monotonic utilities, the problem already becomes polynomial parity argument, directed version-hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus k-splitting, with which we wish to divide the resource into k parts in possibly unequal ratios and provide some consequences of our results on the problem of computing small agreeable sets.

Constant Inapproximability for PPA

“…In [Goldberg et al, 2020] the problem under study deviates slightly from the typical consensus halving problem. There are (divisible) items and they are not presented in a linear order, but rather unordered, with agents having linear and additively separable utilities over them.…”

Two's company, three's a crowd: Consensus-halving for a constant number of agents

“…Batziou et al [12] showed that the corresponding strong approximation problem (with measures represented by algebraic circuits) is complete for BU. A version of Consensus-Halving with divisible items was studied by Goldberg et al [46], who proved that the problem is polynomial-time solvable for additive utilities, but PPAD-hard for slightly more general utilities. Very recently, Deligkas et al [30] showed the PPA-completeness of the related Pizza Sharing problem [50], via a reduction from Consensus-Halving.…”

On the Complexity of Equilibrium Computation in First-Price Auctions

“…We are now ready to show PPAD-hardness. To do this, we reduce from a slightly modi ed version of G studied by Goldberg et al [2020], that we call G [−1,1] . is modi ed version operates on and 𝐺 [−1,1] ×−𝜁 (where the gates truncate to [−1, 1], and 𝜁 ∈ [0, 1]).…”

“…is modi ed version operates on and 𝐺 [−1,1] ×−𝜁 (where the gates truncate to [−1, 1], and 𝜁 ∈ [0, 1]). Goldberg et al [2020] proved that 𝜀 -G [−1,1] is PPAD-hard for some su ciently small constant 𝜀 > 0. We now set 𝜀 := 𝜀 /50.…”