Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

Incomplete Information VCG Contracts for Common Agency

We study contract design for welfare maximization in the well-known “common agency” model introduced in 1986 by Bernheim and Whinston. This model combines the challenges of coordinating multiple principals with the fundamental challenge of contract design: that principals have incomplete information of the agent’s choice of action. Our goal is to design contracts that satisfy truthfulness of the principals, welfare maximization by the agent, and two fundamental properties of individual rationality (IR) for the principals and limited liability (LL) for the agent. Our results reveal an inherent impossibility. Whereas for every common agency setting there exists a truthful and welfare-maximizing contract, which we refer to as “incomplete information Vickrey–Clarke–Groves contracts,” there is no such contract that also satisfies IR and LL for all settings. As our main results, we show that the class of settings for which there exists a contract that satisfies truthfulness, welfare maximization, LL, and IR is identifiable by a polynomial-time algorithm. Furthermore, for these settings, we design a polynomial-time computable contract: given valuation reports from the principals, it returns, if possible for the setting, a payment scheme for the agent that constitutes a contract with all desired properties. We also give a sufficient graph-theoretic condition on the population of principals that ensures the existence of such a contract and two truthful and welfare-maximizing contracts, in which one satisfies LL and the other one satisfies IR.</p

Unified Fair Allocation of Goods and Chores via Copies

We consider fair allocation of indivisible items in a model with goods, chores, and copies, as a unified framework for studying: (1) the existence of EFX and other solution concepts for goods with copies; (2) the existence of EFX and other solution concepts for chores. We establish a tight relation between these issues via two conceptual contributions: First, a refinement of envy-based fairness notions that we term envy without commons (denoted EFX WC when applied to EFX). Second, a formal duality theorem relating the existence of a host of (refined) fair allocation concepts for copies to their existence for chores. We demonstrate the usefulness of our duality result by using it to characterize the existence of EFX for chores through the dual environment, as well as to prove EFX existence in the special case of leveled preferences over the chores. We further study the hierarchy among envy-freeness notions without commons and their α-MMS guarantees, showing, for example, that any EFX WC allocation guarantees at least 11 4 -MMS for goods with copies.</p