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Register a new userPopular Branchings and Their Dual Certificates
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Added by
Ulrike Schmidt-Kraepelin
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Let $G$ be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching $B$ is popular if $B$ does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if $G$ admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that approximately popular branchings always exist.
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We propose an algorithm-independent framework to equip existing optimization methods with primal-dual certificates. Such certificates and corresponding rate of convergence guarantees are important for practitioners to diagnose progress, in particular in machine learning applications. We obtain new primal-dual convergence rates, e.g., for the Lasso as well as many L1, Elastic Net, group Lasso and TV-regularized problems. The theory applies to any norm-regularized generalized linear model. Our approach provides efficiently computable duality gaps which are globally defined, without modifying the original problems in the region of interest.
A matching in a bipartite graph $G:=(X + Y,E)$ is said to be envy-free if no unmatched vertex in $X$ is adjacent to a mathced vertex in $Y$. Every perfect matching is envy-free, but envy-free matchings may exist even when perfect matchings do not. We provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources (cakes) or discrete ones. In particular, we show a symmetric algorithm for proportional cake-cutting, an algorithm for $1$-out-of-$(2n-2)$ maximin-share allocation of discrete goods, and an algorithm for $1$-out-of-$lfloor 2n/3rfloor$ maximin-share allocation of discrete bads (chores) among $n$ agents.
In this paper, we give a simple characterization of a set of popular matchings defined by preference lists with ties. By employing our characterization, we propose a polynomial time algorithm for finding a minimum cost popular matching.
In a classical online decision problem, a decision-maker who is trying to maximize her value inspects a sequence of arriving items to learn their values (drawn from known distributions), and decides when to stop the process by taking the current item. The goal is to prove a prophet inequality: that she can do approximately as well as a prophet with foreknowledge of all the values. In this work, we investigate this problem when the values are allowed to be correlated. Since non-trivial guarantees are impossible for arbitrary correlations, we consider a natural linear correlation structure introduced by Bateni et al. [ESA 2015] as a generalization of the common-base value model of Chawla et al. [GEB 2015]. A key challenge is that threshold-based algorithms, which are commonly used for prophet inequalities, no longer guarantee good performance for linear correlations. We relate this roadblock to another augmentations challenge that might be of independent interest: many existing prophet inequality algorithms are not robust to slight increase in the values of the arriving items. We leverage this intuition to prove bounds (matching up to constant factors) that decay gracefully with the amount of correlation of the arriving items. We extend these results to the case of selecting multiple items by designing a new $(1+o(1))$ approximation ratio algorithm that is robust to augmentations.
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