Online combinatorial assignment in independence systems

Abstract

We consider an online multi-weighted generalization of several classic online optimization problems called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online one by one. Upon arrival, each agent reveals a weight function over the elements of the ground set. If the independence system is given by the matchings of a hypergraph, we recover the combinatorial auction problem, where every node represents an item to be sold, and every edge represents a bundle of items. For combinatorial auctions, Kesselheim et al. showed upper bounds of O(loglog(k)/log(k))documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(log log (k)/log (k))$$end{document} and O(loglog(n)/log(n))documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(log log (n)/log (n))$$end{document} on the competitiveness of any online algorithm, even in the random order model, where k is the maximum bundle size and n is the number of items. We provide an exponential improvement by giving upper bounds of O(log(k)/k)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(log (k)/k)$$end{document}, and O(log(n)/n)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$O(log (n)/sqrt{n})$$end{document} for the prophet IID setting. Furthermore, using linear programming, we provide new and improved guarantees for the k-bounded online combinatorial auction problem (i.e., bundles of size at most k). We show a (1-e-k)/kdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$(1-e{-k})/k$$end{document}-competitive algorithm in the prophet IID model, a 1/(k+1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$1/(k+1)$$end{document}-competitive algorithm in the prophet-secretary model using a single sample per agent, and a k-k/(k-1)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$k{-k/(k-1)}$$end{document}-competitive algorithm in the secretary model. Our algorithms run in polynomial time and work in more general independence systems where the offline combinatorial assignment problem admits the existence of a polynomial-time randomized algorithm that we call certificate sampler. These systems include some classes of matroids, matroid intersections, and matchoids.