Approximate Submodularity in Network Design Problems

57 Pages Posted: 13 May 2021 Last revised: 24 Oct 2022

See all articles by Levi DeValve

Levi DeValve

University of Chicago - Booth School of Business

Saša Pekeč

Fuqua School of Business, Duke University

Yehua Wei

Decision Sciences Area, Fuqua School of Business, Duke University

Date Written: June 2, 2022

Abstract

Network design problems, such as flexibility design, are ubiquitous in modern marketplaces where firms constantly innovate new ways to match supply and demand. We develop a primal-dual based approach to analyze the flexibility design problem, i.e., allocating subsets of limited resources to each of multiple demand classes, and establish that the problem possesses a novel structural property. The property, which we call cover modularity, can be interpreted as an approximate form of submodularity, in the sense that local changes in the objective function can be used to bound global changes. We use this structure to analyze a class of greedy heuristics and establish the first constant factor approximation guarantee for solving the general flexibility design problem. Further, we identify a significant practical byproduct of our primal-dual analysis: the dual solutions we construct can be used as surrogates to guide the heuristics, leading to order of magnitude gains in computational efficiency without loss of optimization performance. Finally, we extend our analysis by demonstrating the presence of cover modularity in a general class of linear programming formulations, indicating applicability of our approach to a wide range of network design problems distinct from flexibility design.

Keywords: flexibility design, network design, approximate submodularity, primal-dual, approximation algorithm

Suggested Citation

DeValve, Levi and Pekeč, Saša and Wei, Yehua, Approximate Submodularity in Network Design Problems (June 2, 2022). Available at SSRN: https://ssrn.com/abstract=3844987 or http://dx.doi.org/10.2139/ssrn.3844987

Levi DeValve (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S Woodlawn Ave
Chicago, IL 60637

HOME PAGE: http://faculty.chicagobooth.edu/levi-devalve

Saša Pekeč

Fuqua School of Business, Duke University ( email )

Yehua Wei

Decision Sciences Area, Fuqua School of Business, Duke University ( email )

100 Fuqua Drive
Durham, NC 27708-0204
United States

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