Author:
Dorit Hochbaum
Affiliation:
Department of Industrial Engineering and Operations Research, University of California, Berkeley, U.S.A.
Keyword(s):
Parametric Flow, Maximum Diversity, Quadratic Knapsack, Efficient Frontier, Text Summarization.
Abstract:
This paper introduces new techniques for any NP-hard problems formulated as monotone integer programming (IPM) with a budget constraint “budgeted IPM”. Problems of this type have diverse applications, including maximizing team collaboration, the maximum diversity problem, facility dispersion, threat detection, minimizing conductance, clustering, and pattern recognition. We present a unified framework for effective algorithms for budgeted IPM problems based on the Langrangian relaxation of the budget constraint. It is shown that all optimal solutions for all values of the Lagrange multiplier are generated very efficiently, and the piecewise linear concave envelope (convex, for minimization problems) of these solutions has breakpoints that are optimal solutions for the respective budgets. This is used to derive high quality upper and lower bounds for budgets that do not correspond to breakpoints. We show that for all these problems, the weight “perturbation” concept, that was successfu
l for the problem of maximum diversity in enhancing the number and distribution of breakpoints, is applicable. Furthermore, the insights derived from this efficient frontier of solutions, lead to the result that all the respective ratio problems have a solution at the “first” breakpoint, which generalizes the concept of maximum density subgraph.
(More)