Lesson 3: Network Algorithms and Approximations by Mohammad Hajiaghayi: Maximum Coverage
This session explores the maximum coverage problem, also known as the budgeted maximum coverage problem, where the goal is to maximize the total weight of covered elements within a given budget. The input is similar to the set cover problem, with sets and a universe of elements, but now each set has a cost, and each element has a weight. The objective is to select a collection of sets within the budget constraint while maximizing the total weight of the covered elements.
The session discusses the greedy algorithm for maximum coverage, which selects sets that maximize the weight of uncovered elements at each step. This algorithm achieves a 1-1/e approximation for the unit cost case, where all sets have the same cost. However, for general cost functions, the greedy algorithm may not perform well.
The session also covers the concept of hard-capacitated problems, where the budget constraint cannot be violated, and the challenges they pose in algorithm design. It introduces configuration LPs as a potential approach to address these challenges, where variables are defined for configurations of sets within the budget.
Finally, the session discusses the limitations of approximation algorithms for maximum coverage, highlighting that it is not possible to achieve an approximation factor better than 1-1/e unless NP has a deterministic time of n to the log log n.
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