Fair colorful k-center clustering

An instance of colorfulk-center consists of points in a metric space that are colored red or blue, along with an integer k and a coverage requirement for each color. The goal is to find the smallest radius $$\rho $$ ρ such that there exist balls of radius $$\rho $$ ρ around k of the points that meet the coverage requirements. The motivation behind this problem is twofold. First, from fairness considerations: each color/group should receive a similar service guarantee, and second, from the algorithmic challenges it poses: this problem combines the difficulties of clustering along with the subset-sum problem. In particular, we show that this combination results in strong integrality gap lower bounds for several natural linear programming relaxations. Our main result is an efficient approximation algorithm that overcomes these difficulties to achieve an approximation guarantee of 3, nearly matching the tight approximation guarantee of 2 for the classical k-center problem which this problem generalizes. algorithms either opened more than k centers or only worked in the special case when the input points are in the plane.

Machine Teaching for Inverse Reinforcement Learning: Algorithms and Applications

Inverse reinforcement learning (IRL) infers a reward function from demonstrations, allowing for policy improvement and generalization. However, despite much recent interest in IRL, little work has been done to understand the minimum set of demonstrations needed to teach a specific sequential decisionmaking task. We formalize the problem of finding maximally informative demonstrations for IRL as a machine teaching problem where the goal is to find the minimum number of demonstrations needed to specify the reward equivalence class of the demonstrator. We extend previous work on algorithmic teaching for sequential decision-making tasks by showing a reduction to the set cover problem which enables an efficient approximation algorithm for determining the set of maximallyinformative demonstrations. We apply our proposed machine teaching algorithm to two novel applications: providing a lower bound on the number of queries needed to learn a policy using active IRL and developing a novel IRL algorithm that can learn more efficiently from informative demonstrations than a standard IRL approach.

Abstract Earliest Arrival Transshipment with Network Reconfiguration

The abstract flow model is the generalization of network flow model which deals with the flow paths (routes) satisfying the switching property. Contraflow model increases the flow value by reversing the required arc directions from the sources to the sinks. In this paper, we integrate the concepts of abstract flow and contraflow to introduce abstract earliest arrival transshipment contraflow model in multi-terminal abstract network. The abstract contraflow on multi-terminal dynamic network is NP-Complete. We present an efficient approximation algorithm to solve the problem. This approach satisfies the demand of sinks by sending optimal flow at every possible time point and seeks to eliminate the crossing conflicts.