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Given a directed graph $G = (V, E)$, the $k$-path partition problem is to find a minimum collection of vertex-disjoint directed paths each of order at most $k$ to cover all the vertices of $V$. The problem has various applications in facility location, network monitoring, transportation and others. Its special case on undirected graphs has received much attention recently, but the general directed version is seemingly untouched in the literature. We present the first $k/2$-approximation algorithm, for any $k ge 3$, based on a novel concept of augmenting path to minimize the number of singletons in the partition. When $k ge 7$, we present an improved $(k+2)/3$-approximation algorithm based on the maximum path-cycle cover followed by a careful $2$-cycle elimination process. When $k = 3$, we define the second novel kind of augmenting paths and propose an improved $13/9$-approximation algorithm.
The restless bandit problem is one of the most well-studied generalizations of the celebrated stochastic multi-armed bandit problem in decision theory. In its ultimate generality, the restless bandit problem is known to be PSPACE-Hard to approximate
We give new approximation algorithms for the submodular joint replenishment problem and the inventory routing problem, using an iterative rounding approach. In both problems, we are given a set of $N$ items and a discrete time horizon of $T$ days in
In the Priority Steiner Tree (PST) problem, we are given an undirected graph $G=(V,E)$ with a source $s in V$ and terminals $T subseteq V setminus {s}$, where each terminal $v in T$ requires a nonnegative priority $P(v)$. The goal is to compute a min
In this paper, we consider several finite-horizon Bayesian multi-armed bandit problems with side constraints which are computationally intractable (NP-Hard) and for which no optimal (or near optimal) algorithms are known to exist with sub-exponential
$ $In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In $k$-clustering, opening $k$ facilities induces an assignment cost v