NP-Hardness and a PTAS for Maximum Interval Multicover, with Applications to Dense Subgraphs
Given intervals and points on a line, each point with a demand, and a budget $K$, Maximum Interval Multicover asks for at most $K$ intervals that satisfy as many points as possible, a point being satisfied if it lies in at least as many chosen intervals as its demand. One application is committee elections in which each candidate is approved by consecutive voters and voters have individual demands. Xu, Ran and Zhang introduced the problem and gave a $1/2$-approximation; the objective is not submodular when demands exceed one. Li, Tu, Zhang and Cao gave a polynomial-time approximation scheme (PTAS) for the minimization counterpart, unweighted partial interval multicover, and asked for its complexity. We prove that both problems are NP-hard, even for intervals of equal length; as their optima are integers bounded by the input size, neither has a fully polynomial-time approximation scheme unless $\mathrm{P}=\mathrm{NP}$. We then give a PTAS for Maximum Interval Multicover, also for arbitrary nonnegative nondecreasing rewards for the coverage of each point. Building on a decomposition from Nonner's PTAS for Densest $k$-Subgraph on interval graphs, our scheme cuts the line into pieces at a small loss and solves every piece exactly. The key dynamic program handles intervals crossing a common point by choosing an interleaving of their left and right endpoints rather than sweeping along the line; it needs only two counters to track these intervals. The same algorithm yields PTASes on convex bipartite graphs for Bipartite Dense Subgraph, improving on the ratio $2/3$ of Backer and Keil for the subclass of bipartite permutation graphs, and on interval graphs for Maximum $k$-Vertex Cover and weighted clique counts.