Randomized Rounding over Dynamic Programs

We show that under mild assumptions for a problem whose solutions admit a dynamic programming-like recurrence relation, we can still find a solution under additional packing constraints, which need to be satisfied approximately. The number of additional constraints can be very large, for example, polynomial in the problem size. Technically, we reinterpret the dynamic programming subproblems and their solutions as a network design problem. Inspired by techniques from, for example, the Directed Steiner Tree problem, we construct a strong LP relaxation, on which we then apply randomized rounding. Our approximation guarantees on the packing constraints have roughly the form of a $(n^{\epsilon} \mathrm{polylog}\ n)$-approximation in time $n^{O(1/\epsilon)}$, for any $\epsilon > 0$. By setting $\epsilon=\log \log n/\log n$, we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting $\epsilon$ as a constant, an $n^\epsilon$-approximation in polynomial time. While there are necessary assumptions on the form of the DP, it is general enough to capture many textbook dynamic programs from Shortest Path to Longest Common Subsequence. Our algorithm then implies that we can impose additional constraints on the solutions to these problems. This allows us to model various problems from the literature in approximation algorithms, many of which were not thought to be connected to dynamic programming. In fact, our result can even be applied indirectly to some problems that involve covering instead of packing constraints, for example, the Directed Steiner Tree problem, or those that do not directly follow a recurrence relation, for example, variants of the Matching problem.