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Randomized Rounding over Dynamic Programs

Etienne Bamas, Shi Li, Lars Rohwedder

TL;DR

This work introduces Additive-DP, a DP-like framework where many global packing or covering constraints are imposed on DP subproblem solutions. It reduces Additive-DP to a Flexible Tree Labeling problem and then to a Perfect Binary Tree Labeling, enabling a polynomial-size LP relaxation and randomized rounding that approximately satisfies packing constraints. The main result provides a tunable, multi-parameter approximation in time $({\Delta}{\Phi})^{O(1/\varepsilon)}$ with a constraint-violation factor $\alpha = O\left(\frac{\Delta^{\varepsilon}}{\varepsilon^2}\log m\right)$, plus extensions to cost-preserving rounding. The framework yields polylogarithmic approximations in quasi-polynomial time for broad classes of DP-like problems and offers practical results for Robust Shortest Path, Directed Steiner Tree variants, Longest Common Subsequence, Santa Claus, and robust augmentation problems, significantly broadening the reach of DP-based approximation methods.

Abstract

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 $(n^ε \mathrm{polylog}\ n)$-approximation in time $n^{O(1/ε)}$, for any $ε> 0$. By setting $ε=\log \log n/\log n$, we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting $ε$ as a constant, an $n^ε$-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.

Randomized Rounding over Dynamic Programs

TL;DR

This work introduces Additive-DP, a DP-like framework where many global packing or covering constraints are imposed on DP subproblem solutions. It reduces Additive-DP to a Flexible Tree Labeling problem and then to a Perfect Binary Tree Labeling, enabling a polynomial-size LP relaxation and randomized rounding that approximately satisfies packing constraints. The main result provides a tunable, multi-parameter approximation in time with a constraint-violation factor , plus extensions to cost-preserving rounding. The framework yields polylogarithmic approximations in quasi-polynomial time for broad classes of DP-like problems and offers practical results for Robust Shortest Path, Directed Steiner Tree variants, Longest Common Subsequence, Santa Claus, and robust augmentation problems, significantly broadening the reach of DP-based approximation methods.

Abstract

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 -approximation in time , for any . By setting , we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting as a constant, an -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.

Paper Structure

This paper contains 35 sections, 24 theorems, 72 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Applications to packing problems
  3. Applications to covering problems
  4. Applications via augmentation
  5. Overview of our algorithm
  6. Related works
  7. Overview of the paper
  8. Reduction from Additive-DP to Perfect Binary Tree Labeling
  9. Rephrasing Additive-DP as the Flexible Tree Labeling Problem
  10. Making set size at most 2 in allowed pairs.
  11. Making the height of a valid label tree small
  12. Final cleaning
  13. Perfect Binary Tree Labeling without costs
  14. Collapsing the tree $T$
  15. Relaxation for Perfect Tree Labeling
  16. ...and 20 more sections

Key Result

Theorem 1.1

Suppose we are given an Additive-DP instance. Let ${\Phi} := |\mathcal{I}_\mathrm{base}| + \sum_{I \in \mathcal{I}\setminus \mathcal{I}^\mathrm{base}, C \in [k_I]} (\ell(I, C)+1)$. Suppose we are given a promised upper bound ${\Delta}$ on $\mathrm{size}_{I^\circ}(x^*)$ for the optimum solution $x^*

Figures (1)

  • Figure 1: Example of decomposition tree. Inside each rectangle is a piece on which tree-decompose is called. Portals are dashed. Labels are the numbers, with $2,5,6$ being base labels. The tripel below each piece is the corresponding label of the new FTL instance.

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 33 more