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Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms

Authors Susanna Caroppo , Giordano Da Lozzo , Giuseppe Di Battista , Michael T. Goodrich , Martin Nöllenburg

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ACM Subject Classification
  • Theory of computation → Dynamic programming
  • Theory of computation → Quantum complexity theory
Keywords
  • Dynamic Programming
  • Quantum Algorithms
  • Quantum Random Access Memory

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Abstract

We introduce a quantum dynamic programming framework that allows us to directly extend to the quantum realm a large body of classical dynamic programming algorithms. The corresponding quantum dynamic programming algorithms retain the same space complexity as their classical counterpart, while achieving a computational speedup. For a combinatorial (search or optimization) problem P and an instance I of P, such a speedup can be expressed in terms of the average degree δ of the {dependency digraph} G_𝒫(I) of I, determined by a recursive formulation of P. The nodes of this graph are the subproblems of P induced by I and its arcs are directed from each subproblem to those on whose solution it relies. In particular, our framework allows us to solve the considered problems in Õ(|V(G_𝒫(I))| √δ) time. As an example, we obtain a quantum version of the Bellman-Ford algorithm for computing shortest paths from a single source vertex to all the other vertices in a weighted n-vertex digraph with m edges that runs in Õ(n√{nm}) time, which improves the best known classical upper bound when m ∈ Ω(n^{1.4}).

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Susanna Caroppo, Giordano Da Lozzo, Giuseppe Di Battista, Michael T. Goodrich, and Martin Nöllenburg. Quantum Speedups for Polynomial-Time Dynamic Programming Algorithms. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.14

Author Details

Susanna Caroppo
  • Roma Tre University, Rome, Italy
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Giuseppe Di Battista
  • Roma Tre University, Rome, Italy
Michael T. Goodrich
  • University of California, Irvine, CA, USA
Martin Nöllenburg
  • TU Wien, Austria

Funding

This research was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND), and the U.S. NSF under grant 2212129.

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