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Branch-and-Bound Algorithms as Polynomial-time Approximation Schemes

Koppány István Encz, Monaldo Mastrolilli, Eleonora Vercesi

Abstract

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B\&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025

Branch-and-Bound Algorithms as Polynomial-time Approximation Schemes

Abstract

Branch-and-bound algorithms (B&B) and polynomial-time approximation schemes (PTAS) are two seemingly distant areas of combinatorial optimization. We intend to (partially) bridge the gap between them while expanding the boundary of theoretical knowledge on the B\&B framework. Branch-and-bound algorithms typically guarantee that an optimal solution is eventually found. However, we show that the standard implementation of branch-and-bound for certain knapsack and scheduling problems also exhibits PTAS-like behavior, yielding increasingly better solutions within polynomial time. Our findings are supported by computational experiments and comparisons with benchmark methods. This paper is an extended version of a paper accepted at ICALP 2025

Paper Structure

This paper contains 11 sections, 22 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A B&B PTAS for the Multi-Knapsack Problem
  3. Proof of Theorem \ref{['multi_knapsack_ptas']}
  4. A B&B EPTAS for the Unrelated Machine Scheduling Problem
  5. Proof of Theorem \ref{['unrel_machine_ptas']}
  6. A B&B FPTAS for the Uniform Machine Scheduling Problem
  7. Proof of Theorem \ref{['fptas']}
  8. A B&B PTAS for the Identical Machine Scheduling Problem with a non-constant number of machines
  9. Computational Experiments
  10. Multiple knapsack problem
  11. Unrelated machine scheduling problem

Key Result

Theorem 1

For every fixed $0< \alpha < 1$, the algorithm $A^{\text{knap}}_{\alpha}$ returns an $\alpha$-approximate solution to the multiple knapsack problem, after processing $O(n^{c_{\alpha,m}+1} \cdot m^{c_{\alpha,m}})$-many nodes in the branching tree for some constant $c_{\alpha, m}$ that depends on $\al

Figures (3)

  • Figure 1: Depiction of a general step of our B&B for the multi-knapsack problem. On the left is the B&B tree built so far. White nodes correspond to active nodes. On the right is an example of the branching phase. The node in consideration is the black one. On the top left corner is the corresponding sub-problem, with the black portions of the knapsacks representing the items fixed so far. On the top right, the fractional optimal solution, with $m$ critical items highlighted in grey. The most profitable critical item is fixed on each of the $m+1$ branches, on the $i$-th branch in knapsack $i$ and on the $(m+1)$-th branch it is disposed of.
  • Figure 2: The relationship between the three trees. For $i=2$, $v \sim_{(i-1)\varepsilon} u$, but $u \not \in F^{\text{sim-prof}}$. $u' \sim_{\varepsilon} u$ and $u' \in F^{\text{sim-prof}}$, so $v \sim_{i\varepsilon} u'$.
  • Figure 3: The notion of $\varepsilon$-similar (left), and $\varepsilon$-equivalent profiles (right). Each dot represents a potential profile vector with two coordinates. The first coordinate corresponds to the $x$-axis, the second to the $y$-axis. Profiles in the same cell (left) or the same grid point (right) are $\varepsilon$-similar/equivalent; one of them can be discarded without decreasing worst-case guarantees beyond $(1+\varepsilon)$ unless they are in the gray area. The number of cells/grid points only depends on $\varepsilon$.

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Proposition 1: Martello, Toth; martello_toth
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 22 more