Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.06869

Grouping Strategies on Two-Phase Methods for Bi-objective Combinatorial Optimization

Felipe O. Mota·Luís Paquete·Daniel Vanderpooten
View original source ↗Read source document ↗Export metadata ↓

Abstract

Two-phase methods are commonly used to solve bi-objective combinatorial optimization problems. In the first phase, all extreme supported nondominated points are generated through a dichotomic search. This phase also allows the identification of search zones that may contain other nondominated points. The second phase focuses on exploring these search zones to locate the remaining points, which typically accounts for most of the computational cost. Ranking algorithms are frequently employed to explore each zone individually, but this approach leads to redundancies, causing multiple visits to the same solutions. To mitigate these redundancies, we propose several strategies that group adjacent zones, allowing a single run of the ranking algorithm for the entire group. Additionally, we explore an implicit grouping approach based on a new concept of coverage. Our experiments on the Bi-Objective Spanning Tree Problem demonstrate the beneficial impact of these grouping strategies when combined with coverage.

Explore related subjects

cs.DScs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Felipe O. Mota, Luís Paquete, Daniel Vanderpooten. 2025-04-09. Grouping Strategies on Two-Phase Methods for Bi-objective Combinatorial Optimization. https://arxiv.org/abs/2504.06869

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗
arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗