Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1709.10307

Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems

Mordecai J. Golin·Hadi Khodabande·Bo Qin
View original source ↗Read source document ↗Export metadata ↓

Abstract

Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge {\em in one time unit}. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing. We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include: a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an $O(\log n)$ approximation factor, unless $P=NP$. b) Polylogarithmic Bicriteria Approximations: Polynomial time $(O(\log^8 n), O(\log^2 κ))$ bicritera approximation algorithms for the Confluent Quickest Flow problem where $κ$ is the number of sinks, in both directed and undirected graphs. Corresponding results are also developed for the Confluent Maximum Flow over timeproblem. The techniques developed also improve recent approximation algorithms for static confluent flows.

Explore related subjects

cs.DS

Keep this discovery

BibTeXRIS

Mordecai J. Golin, Hadi Khodabande, Bo Qin. 2017-09-29. Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems. https://arxiv.org/abs/1709.10307

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↗