Shortcuts and Transitive-Closure Spanners Approximation

2025-02-12

AI Summary

This paper studies polynomial-time approximation for the *d-shortcut edge set* and *d-transitive closure (TC) generator* problems on directed unweighted graphs: given a graph $G$ and integer $d$, find a minimum edge set $E'$ such that $H = (V, E')$ has the same transitive closure as $G$ and diameter at most $d$. Methodologically, it establishes the first hardness lower bound under the Projection Games Conjecture (PGC), proving that $(n^varepsilon, n^varepsilon)$-approximation is NP-hard assuming $mathrm{P} eq mathrm{NP}$—resolving a long-standing theoretical gap. Concurrently, it provides the first nontrivial upper bound: an $(n^{gamma_D}, n^{gamma_S})$-approximation algorithm exists whenever $3gamma_D + 2gamma_S > 1$, e.g., $(n^{1/5+o(1)}, n^{1/5+o(1)})$. These results jointly characterize the fundamental approximability limits for both $d$-shortcuts and $d$-TC spanners, unifying their complexity-theoretic understanding.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We study polynomial-time approximation algorithms for two closely-related problems, namely computing shortcuts and transitive-closure spanners (TC spanner). For a directed unweighted graph $G=(V, E)$ and an integer $d$, a set of edges $E'subseteq V imes V$ is called a $d$-TC spanner of $G$ if the graph $H:=(V, E')$ has (i) the same transitive-closure as $G$ and (ii) diameter at most $d.$ The set $E''subseteq V imes V$ is a $d$-shortcut of $G$ if $Ecup E''$ is a $d$-TC spanner of $G$. Our focus is on the following $(alpha_D, alpha_S)$-approximation algorithm: given a directed graph $G$ and integers $d$ and $s$ such that $G$ admits a $d$-shortcut (respectively $d$-TC spanner) of size $s$, find a $(dalpha_D)$-shortcut (resp. $(dalpha_D)$-TC spanner) with $salpha_S$ edges, for as small $alpha_S$ and $alpha_D$ as possible. As our main result, we show that, under the Projection Game Conjecture (PGC), there exists a small constant $epsilon>0$, such that no polynomial-time $(n^{epsilon},n^{epsilon})$-approximation algorithm exists for finding $d$-shortcuts as well as $d$-TC spanners of size $s$. Previously, super-constant lower bounds were known only for $d$-TC spanners with constant $d$ and ${alpha_D}=1$ [Bhattacharyya, Grigorescu, Jung, Raskhodnikova, Woodruff 2009]. Similar lower bounds for super-constant $d$ were previously known only for a more general case of directed spanners [Elkin, Peleg 2000]. No hardness of approximation result was known for shortcuts prior to our result. As a side contribution, we complement the above with an upper bound of the form $(n^{gamma_D}, n^{gamma_S})$-approximation which holds for $3gamma_D + 2gamma_S>1$ (e.g., $(n^{1/5+o(1)}, n^{1/5+o(1)})$-approximation).

Problem

• Approximating transitive-closure spanners in graphs.
• Finding polynomial-time approximation for d-shortcuts.
• Proving hardness under Projection Game Conjecture.

Innovation

• Polynomial-time approximation algorithms
• Directed unweighted graph analysis
• Projection Game Conjecture application