Shortcuts and Transitive-Closure Spanners Approximation

TL;DR

This work studies polynomial-time bicriteria approximations for two linked graph problems: computing $d$-shortcuts and building $d$-TC spanners. It formalizes the problem as finding a $(dα_D)$-shortcut with $sα_S$ edges when a $(d,s)$-shortcut exists, and similarly for TC-spanners, establishing a strong conditional hardness under the Projection Games Conjecture that no polynomial-time algorithm can achieve $(n^{ε},n^{ε})$-approximation for either problem. The authors also provide a matching upper-bound framework yielding $(n^{γ_D},n^{γ_S})$-approximations for all $3γ_D+2γ_S>1$, and develop a deep reduction toolkit based on Label Cover, PGC, and a Steiner-variant intermediate problem MinStShC, augmented by a compact boosting gadget $O^*$ to amplify density. These results effectively rule out subpolynomial bicriteria approximations for both shortcuts and TC-spanners under standard complexity assumptions, guiding future efforts toward structure-exploiting graph families and more nuanced approximation goals. The findings illuminate fundamental limits of universal shortcut-based parallel reachability approaches and connect spanner hardness to classic labeling problems via robust reductions.

Abstract

We study polynomial-time approximation algorithms for two closely-related problems, namely computing shortcuts and transitive-closure spanners (TC spanners). For a directed unweighted graph $G=(V, E)$ and an integer $d$, a set of edges $E'\subseteq V\times V$ 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 $E''\subseteq V\times V$ is a $d$-shortcut of $G$ if $E\cup E''$ is a $d$-TC spanner of $G$. Our focus is on the following $(α_D, α_S)$-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 $(dα_D)$-shortcut (resp. $(dα_D)$-TC spanner) with $sα_S$ edges, for as small $α_S$ and $α_D$ as possible. As our main result, we show that, under the Projection Game Conjecture (PGC), there exists a small constant $ε>0$, such that no polynomial-time $(n^ε,n^ε)$-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 ${α_D}=1$ [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 $(n^{γ_D}, n^{γ_S})$-approximation which holds for $3γ_D + 2γ_S > 1$ (e.g., $(n^{1/5+o(1)}, n^{1/5+o(1)})$-approximation).

Figures (4)

• Figure 1: A high-level structure of our reduction. The boosting step handles the instances of the intermediate problem.
• Figure 2: In this example, $|A|=|B|=|\mathcal{L}|=\rho-1=3$. Suppose $A=\{1,2,3\},B=\{1,2,3\},\mathcal{L}=\{1,2,3\}$, then in this example we have $E=\{(2,2)\}$ and $\pi_{(2,2)}=\{(1,2),(2,3),(3,1)\}$, which corresponds to the three edges in the middle.
• Figure 3: Suppose the graph above is the graph $G_{geo}$ with $\Delta=2$ (notice that for ease of explanation, the graph does not satisfy properties described in \ref{['lem:geograph']}). The graph below shows how we substitute each node $v$ by $G_v$. If $v$ is in the even layer, then $G_v$ is a copy of $G_{inr}$; otherwise, if $v$ is not in the first or last layer, $G_v$ is a star graph.
• Figure 4: $G^A_{i,j,k}$ is one copy of the graph described in \ref{['lem:geograph']} in the $i$-th batch (from right to left), $j$-th fan (from top to bottom) and $k$-th piece. For different $k$ and fixed $i,j$, $G^A_{i,j,k}$ shares the same first layer nodes (which are $s^A_{(i,j),\ell}$), but have different last layer nodes (which are $t^A_{(i,j,k),\ell}$). By changing $A$ to $B$ we get another side of the graph. Each dashed rectangle specifies the graph similar to the middle part in \ref{['fig:minrepgraph']} according to the input LabelCover instance $\mathcal{I}$. Fix $j,k$, for any $i,\ell$, we have $t^A_{(i,j,k),\ell}$ and $t^B_{(\ell,j,k),i}$ in the same dashed rectangle.

Theorems & Definitions (59)

• theorem 1.1: Conditional lower bound; informal
• definition 2.1: TC spanner
• definition 2.2: $(\alpha_D,\alpha_S)$- MinTC
• definition 2.3: Shortcut
• definition 2.4: $(\alpha_D,\alpha_S)$- MinShC
• lemma 2.5
• proof
• lemma 2.6
• proof
• definition 2.7: LabelCover