Greedy Algorithms for Shortcut Sets and Hopsets

TL;DR

This work investigates shortcut sets and exact hopsets, introducing simple greedy, potential-based algorithms that complement the dominant sampling-based techniques. The core idea is to optimize a potential $\phi(H)$ that captures remaining long-distance pairs, driving edge additions that rapidly shrink the diameter or hopbound gap. The authors prove that, for shortcut sets, the greedy method matches the state-of-the-art size-hobound trade-offs up to polylog factors, yielding bounds like $|H| \le \tilde{O}(\frac{n^2}{\beta^3} + \frac{n^{3/2}}{\beta^{3/2}})$; for exact hopsets, the greedy approach achieves existentially optimal or near-optimal sizes under mild assumptions, via a link to $\mathsf{exopt}(n,m,\beta)$. They further present faster deterministic algorithms by combining chain covers with greedy set cover techniques, achieving near-linear time for $\sqrt{n}$-diameter shortcut sets and $\tilde{O}(m n^{2/3})$-time constructions for $O(n^{1/3})$-hopsets. The work also introduces novel lower-bound framing through perturbations that enforce unique shortest paths, offering a new lens on extremal bounds and instance-optimality questions. Overall, the paper advances both extremal bounds and practical deterministic algorithms for hopsets and shortcut sets, with implications for parallel and dynamic graph algorithms.

Abstract

We explore the power of greedy algorithms for hopsets and shortcut sets. In particular, we propose simple greedy algorithms that, given an input graph $G$ and a parameter $β$, compute a shortcut set or an exact hopset $H$ of hopbound at most $β$, and we prove the following guarantees about the size $|H|$ of the output: For shortcut sets, we prove the bound $$|H| \le \tilde{O}\left( \frac{n^2}{β^3} + \frac{n^{3/2}}{β^{3/2}} \right).$$ This matches the current state-of-the-art upper bound by Kogan and Parter [SODA '22]. For exact hopsets of $n$-node, $m$-edge weighted graphs, the size of the output hopset is existentially optimal up to subpolynomial factors, under some technical assumptions. Despite their simplicity and conceptual implications, these greedy algorithms are slower than existing sampling-based approaches. Our second set of results focus on faster deterministic algorithms that are based on a certain greedy set cover approximation algorithm on paths in the transitive closure. One consequence is a deterministic algorithm that takes $O(mn^{2/3})$ time to compute a shortcut set of size $\tilde{O}(n)$ and hopbound $O(n^{1/3})$.

Figures (3)

• Figure 1: The (red) valid path $P$ passes through $L$ chains in the $\ell$-chain cover. The shortcut edge between $v^{2}_{L/3}$ and $v^{1}_{2L/3}$ reduces the normalized distance between the first $L/3$ vertices on the path to the last $L/3$ chains on the path by $L/3$.
• Figure 2: The short path constructed in the proof of \ref{['lem:setcover-gives-shortcuts']}. Every red edge represents a segment of the path that has length at most $D$ (or even constant for the parts that follow a chain). $C_\mathrm{pre}$ and $C_\mathrm{suf}$ are included in the set cover $C'$. The segments between $v'$ and $v_\mathrm{pre}$ as well as between $v_\mathrm{suf}$ and $w$ are suffixes covered by $C'$.
• Figure 3: Result of computing $\mathop{\mathrm{\varphi}}\nolimits^C(v)$ for the nodes in $T_3^C$. Observe that multiple nodes from the chain $C'$ appear in $T_3^C$ and only those occurrences that are not a descendant of another node from $C'$ in $T^C_3$ are assigned non-zero values.

Theorems & Definitions (64)

• Theorem 1.1: First Main Result
• Theorem 1.3: Second Main Result; See also \ref{['thm:hsexopt']}
• Theorem 1.5: Third Main Result
• Definition 2.1: Hopdistance
• Definition 2.2: Hopsets
• Definition 2.3: Shortcut Sets
• Lemma 2.4: Lemma 1.3 of Raskhodnikova10
• Corollary 2.5
• proof
• Definition 2.6: $\ell$-Cover; Definition 2.1 of kogan_beating_2022