Worst-Case to Expander-Case Reductions: Derandomized and Generalized

TL;DR

The paper addresses the long-standing question of whether expanders are inherently easier or harder for core graph problems by developing derandomized and generalized self-reductions (Direct-WTERs) that transform arbitrary inputs into $\,\Omega(1)$-expanders with controlled blowups. The authors introduce a two-layer, deterministic core gadget that replaces the prior randomized approach, enabling deterministic Direct-WTERs and dynamic (DD-WTERs) variants across multiple models, including Fully Dynamic, CONGESTED-CLIQUE, and sublinear-MPC, and for a broad set of problems (Max-Cut, Densest Subgraph, OMv-related problems, Matching, Clique variants, and more). Key contributions include a derandomization of AW’s core gadget, a dynamic core gadget with amortized $O(1)$ update time, and a suite of reductions that lift OMv-based lower bounds to expanders, thereby arguing that expander-decomposition techniques are largely ineffective against problems admitting Direct-WTERs. The results yield near-optimal blowups (linear in $n$ and $m$) and preserve important graph properties (bipartiteness, degree bounds), with practical implications for understanding worst-case versus expander-case complexity and for designing robust deterministic algorithms in dynamic and distributed settings. Overall, the work strengthens the negative view of expander decompositions as a derandomization tool and broadens the applicability of self-reductions to multiple computation models and problem families.

Abstract

A recent paper by Abboud and Wallheimer [ITCS 2023] presents self-reductions for various fundamental graph problems, which transform worst-case instances to expanders, thus proving that the complexity remains unchanged if the input is assumed to be an expander. An interesting corollary of their self-reductions is that if some problem admits such reduction, then the popular algorithmic paradigm based on expander-decompositions is useless against it. In this paper, we improve their core gadget, which augments a graph to make it an expander while retaining its important structure. Our new core construction has the benefit of being simple to analyze and generalize while obtaining the following results: 1. A derandomization of the self-reductions, showing that the equivalence between worst-case and expander-case holds even for deterministic algorithms, and ruling out the use of expander-decompositions as a derandomization tool. 2. An extension of the results to other models of computation, such as the Fully Dynamic model and the Congested Clique model. In the former, we either improve or provide an alternative approach to some recent hardness results for dynamic expander graphs by Henzinger, Paz, and Sricharan [ESA 2022]. In addition, we continue this line of research by designing new self-reductions for more problems, such as Max-Cut and dynamic Densest Subgraph, and demonstrating that the core gadget can be utilized to lift lower bounds based on the OMv Conjecture to expanders.

Figures (4)

• Figure 1: The core gadget augments $G$ with $O(m+n)$ edges connected to a bipartite, $d$-regular, $\phi d$-edge expander, resulting in an $\Omega(1)$-expander $G_{exp}$.
• Figure 2: An illustration for the proof of \ref{['lem:coregadget']}. In the first case, we deal with cuts whose volume is mostly concentrated in $X$ and show that there are many short-dashed edges because $X$ is an expander, and the degrees in $L$ are bounded. In the second case, we deal with cuts whose volume is mostly concentrated in $G$ and show that, in this case, there must be many long-dashed edges.
• Figure 3: A Direct-WTER for the Max-Cut problem.
• Figure 4: Our Direct-WTER for $k$-Clique Detection, $k$-Clique Counting, and Max Clique. The dashed edges correspond to edges in $G$.

Theorems & Definitions (35)

• Theorem 2
• Theorem 3
• Definition 6
• Definition 7
• Lemma 9
• proof : Proof of \ref{['lem:coregadget']}
• Lemma 10
• proof
• Proposition 11
• proof