New Lower Bounds in Merlin-Arthur Communication and Graph Streaming Verification

TL;DR

The paper advances the understanding of Merlin-Arthur and annotated streaming verification by introducing the non-trivial OMA complexity measure and proving new lower bounds via the Equals-Index problem. It establishes that the non-trivial OMA complexity $\widehat{MA}^{\to}$ for eq{-}idx is exponentially larger than its classical one-way or standard MA complexity, even yielding explicit functions with gaps $\widehat{MA}^{\to}(f)=\exp(\Omega(MA^{\to}(f)))$ and $\widehat{MA}^{\to}(f)=\exp(\Omega(R^{\to}(f)))$. These results propagate to annotated streaming, delivering strong lower bounds for Distinct Items and graph problems under SGTurnstile streams, while also developing strong $\ell_0$-samplers that enable near-optimal classical streaming algorithms in such regimes. The work also develops a suite of annotated streaming schemes for $k$-vertex-connectivity and $k$-edge-connectivity, including layering-based proofs and AM-model variants, highlighting a separation between classical streaming and annotated streaming under complex streams. Overall, the paper delivers canonical hard problems, explicit exponential gaps, and robust algorithmic tools (strong $\ell_0$- samplers, layering) that deepen our understanding of verification in both communication and streaming models and open avenues for efficient non-trivial certificates in graph problems.

Abstract

We show new lower bounds in the \emph{Merlin-Arthur} (MA) communication model and the related \emph{annotated streaming} or stream verification model. The MA communication model is an enhancement of the classical communication model, where in addition to the usual players Alice and Bob, there is an all-powerful but untrusted player Merlin who knows their inputs and tries to convince them about the output. Most functions have MA protocols with total communication significantly smaller than what would be needed without Merlin. We focus on the online MA (OMA) model, which is the MA analogue of one-way communication, and introduce the notion of \emph{non-trivial-OMA} complexity of a function. This is the minimum total communication needed by any non-trivial OMA protocol computing that function, where a trivial OMA protocol is one where Alice sends Bob roughly as many bits as she would have sent without Merlin. We prove a lower bound on the non-trivial-OMA complexity of a natural function \emph{Equals-Index} (basically the well-known Index problem on large domains) and identify it as a canonical problem for proving strong lower bounds on this complexity: reductions from it (i) reproduce and/or improve upon the lower bounds for all functions that were previously known to have large non-trivial-OMA complexity, (ii) exhibit the first explicit functions whose non-trivial-OMA complexity is superlinear, and even exponential, in their classical one-way complexity, and (iii) show functions on input size $n$ for which this complexity is as large as $n/\log n$. While exhibiting a function with $ω(\sqrt{n})$ (standard) OMA complexity is a longstanding open problem, we did not even know of any function with $ω(\sqrt{n})$ non-trivial-OMA complexity. We further extend the lower bounds to a related streaming model called annotated streaming.

Figures (5)

• Figure 1: An illustration of the cut $S$ in $G$ and $H$. After $X$ is deleted, there are no edges between $S$ and $T$ in $H$, while $G$ has at least one edge $e=(s,t)$ between $S$ and $T$, to ensure its $k$-edge-connectivity as $\left\vert{X}\right\vert < k$.
• Figure 2: An illustration of the $k$ edge-disjoint $s$-$t$ paths $P_1(X),P_2(X),\ldots,P_k(X)$. Note that the paths can share vertices but are edge-disjoint.
• Figure 3: An illustration of a good graph $G_{i^*}$ wherein the edge $e=(s,t)$ is sampled and all the edges in set $X$ are not. Thus, none of the $s$-$t$ paths, except for the edge $e$, exist in $G_{i^*}$. Therefore, the spanning forest $T_{i^*}$ necessarily contains the edge $e=(s,t)$.
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Theorems & Definitions (129)

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