arXiv subjects

Benjamin Jauregui

Publications and source records attributed to Benjamin Jauregui.

6 recordsLinked to original sources

arXiv2026

A Simple Construction of Locally Checkable Problems Filling the LOCAL Complexity Gaps in Graphs with Arbitrary Large Degrees

We show that the complexity gaps in the round complexities of locally checkable labeling (LCL) problems are not due to the fact that solutions to LCL problems must be locally checkable, but solely to the fact that LCL problems are defined only for graphs of maximum degree upper bounded by some arbitrary yet constant value $\Delta$. Specifically, we show that there are infinitely many locally checkable problems (i.e., problems whose solutions can be checked locally) whose round complexities belongs to the two intervals $[\omega(1),o(\log\log^\star n)]$ and $[\omega(\log^\star n),o(\log n)]$ whenever these problems are considered in networks with unbounded maximum degrees. This extends the previous results by Schmid (arXiv, 2026), which hold for the polynomial regime only, and by Bousquet, Feuilloley, and Pierron (OPODIS, 2025), which hold for trees only. All our upper bounds are obtained using deterministic algorithms that can be run under the port-numbering model, which is a weak variant of LOCAL, without any a priori information on the number of nodes in the network. Instead, our lower bounds apply to randomized LOCAL, and quantum LOCAL, even if nodes have identifiers in $[1,n]$, and even if they know the exact number of nodes in the network. They even hold under randomized online LOCAL, a strong variant of the LOCAL model. Finally, our lower bounds hold even for trees. Our results are obtained using two main ingredients. The first one is the analysis of a new locally checkable problem called Increasing Degree, parameterized by a function $f:\mathbb{N}\to\mathbb{N}$. Different round complexities can be obtained by tuning the function $f$ accordingly. Our second tool is a general Translation Theorem that enables to transfer results from a given range of complexities to results for a range of lower complexities.

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arXiv2026

Distributed Statistical Zero-Knowledge Proofs via Sumcheck

We study distributed zero-knowledge proofs, introduced by Bick, Kol, and Oshman (SODA 2022). While distributed interactive proofs have advanced rapidly, general-purpose techniques for distributed zero-knowledge remain limited and mostly problem-specific. We address this gap by introducing distributed statistical zero-knowledge, requiring that each node's view be simulatable within negligible statistical distance, and by lifting the classical Sumcheck protocol (Lund, Fortnow, Karloff, and Nisan, FOCS 1990) into a modular primitive for distributed zero-knowledge proofs. Our main contribution is a distributed zero-knowledge implementation of Sumcheck. Given oracle access to a polynomial F over a finite field $\mathbb{F}$ with N variables, we design a protocol verifying claims of the form $\sum_{x\in\mathbb{F}} F(x)=a$ using $O(N)$ rounds of $O(\log |\mathbb{F}|)$-bit messages, while achieving statistical zero-knowledge and small soundness error. We apply this primitive to two problems. For non-k-colorability, we obtain an $O(n)$-round distributed statistical zero-knowledge proof deciding whether a graph is not k-colorable, for any constant k, using $O(log^{1+o(1)} n)$-bit messages. This is the first nontrivial distributed interactive proof for this problem, even without zero-knowledge guarantees. For Subgraph Counting, we obtain an $O(k \log n)$-round, $O(k \log n)$-bit distributed statistical zero-knowledge proof for counting copies of a given k-node pattern, improving previous distributed interactive proofs while additionally providing statistical zero-knowledge. Finally, we show that additional round compression of Sumcheck is problem-dependent: for non-3-colorability on constant-degree graphs, we prove a lower bound excluding $o(n/\log n)$ rounds under polynomial-time local computation.

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arXiv2025

Deterministic Distributed DFS and Other Problems via Cycle Separators in Planar Graphs

One of the most basic techniques in algorithm design consists of breaking a problem into subproblems and then proceeding recursively. In the case of graph algorithms, one way to implement this approach is through separator sets. Given a graph $G=(V,E)$, a subset of nodes $S \subseteq V$ is called a separator set of $G$ if the size of each connected component of $G-S$ is at most $2/3 \cdot |V|$. The most useful separator sets are those that satisfy certain restrictions of cardinality or structure. For over 40 years, various efficient algorithms have been developed for computing separators of different kinds, particularly in planar graphs. Separator sets, combined with a divide and conquer approach, have been fundamental in the design of efficient algorithms in various settings. In this work, we present the first deterministic algorithm in the distributed CONGEST model that recursively computes a cycle separator in planar graphs in $\tilde{\mathcal{O}}(D)$ rounds. This result, as in the centralized setting, has significant implications for distributed planar algorithms. In fact, from this result, we can construct a deterministic algorithm that computes a DFS tree in $\tilde{\mathcal{O}}(D)$ rounds. This matches both the best-known randomized algorithm of Ghaffari and Parter (DISC'17) and, up to polylogarithmic factors, the trivial lower bound of $\Omega(D)$ rounds. Besides DFS, our deterministic cycle separator algorithm can be used to derandomize several planar-graph algorithms whose only randomized ingredient is the computation of a cycle separator, such as maximum flow (Abd-Elhaleem, Dory, Parter and Weimann, PODC'25), single-source shortest path (Li and Parter, STOC'19), and reachability (Parter, DISC'20).

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arXiv2025

Strong and Hiding Distributed Certification of Bipartiteness

In this paper, we study the problem of certifying whether a graph is bipartite (i.e. $2$-colorable) with a locally checkable proof (LCP) that is able to hide a $2$-coloring from the verifier. More precisely, we say an LCP for $2$-coloring is hiding if, in a yes-instance, it is possible to assign certificates to nodes without revealing an explicit $2$-coloring. Motivated by the search for promise-free separations of extensions of the LOCAL model in the context of locally checkable labeling (LCL) problems, we also require the LCPs to satisfy what we refer to as the strong soundness property. This is a strengthening of soundness that requires that, in a no-instance (i.e., a non-$2$-colorable graph) and for every certificate assignment, the subset of accepting nodes must induce a $2$-colorable subgraph. We show that strong and hiding LCPs for $2$-coloring exist in specific graph classes and requiring only $O(\log n)$-sized certificates. Furthermore, when the input is promised to be a cycle or contains a node of degree $1$, we show the existence of strong and hiding LCPs even in an anonymous network and with constant-size certificates. Despite these upper bounds, we prove that there are no strong and hiding LCPs for $2$-coloring in general, unless the algorithm has access to node identifiers and uses certificates of size~$\omega(1)$. Furthermore, in anonymous networks, the lower bound holds regardless of the certificate size. The proof relies on a Ramsey-type result as well as an argument about the realizability of subgraphs of the neighborhood graph consisting of the accepting views of an LCP. Along the way, we also give a characterization of the hiding property for the general $k$-coloring problem that appears to be a key component for future investigations in this context.

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arXiv2021

Distributed Interactive Proofs for the Recognition of Some Geometric Intersection Graph Classes

A graph $G=(V,E)$ is a geometric intersection graph if every node $v \in V$ is identified with a geometric object of some particular type, and two nodes are adjacent if the corresponding objects intersect. Geometric intersection graph classes have been studied from both the theoretical and practical point of view. On the one hand, many hard problems can be efficiently solved or approximated when the input graph is restricted to a geometric intersection class of graphs. On the other hand, these graphs appear naturally in many applications such as sensor networks, scheduling problems, and others. Recently, in the context of distributed certification and distributed interactive proofs, the recognition of graph classes has started to be intensively studied. Different results related to the recognition of trees, bipartite graphs, bounded diameter graphs, triangle-free graphs, planar graphs, bounded genus graphs, $H$-minor free graphs, etc., have been obtained. The goal of the present work is to design efficient distributed protocols for the recognition of relevant geometric intersection graph classes, namely permutation graphs, trapezoid graphs, circle graphs, and polygon-circle graphs. More precisely, for the two first classes, we give proof labeling schemes recognizing them with logarithmic-sized certificates. For the other two classes, we give three-round distributed interactive protocols that use messages and certificates of size $\mathcal{O}(\log n)$. Finally, we provide logarithmic lower-bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes.

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arXiv2018

Distributed Treewidth Computation and Courcelle's Theorem in the CONGEST Model

Algorithmic meta-theorems, stating that graph properties expressible in some particular logic can be decided efficiently in graph classes having some specific structural properties, are now standard in sequential graph algorithms. One of the most classic examples is Courcelle's theorem: all properties expressible in Monadic Second-Order logic (MSO) are decidable in linear time in graphs of bounded treewidth. We provide here a distributed version of Courcelle's theorem, in the standard CONGEST model for distributed computing: For any MSO formula $\varphi$ and any constant $k$, there is a CONGEST algorithm that, given an input communication network $G$ of treewidth at most $k$ and of diameter $D$, decides if $G$ satisfies property $\varphi$ in $\tilde O(D)$ rounds. Simple examples show that the dependency on $D$ is unavoidable. Also, if we drop the assumption of bounded treewidth, deciding MSO properties such as 3-colorability are known to require $\tilde{\Omega}(n^2)$ rounds in the CONGEST model. Our results extend to optimization problems (e.g., computing a maximum size independent set, or a minimum dominating set) and counting (e.g. triangle counting). As usual, the $\tilde{O}$ notation hides polylogarithmic factors in $n$; here it also hides a constant factor depending on $k$ and on the MSO formula $\varphi$. We also give a distributed algorithm producing a linear approximation for treewidth: For any $k$, it decides that the treewidth of the input network $G$ is larger than $k$ or computes a tree decomposition of width $O(k)$ and depth $O(\log n)$, in $\tilde O(k^{O(k)} D)$ rounds in CONGEST. Our algorithms make use of the low-congestion shortcuts framework introduced by Ghaffari and Haeupler [SODA 2016], and our main technical tool is an $\tilde O(k^4 D)$ algorithm for computing $(s,t)$-vertex separators of size at most $k+1$ in graphs of treewidth at most $k$.

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