arXiv subjects

Ioan Todinca

Publications and source records attributed to Ioan Todinca.

At least 19 recordsLinked to original sources

arXiv2026

What Can Be Computed Locally Revisited: First-Order Logic on Sparse Graphs in Distributed Computing

The question of 'what can be computed locally?' lies at the heart of distributed computing in networks. As established in Naor and Stockmeyer's seminal paper (STOC 1993), this question is undecidable, even for graph problems whose solutions can be checked locally. In this paper, we adopt a novel perspective on the question, by asking for which classes $Π$ of problems, and for which classes $\mathcal{G}$ of graphs, all problems in $Π$ can be solved efficiently in a distributed manner in all graphs of $\mathcal{G}$. This paper focuses on two natural candidates for such an approach, namely the class of problems expressible in first-order logic (FO), because of their intrinsic form of locality thanks to Gaifman's theorem, and the class of graphs with bounded expansion, because they form a large class of graphs encompassing, e.g., planar, bounded-treewidth, and bounded-degree graphs. The starting point of our work is the decade-old open question of Nešetřil and Ossona de Mendez (Distributed Computing 2016) on the distributed complexity of local FO formula on graphs of bounded expansion, in the standard CONGEST model of distributed computing. A formula $φ(x)$ is local if the satisfaction of $φ(x)$ depends only on the $r$-neighborhood of its free variable $x$, for some fixed $r$. For instance, the formula '$x$ belongs to a triangle' is local. We resolve the open problem positively by showing that, for every local FO formula $φ(x)$, and for every graph class $\mathcal{G}$ of bounded expansion, there exists a deterministic algorithm that identifies, for every $n$-vertex graph $G\in \mathcal{G}$, all vertices $v$ of $G$ such that $G\models φ(v)$, in $O(\log n)$ rounds. When dropping the locality condition, we show that $O(D+\log n)$ rounds are sufficient for deciding any FO formula $φ$ on graphs of bounded expansion.

cs.DS↗

arXiv2025

Even-Cycle Detection in the Randomized and Quantum CONGEST Model

We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the \CONGEST{} model by a randomized Monte-Carlo distributed algorithm with one-sided error probability $1/3$. This matches the best round-complexities of previously known algorithms for $k\in\{2,3,4,5\}$ by Drucker et al. [PODC'14] and Censor-Hillel et al. [DISC'20], but improves the complexities of the known algorithms for $k>5$ by Eden et al. [DISC'19], which were essentially of the form $\tilde O(n^{1-2/k^2})$. Our algorithm uses colored BFS-explorations with threshold, but with an original \emph{global} approach that enables to overcome a recent impossibility result by Fraigniaud et al. [SIROCCO'23] about using colored BFS-exploration with \emph{local} threshold for detecting cycles. We also show how to quantize our algorithm for achieving a round-complexity $\tilde O(n^{\frac{1}{2}-\frac{1}{2k}})$ in the quantum setting for deciding $C_{2k}$ freeness. Furthermore, this allows us to improve the known quantum complexities of the simpler problem of detecting cycles of length \emph{at most}~$2k$ by van Apeldoorn and de Vos [PODC'22]. Our quantization is in two steps. First, the congestion of our randomized algorithm is reduced, to the cost of reducing its success probability too. Second, the success probability is boosted using a new quantum framework derived from sequential algorithms, namely Monte-Carlo quantum amplification.

cs.DC↗

arXiv2025

Tight Lieb-Robinson Bound for approximation ratio in Quantum Annealing

Quantum annealing (QA) holds promise for optimization problems in quantum computing, especially for combinatorial optimization. This analog framework attracts attention for its potential to address complex problems. Its gate-based homologous, QAOA with proven performance, has brought lots of attention to the NISQ era. Several numerical benchmarks try to classify these two metaheuristics however, classical computational power highly limits the performance insights. In this work, we introduce a new parametrized version of QA enabling a precise 1-local analysis of the algorithm. We develop a tight Lieb-Robinson bound for regular graphs, achieving the best-known numerical value to analyze QA locally. Studying MaxCut over cubic graph as a benchmark optimization problem, we show that a linear-schedule QA with a 1-local analysis achieves an approximation ratio over 0.7020, outperforming any known 1-local algorithms.

quant-ph↗

arXiv2025

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 $φ$ 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 $φ$ 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 $\tildeΩ(n^2)$ 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 $φ$. 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 $k$.

cs.DS↗

arXiv2025

Deterministic Even-Cycle Detection in Broadcast CONGEST

We show that, for every $k\geq 2$, $C_{2k}$-freeness can be decided in $O(n^{1-1/k})$ rounds in the Broadcast CONGEST model, by a deterministic algorithm. This (deterministic) round-complexity is optimal for $k=2$ up to logarithmic factors thanks to the lower bound for $C_4$-freeness by Drucker et al. [PODC 2014], which holds even for randomized algorithms. Moreover it matches the round-complexity of the best known randomized algorithms by Censor-Hillel et al. [DISC 2020] for $k\in\{3,4,5\}$, and by Fraigniaud et al. [PODC 2024] for $k\geq 6$. Our algorithm uses parallel BFS-explorations with deterministic selections of the set of paths that are forwarded at each round, in a way similar to what was done for the detection of odd-length cycles, by Korhonen and Rybicki [OPODIS 2017]. However, the key element in the design and analysis of our algorithm is a new combinatorial result bounding the "local density" of graphs without $2k$-cycles, which we believe is interesting on its own.

cs.DC↗

arXiv2025

Polynomial kernels for edge modification problems towards block and strictly chordal graphs

We consider edge modification problems towards block and strictly chordal graphs, where one is given an undirected graph $G = (V,E)$ and an integer $k \in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to obtain a block graph or a strictly chordal graph. The completion and deletion variants of these problems are defined similarly by only allowing edge additions for the former and only edge deletions for the latter. Block graphs are a well-studied class of graphs and admit several characterizations, e.g. they are diamond-free chordal graphs. Strictly chordal graphs, also referred to as block duplicate graphs, are a natural generalization of block graphs where one can add true twins of cut-vertices. Strictly chordal graphs are exactly dart and gem-free chordal graphs. We prove the NP-completeness for most variants of these problems and provide $O(k^2)$ vertex-kernels for Block Graph Editing and Block Graph Deletion, $O(k^3)$ vertex-kernels for Strictly Chordal Completion and Strictly Chordal Deletion and a $O(k^4)$ vertex-kernel for Strictly Chordal Editing.

cs.DS↗

arXiv2024

Distributed Model Checking on Graphs of Bounded Treedepth

We establish that every monadic second-order logic (MSO) formula on graphs with bounded treedepth is decidable in a constant number of rounds within the CONGEST model. To our knowledge, this marks the first meta-theorem regarding distributed model-checking. Various optimization problems on graphs are expressible in MSO. Examples include determining whether a graph $G$ has a clique of size $k$, whether it admits a coloring with $k$ colors, whether it contains a graph $H$ as a subgraph or minor, or whether terminal vertices in $G$ could be connected via vertex-disjoint paths. Our meta-theorem significantly enhances the work of Bousquet et al. [PODC 2022], which was focused on distributed certification of MSO on graphs with bounded treedepth. Moreover, our results can be extended to solving optimization and counting problems expressible in MSO, in graphs of bounded treedepth.

cs.DS↗

arXiv2023

Anti-crossings occurrence as exponentially closing gaps in Quantum Annealing

This paper explores the phenomenon of avoided level crossings in quantum annealing, a promising framework for quantum computing that may provide a quantum advantage for certain tasks. Quantum annealing involves letting a quantum system evolve according to the Schrödinger equation, with the goal of obtaining the optimal solution to an optimization problem through measurements of the final state. However, the continuous nature of quantum annealing makes analytical analysis challenging, particularly with regard to the instantaneous eigenenergies. The adiabatic theorem provides a theoretical result for the annealing time required to obtain the optimal solution with high probability, which is inversely proportional to the square of the minimum spectral gap. Avoided level crossings can create exponentially closing gaps, which can lead to exponentially long running times for optimization problems. In this paper, we use a perturbative expansion to derive a condition for the occurrence of an avoided level crossing during the annealing process. We then apply this condition to the MaxCut problem on bipartite graphs. We show that no exponentially small gaps arise for regular bipartite graphs, implying that QA can efficiently solve MaxCut in that case. On the other hand, we show that irregularities in the vertex degrees can lead to the satisfaction of the avoided level crossing occurrence condition. We provide numerical evidence to support this theoretical development, and discuss the relation between the presence of exponentially closing gaps and the failure of quantum annealing.

quant-ph↗

arXiv2023

Distributed Certification for Classes of Dense Graphs

A proof-labeling scheme (PLS) for a boolean predicate $Π$ on labeled graphs is a mechanism used for certifying the legality with respect to $Π$ of global network states in a distributed manner. In a PLS, a certificate is assigned to each processing node of the network, and the nodes are in charge of checking that the collection of certificates forms a global proof that the system is in a correct state, by exchanging the certificates once, between neighbors only. The main measure of complexity is the size of the certificates. Many PLSs have been designed for certifying specific predicates, including cycle-freeness, minimum-weight spanning tree, planarity, etc. In 2021, a breakthrough has been obtained, as a meta-theorem stating that a large set of properties have compact PLSs in a large class of networks. Namely, for every $\mathrm{MSO}_2$ property $Π$ on labeled graphs, there exists a PLS for $Π$ with $O(\log n)$-bit certificates for all graphs of bounded tree-depth. This result has been extended to the larger class of graphs with bounded {tree-width}, using certificates on $O(\log^2 n)$ bits. We extend this result even further, to the larger class of graphs with bounded clique-width, which, as opposed to the other two aforementioned classes, includes dense graphs. We show that, for every $\mathrm{MSO}_1$ property $Π$ on labeled graphs, there exists a PLS for $Π$ with $O(\log^2 n)$ bit certificates for all graphs of bounded clique-width.

cs.DC↗

arXiv2023

On graphs coverable by k shortest paths

We show that if the edges or vertices of an undirected graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is upper-bounded by a single-exponential function of $k$. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph $G$ and a set of $k$ pairs of vertices called terminals, asks whether $G$ can be covered by $k$ shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph $G$ and a set of $k$ terminals, asks whether there exist $\binom{k}{2}$ shortest paths covering $G$, each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter $k$.

cs.DM↗

arXiv2023

Energy-Efficient Distributed Algorithms for Synchronous Networks

We study the design of energy-efficient algorithms for the LOCAL and CONGEST models. Specifically, as a measure of complexity, we consider the maximum, taken over all the edges, or over all the nodes, of the number of rounds at which an edge, or a node, is active in the algorithm. We first show that every Turing-computable problem has a CONGEST algorithm with constant node-activation complexity, and therefore constant edge-activation complexity as well. That is, every node (resp., edge) is active in sending (resp., transmitting) messages for only $O(1)$ rounds during the whole execution of the algorithm. In other words, every Turing-computable problem can be solved by an algorithm consuming the least possible energy. In the LOCAL model, the same holds obviously, but with the additional feature that the algorithm runs in $O(\mbox{poly}(n))$ rounds in $n$-node networks. However, we show that insisting on algorithms running in $O(\mbox{poly}(n))$ rounds in the CONGEST model comes with a severe cost in terms of energy. Namely, there are problems requiring $Ω(\mbox{poly}(n))$ edge-activations (and thus $Ω(\mbox{poly}(n))$ node-activations as well) in the CONGEST model whenever solved by algorithms bounded to run in $O(\mbox{poly}(n))$ rounds. Finally, we demonstrate the existence of a sharp separation between the edge-activation complexity and the node-activation complexity in the CONGEST model, for algorithms bounded to run in $O(\mbox{poly}(n))$ rounds. Specifically, under this constraint, there is a problem with $O(1)$ edge-activation complexity but $\tildeΩ(n^{1/4})$ node-activation complexity.

cs.DC↗

arXiv2023

On the Power of Threshold-Based Algorithms for Detecting Cycles in the CONGEST Model

It is known that, for every $k\geq 2$, $C_{2k}$-freeness can be decided by a generic Monte-Carlo algorithm running in $n^{1-1/Θ(k^2)}$ rounds in the CONGEST model. For $2\leq k\leq 5$, faster Monte-Carlo algorithms do exist, running in $O(n^{1-1/k})$ rounds, based on upper bounding the number of messages to be forwarded, and aborting search sub-routines for which this number exceeds certain thresholds. We investigate the possible extension of these threshold-based algorithms, for the detection of larger cycles. We first show that, for every $k\geq 6$, there exists an infinite family of graphs containing a $2k$-cycle for which any threshold-based algorithm fails to detect that cycle. Hence, in particular, neither $C_{12}$-freeness nor $C_{14}$-freeness can be decided by threshold-based algorithms. Nevertheless, we show that $\{C_{12},C_{14}\}$-freeness can still be decided by a threshold-based algorithm, running in $O(n^{1-1/7})= O(n^{0.857\dots})$ rounds, which is faster than using the generic algorithm, which would run in $O(n^{1-1/22})\simeq O(n^{0.954\dots})$ rounds. Moreover, we exhibit an infinite collection of families of cycles such that threshold-based algorithms can decide $\mathcal{F}$-freeness for every $\mathcal{F}$ in this collection.

cs.DC↗

arXiv2022

Computing Power of Hybrid Models in Synchronous Networks

During the last two decades, a small set of distributed computing models for networks have emerged, among which LOCAL, CONGEST, and Broadcast Congested Clique (BCC) play a prominent role. We consider hybrid models resulting from combining these three models. That is, we analyze the computing power of models allowing to, say, perform a constant number of rounds of CONGEST, then a constant number of rounds of LOCAL, then a constant number of rounds of BCC, possibly repeating this figure a constant number of times. We specifically focus on 2-round models, and we establish the complete picture of the relative powers of these models. That is, for every pair of such models, we determine whether one is (strictly) stronger than the other, or whether the two models are incomparable. The separation results are obtained by approaching communication complexity through an original angle, which may be of independent interest. The two players are not bounded to compute the value of a binary function, but the combined outputs of the two players are constrained by this value. In particular, we introduce the XOR-Index problem, in which Alice is given a binary vector $x\in\{0,1\}^n$ together with an index $i\in[n]$, Bob is given a binary vector $y\in\{0,1\}^n$ together with an index $j\in[n]$, and, after a single round of 2-way communication, Alice must output a boolean $\textrm{out}_A$, and Bob must output a boolean $\textrm{out}_B$, such that $\mbox{out}_A\land\mbox{out}_B = x_j\oplus y_i$. We show that the communication complexity of XOR-Index is $Ω(n)$ bits.

cs.DC↗

arXiv2022

On constant-time quantum annealing and guaranteed approximations for graph optimization problems

Quantum Annealing (QA) is a computational framework where a quantum system's continuous evolution is used to find the global minimum of an objective function over an unstructured search space. It can be seen as a general metaheuristic for optimization problems, including NP-hard ones if we allow an exponentially large running time. While QA is widely studied from a heuristic point of view, little is known about theoretical guarantees on the quality of the solutions obtained in polynomial time. In this paper we use a technique borrowed from theoretical physics, the Lieb-Robinson (LR) bound, and develop new tools proving that short, constant time quantum annealing guarantees constant factor approximations ratios for some optimization problems when restricted to bounded degree graphs. Informally, on bounded degree graphs the LR bound allows us to retrieve a (relaxed) locality argument, through which the approximation ratio can be deduced by studying subgraphs of bounded radius. We illustrate our tools on problems MaxCut and Maximum Independent Set for cubic graphs, providing explicit approximation ratios and the runtimes needed to obtain them. Our results are of similar flavor to the well-known ones obtained in the different but related QAOA (quantum optimization algorithms) framework. Eventually, we discuss theoretical and experimental arguments for further improvements.

quant-ph↗

arXiv2021

A Meta-Theorem for Distributed Certification

Distributed certification, whether it be proof-labeling schemes, locally checkable proofs, etc., deals with the issue of certifying the legality of a distributed system with respect to a given boolean predicate. A certificate is assigned to each process in the system by a non-trustable oracle, and the processes are in charge of verifying these certificates, so that two properties are satisfied: completeness, i.e., for every legal instance, there is a certificate assignment leading all processes to accept, and soundness, i.e., for every illegal instance, and for every certificate assignment, at least one process rejects. The verification of the certificates must be fast, and the certificates themselves must be small. A large quantity of results have been produced in this framework, each aiming at designing a distributed certification mechanism for specific boolean predicates. This paper presents a "meta-theorem", applying to many boolean predicates at once. Specifically, we prove that, for every boolean predicate on graphs definable in the monadic second-order (MSO) logic of graphs, there exists a distributed certification mechanism using certificates on $O(\log^2n)$ bits in $n$-node graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors.

cs.DC↗

arXiv2021

A cubic vertex-kernel for Trivially Perfect Editing

We consider the Trivially Perfect Editing problem, where one is given an undirected graph $G = (V,E)$ and a parameter $k \in \mathbb{N}$ and seeks to edit (add or delete) at most $k$ edges from $G$ to obtain a trivially perfect graph. The related Trivially Perfect Completion and Trivially Perfect Deletion problems are obtained by only allowing edge additions or edge deletions, respectively. Trivially perfect graphs are both chordal and cographs, and have applications related to the tree-depth width parameter and to social network analysis. All variants of the problem are known to be NP-Complete and to admit so-called polynomial kernels. More precisely, the existence of an $O(k^3)$ vertex-kernel for Trivially Perfect Completion was announced by Guo (ISAAC 2007) but without a stand-alone proof. More recently, Drange and Pilipczuk (Algorithmica 2018) provided $O(k^7)$ vertex-kernels for these problems and left open the existence of cubic vertex-kernels. In this work, we answer positively to this question for all three variants of the problem.

cs.DS↗

arXiv2020

Local Certification of Graphs with Bounded Genus

Naor, Parter, and Yogev [SODA 2020] recently designed a compiler for automatically translating standard centralized interactive protocols to distributed interactive protocols, as introduced by Kol, Oshman, and Saxena [PODC 2018]. In particular, by using this compiler, every linear-time algorithm for deciding the membership to some fixed graph class can be translated into a $\mathsf{dMAM}(O(\log n))$ protocol for this class, that is, a distributed interactive protocol with $O(\log n)$-bit proof size in $n$-node graphs, and three interactions between the (centralizer) computationally-unbounded but non-trustable prover Merlin, and the (decentralized) randomized computationally-limited verifier Arthur. As a corollary, there is a $\mathsf{dMAM}(O(\log n))$ protocol for the class of planar graphs, as well as for the class of graphs with bounded genus. We show that there exists a distributed interactive protocol for the class of graphs with bounded genus performing just a single interaction, from the prover to the verifier, yet preserving proof size of $O(\log n)$ bits. This result also holds for the class of graphs with bounded demi-genus, that is, graphs that can be embedded on a non-orientable surface of bounded genus. The interactive protocols described in this paper are actually proof-labeling schemes, i.e., a subclass of interactive protocols, previously introduced by Korman, Kutten, and Peleg [PODC 2005]. In particular, these schemes do not require any randomization from the verifier, and the proofs may often be computed a priori, at low cost, by the nodes themselves. Our results thus extend the recent proof-labeling scheme for planar graphs by Feuilloley et al. [PODC 2020], to graphs of bounded genus, and to graphs of bounded demigenus.

cs.DC↗

arXiv2020

On the heapability of finite partial orders

We investigate the partitioning of partial orders into a minimal number of heapable subsets. We prove a characterization result reminiscent of the proof of Dilworth's theorem, which yields as a byproduct a flow-based algorithm for computing such a minimal decomposition. On the other hand, in the particular case of sets and sequences of intervals we prove that this minimal decomposition can be computed by a simple greedy-type algorithm. The paper ends with a couple of open problems related to the analog of the Ulam-Hammersley problem for decompositions of sets and sequences of random intervals into heapable sets.

math.CO↗