Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.12021

Monitoring graph edges via shortest paths: computational complexity and approximation algorithms

Giordano Colli
View original source ↗Read source document ↗Export metadata ↓

Abstract

Edge-Geodetic Sets play a crucial role in network monitoring and optimization, wherein the goal is to strategically place monitoring stations on vertices of a network, represented as a graph, to ensure complete coverage of edges and mitigate faults by monitoring lines of communication. This paper illustrates and explores the Monitoring Edge-Geodetic Set (MEG-set) problem, which involves determining the minimum set of vertices that need to be monitored to achieve geodetic coverage for a given network. The significance of this problem lies in its potential to facilitate efficient network monitoring, enhancing the overall reliability and performance of various applications. In this work, we prove the $\mathcal{NP}$-completeness of the MEG-set optimization problem by showing a reduction from the well-known Vertex Cover problem. Furthermore, we present inapproximability results, proving that the MEG-set optimization problem is $\mathcal{APX}$-Hard and that, if the unique games conjecture holds, the problem is not approximable within a factor of $2-ε$ for any constant $ε> 0$. Despite its $\mathcal{NP}$-hardness, we propose an efficient approximation algorithm achieving an approximation ratio of $O(\sqrt{|V(G)| \cdot \ln{|V(G)|})}$ for the MEG-set optimization problem, based on the well-known Set Cover approximation algorithm, where $|V(G)|$ is the number of nodes of the MEG-set instance.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Giordano Colli. 2025-04-18. Monitoring graph edges via shortest paths: computational complexity and approximation algorithms. https://arxiv.org/abs/2506.12021

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic Rank Speedup Theorems, Revisited

Motivated by fast matrix multiplication and recent connections between asymptotic tensor rank and fine-grained complexity, we revisit classical tools from the matrix multiplication literature and develop a framework for obtaining improved asymptotic rank upper bounds for tensors beyond matrix multiplication. In the 1980s, Coppersmith-Winograd and Strassen discovered a series of speedup theorems for asymptotic rank: in certain regimes, one can extract additional terms from a border rank upper bound on a tensor $T$, and then use these terms to obtain an improved asymptotic rank of $T$. We establish general speedup theorems that subsume these results and enable quantitative improvements. Two representative applications are: (1) The asymptotic rank of the small Coppersmith-Winograd tensor $\mathrm{cw}_q$ is less than its border rank. For instance, we prove the asymptotic rank of $\mathrm{cw}_2$ is smaller than $3.931$, improving on $\underline{\mathrm{R}}(\mathrm{cw}_2)=4$. It is known that if the asymptotic rank of $\mathrm{cw}_2$ equals $3$, this would imply $ω=2$. (2) A general improvement over Strassen's bound: we obtain an upper bound below $d^{2ω/3}$ on the asymptotic rank of any $d\times d\times d$ tensor. To make full use of speedups, we analyze degenerations in which both sides are nontrivial direct sums, a setting where the optimal quantitative bound one can achieve was previously unclear. We do so via an approach we call Strassen calculus: a systematic method for converting such degeneration data into explicit asymptotic rank bounds using Strassen's theory of the asymptotic spectrum.

cs.CC↗
arXiv2026

Lettericity Is NP-Complete

The lettericity of a graph $G$ is the smallest size of a set $Σ$ such that there exist $w_1, \ldots, w_{|V(G)|} \in Σ$ and a decoder $D \subseteq Σ^2$ for which $G$ is isomorphic to the letter graph $(\{1, \ldots, |V(G)|\}, \{ij : 1 \le i < j \le |V(G)|, w_iw_j \in D\})$. It took around two decades of the study of lettericity for, in the simpler case of paths, a closed-form expression for its lettericity to be derived; this suggests that the question of whether the lettericity of an arbitrary graph can be computed in polynomial time is nontrivial. Indeed, this question has been raised repeatedly as an open problem in recent literature. We solve this problem by showing that the lettericity problem on arbitrary graphs is \textsf{NP}-complete (Theorem~10). We also prove that the coloring extension problem --- the same problem as lettericity, with the added condition that if $f$ is the isomorphism mapping from $G$ to the letter graph, $w_{f(v)} = χ(v)$ for a given coloring $χ$ of $G$ --- is \textsf{NP}-complete (Theorem~12). We also resolve the open problem of classifying the complexity of the word extension problem, which is the same problem as lettericity except that the $w_i$ are fixed; we show it to be \textsf{NP}-complete (Theorem~13), which, in tandem with our \textsf{NP}-completeness result for coloring extension, contrasts with the known result that when the constraint of the coloring extension problem and the constraint of the word extension problem are both applied to lettericity, lettericity can be decided in polynomial time. Additionally, we use the reduction in the \textsf{NP}-completeness proof to show that unless the Exponential Time Hypothesis is false, there cannot exist a deterministic algorithm to decide whether the lettericity of an $n$-vertex graph is at most~$k$ in time $2^{o(n)}$, even when $n = 6k$ (Theorem~11).

cs.CC↗
arXiv2026

Ideal Membership in Polynomial Calculus: Complexity and Reductions

The Ideal Membership Problem (IMP) asks whether a polynomial f belongs to an ideal of Q[x_1, ..., x_n]. Polynomial Calculus (PC) certifies membership by deriving f from the generators, and a degree-d derivation needs at most n^O(d) steps. We write PC-IMPd for the problem of producing a degree-bounded PC certificate, and call it solvable when one is guaranteed to exist and can be found in time n^O(d). Over Q, unlike over finite fields, a derivation may need exponentially many bits. We study PC-IMPd on instances arising from constraint satisfaction problems, and ask for which constraint languages L it is solvable. Our main contribution is a reduction framework for PC-IMPd, based on pp-definitions, pp-interpretations, and pp-encodings, that mirrors the algebraic approach to CSP complexity. Solvability is preserved by these constructions and, in the language of algebras, by passing to subalgebras, finite direct powers, and homomorphic images. We obtain new tractable classes over ternary and larger domains: every language closed under the median operation on a finite chain has solvable PC-IMPd, by reduction to the Boolean majority algebra, and in particular so does every language over {0, 1, 2} closed under a fixed-value majority. This also places IMPd(L) in P for such languages, advancing the classification of IMPd over ternary domains. In the process, we settle the last open case of the Boolean dichotomy for IMPd(L) and complete the Boolean classification of PC-IMPd(L) with an unconditional lower bound for an instance of PC-IMP1. A recent PC-to-SoS simulation reduces degree-automatability of Sum-of-Squares (the open problem of finding a degree-d SoS proof in time n^O(d) when one exists) to solvability of PC-IMPd. Each new tractable class therefore yields a family of constraint systems on which SoS proofs are degree-automatable.

cs.CC↗