Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.26006

Structural Complexity of Matching-Match: Dense and Sparse Graphs

Ilie Dumitru·Adrian Miclăuş·Alexandru Popa
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Matching-Match puzzle asks whether the vertices of a fixed graph can be colored so that the multiset of color pairs induced by its edges is exactly a prescribed multiset. We study how the complexity of this realization problem depends on the host graph. On the dense side, we give a polynomial-time algorithm for complete $k$-partite graphs for every fixed number $k$ of parts, with arbitrary precoloring and an arbitrary number of colors. We prove a sharp complement-degree threshold: the problem is polynomial-time solvable when $Δ(\overline G)\le1$, but NP-complete on completely uncolored graphs already when $Δ(\overline G)=2$. This yields a dichotomy for uniform complete multipartite graphs, and connected diameter two already suffices for NP-completeness. We also prove W[1]-hardness on cographs parameterized by the number of colors. On the sparse side, completely uncolored paths and cycles admit a linear-time characterization by Euler trails and circuits, while counting feasible colorings is $\#P$-complete on both classes. Counting is nevertheless polynomial-time solvable on stars and complete graphs, even with arbitrary precoloring. A decomposition-transfer theorem yields NP-completeness already at tree-depth two. A separate path-decomposition reduction gives a maximum-degree threshold between one and two for completely uncolored disconnected host graphs with unrestrictedly many colors. Components with at most two edges are tractable, while a disjoint union of $P_4$'s is NP-complete. Finally, precoloring restores tractability in several cases: star forests are polynomial when every center is precolored, and two broad precoloring regimes on length-two spiders are polynomial even when the number of colors is unbounded.

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ilie Dumitru, Adrian Miclăuş, Alexandru Popa. 2026-09-22. Structural Complexity of Matching-Match: Dense and Sparse Graphs. https://arxiv.org/abs/2609.26006

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

KEEP EXPLORING

Related papers

arXiv2026

Beyond Kruskal: Polynomial-Time Tensor Decomposition under the Lovitz-Petrov Condition

Identifiability criteria certify that a given tensor decomposition is a unique rank decomposition. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition, and verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was previously known under this condition. We give a polynomial-time algorithm for tensor decomposition under the Lovitz--Petrov condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank. This contrasts with an arbitrary tensor decomposition, which certifies only an upper bound on the tensor rank, while determining tensor rank is NP-hard in general.

cs.DS↗
arXiv2026

Poisson Exchange Beyond Submodularity: Effective Approximation Algorithms for Offline and Online Subset Selection over Matroids

Over the past decade, a growing body of research has shown that $γ$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $γ$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/γ)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $ρ_γ=1-\left(γ/(2-γ)\right)^{ \frac{γ^2}{2(1-γ)} }$. In sharp contrast to the previous guarantee, our obtained factor $ρ_γ$ not only strictly improves upon $(1+1/γ)^{-2}$ for every $γ\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $γ\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $α$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-γ}$ and $1-e^{-α}$, respectively. Here, $α\in(0,1]$ denotes the DR ratio.

cs.DS↗
arXiv2026

Approximating Prize-Collecting TSP below 1.556

The prize-collecting traveling salesperson problem is a variant of the metric traveling salesperson problem in which vertices may be left unvisited by paying their associated penalties. The objective is to minimize the length of the tour plus the total penalty of the unvisited vertices. Blauth, Klein, and Nägele gave the previously best-known LP-relative $1.599$-approximation. We show that a simpler version of their algorithm, obtained by omitting the splitting-off preprocessing before the tree decomposition, has an LP-relative approximation ratio of $1.555761$. The improvement comes entirely from a new analysis of the parity-correction step: a simple analysis already gives $1.56$, and the stated factor follows from a numerical parameter search with exact verification.

cs.DS↗