arXiv subjects

Ilie Dumitru

Publications and source records attributed to Ilie Dumitru.

3 recordsLinked to original sources

arXiv2026

Structural Complexity of Matching-Match: Dense and Sparse Graphs

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.

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arXiv2026

Colored Interaction-Profile Realization: Complexity of Matching-Match on Spiders

Network motifs and colored local interaction patterns provide a useful way to describe the structure of complex networks. Motivated by an inverse realization perspective, we study the problem of assigning colors to the vertices of a fixed graph so that its edges realize a prescribed multiset of colored pairwise interactions. This problem is formalized by the Matching-Match Puzzle, introduced by Iburi and Uehara. We investigate how its computational complexity depends on the number of colors and on the structure of the host graph. We first prove that Matching-Match is NP-complete with only two colors, even when no vertex is precolored and the graph has maximum degree three. We then focus on spiders. We show that the problem is W[1]-hard parameterized by the number of colors even on spiders with only the body precolored. In contrast, for spiders whose legs have length at most two, we give a fixed-parameter tractable algorithm parameterized by the number of colors, allowing arbitrary precoloring. Finally, we prove NP-completeness for spiders whose legs all have length exactly three when precoloring is allowed.

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arXiv2026

Maximum Matching-Match: Hardness and Approximation

In this paper, we study \textsc{MaxMMP}, an optimization variant of the Matching-Match Puzzle introduced by Iburi and Uehara (FUN 2024). Given a graph, a partial vertex coloring, and a multiset of colored sticks, the goal is to complete the coloring and assign the sticks to graph edges so as to maximize the number of satisfied edges. We first prove that \textsc{MaxMMP} is APX-hard by an reduction from \textsc{Max-Cut}. The hardness already holds with two colors, no precolored vertices, and only bichromatic sticks. We then give a simple deterministic $\frac{2}{c(c+1)}$-approximation for completely uncolored instances, improving to $\frac{2}{c(c-1)}$ when all sticks are bichromatic. Next, we obtain a randomized $\frac{1-\frac{1}{e}}{2c}$-approximation for arbitrary instances with $c$ colors by reducing the remaining coloring choices to monotone submodular maximization under a partition matroid. On bipartite graphs, the approximation ratio improves to $\frac{1-\frac{1}{e}}{c}$. For every fixed $c$, we further obtain deterministic $\frac{1}{2c}$ and $\frac{1}{c}$-approximations on general and bipartite graphs, respectively, in time $n^{O(c^2)}$. Finally, for every fixed number of colors, we show that \textsc{MaxMMP} can be solved exactly in time $n^{O(c^2)}$ on trees and on cographs.

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