arXiv subjects

Luís Cunha

Publications and source records attributed to Luís Cunha.

10 recordsLinked to original sources

arXiv2026

Complexity of the Swap Median and Swap Closest Problems

Genome rearrangement distances provide a combinatorial framework for comparing genomes represented as permutations. Although the swap distance between two permutations is polynomial-time computable, optimization over multiple permutations is harder. Given three inputs, Swap Median minimizes the sum of distances to them, while Swap Closest minimizes the maximum distance. We prove that the decision version of Swap Median is NP-complete for three permutations, resolving a question open since Eriksen (2007). The reduction considers instances in which a solution exists exactly when the triangle-inequality lower bound is attained. We represent pairs of cycle decompositions by a graph whose vertices correspond to swaps that simultaneously break cycles and whose edges encode incompatible swaps. This graph class contains all 2-subdivision graphs; the hardness of Maximum Independent Set on this class then implies NP-completeness. We extend the construction to show that Swap Closest is also NP-complete for three inputs, strengthening Popov's result for an arbitrary number of permutations. Finally, the same graph-theoretic framework yields approximation procedures: selecting compatible cycle-breaking swaps improves the standard 4/3 metric bound for Swap Median and the standard 2 metric bound for Swap Closest by explicit terms relative to the median lower bound and input diameter, respectively.

cs.CC↗

arXiv2026

Determining subtree movement distance and consensus between cell trees

Tumor mutational heterogeneity can be represented by trees describing the evolutionary history of a tumor. However, noisy sequencing data may create uncertainty in the inferred tree structure, making it important to compare trees and identify the mutations required to transform one into another. We address this problem by defining a tree operation called subtree movement (SBM) and proving that the decision problem associated with the SBM distance is NP-complete. We also establish a close relationship between this distance and the maximum common almost $v$-tree problem (MCAT), which is solvable in polynomial time and provides an upper bound on the SBM distance. For collections of mutation trees, we study two natural consensus formulations: the median problem, which minimizes the sum of distances to the input trees, and the closest problem, which minimizes the maximum distance. We prove that both problems are NP-complete even for only three input trees, considering SBM sequences obtained from MCAT solutions. Finally, we develop algorithms that provide upper bounds for the median and closest problems and evaluate them on synthetic and real datasets. The experiments indicate that the resulting consensus trees summarize the input mutation trees better than any individual tree in the input set.

cs.CC↗

arXiv2025

Closing the complexity gap of the double distance problem

Genome rearrangement has been an active area of research in computational comparative genomics for the last three decades. While initially mostly an interesting algorithmic endeavor, now the practical application by applying rearrangement distance methods and more advanced phylogenetic tasks is becoming common practice, given the availability of many completely sequenced genomes. Several genome rearrangement models have been developed over time, sometimes with surprising computational properties. A prominent example is the fact that computing the reversal distance of two signed permutations is possible in linear time, while for two unsigned permutations it is NP-hard. Therefore one has always to be careful about the precise problem formulation and complexity analysis of rearrangement problems in order not to be fooled. The double distance is the minimum number of genomic rearrangements between a singular and a duplicated genome that, in addition to rearrangements, are separated by a whole genome duplication. At the same time it allows to assign the genes of the duplicated genome to the two paralogous chromosome copies that existed right after the duplication event. Computing the double distance is another example of a tricky hardness landscape: If the distance measure underlying the double distance is the simple breakpoint distance, the problem can be solved in linear time, while with the more elaborate DCJ distance it is NP-hard. Indeed, there is a family of distance measures, parameterized by an even number k, between the breakpoint distance (k=2) and the DCJ distance (k=\infty). Little was known about the hardness border between these extremes; the problem complexity was known only for k=4 and k=6. In this paper, we close the gap, providing a full picture of the hardness landscape when computing the double distance.

cs.CC↗

arXiv2024

Binary Jumbled Indexing: Suffix tree histogram

Given a binary string $ω$ over the alphabet $\{0, 1\}$, a vector $(a, b)$ is a Parikh vector if and only if a factor of $ω$ contains exactly $a$ occurrences of $0$ and $b$ occurrences of $1$. Answering whether a vector is a Parikh vector of $ω$ is known as the Binary Jumbled Indexing Problem (BJPMP) or the Histogram Indexing Problem. Most solutions to this problem rely on an $O(n)$ word-space index to answer queries in constant time, encoding the Parikh set of $ω$, i.e., all its Parikh vectors. Cunha et al. (Combinatorial Pattern Matching, 2017) introduced an algorithm (JBM2017), which computes the index table in $O(n+ρ^2)$ time, where $ρ$ is the number of runs of identical digits in $ω$, leading to $O(n^2)$ in the worst case. We prove that the average number of runs $ρ$ is $n/4$, confirming the quadratic behavior also in the average-case. We propose a new algorithm, SFTree, which uses a suffix tree to remove duplicate substrings. Although SFTree also has an average-case complexity of $Θ(n^2)$ due to the fundamental reliance on run boundaries, it achieves practical improvements by minimizing memory access overhead through vectorization. The suffix tree further allows distinct substrings to be processed efficiently, reducing the effective cost of memory access. As a result, while both algorithms exhibit similar theoretical growth, SFTree significantly outperforms others in practice. Our analysis highlights both the theoretical and practical benefits of the SFTree approach, with potential extensions to other applications of suffix trees.

cs.DS↗

arXiv2024

Finding Patterns in Ambiguity: Interpretable Stress Testing in the Decision~Boundary

The increasing use of deep learning across various domains highlights the importance of understanding the decision-making processes of these black-box models. Recent research focusing on the decision boundaries of deep classifiers, relies on generated synthetic instances in areas of low confidence, uncovering samples that challenge both models and humans. We propose a novel approach to enhance the interpretability of deep binary classifiers by selecting representative samples from the decision boundary - prototypes - and applying post-model explanation algorithms. We evaluate the effectiveness of our approach through 2D visualizations and GradientSHAP analysis. Our experiments demonstrate the potential of the proposed method, revealing distinct and compact clusters and diverse prototypes that capture essential features that lead to low-confidence decisions. By offering a more aggregated view of deep classifiers' decision boundaries, our work contributes to the responsible development and deployment of reliable machine learning systems.

cs.LG↗

arXiv2023

On the Complexity of the Median and Closest Permutation Problems

Genome rearrangements are events where large blocks of DNA exchange places during evolution. The analysis of these events is a promising tool for understanding evolutionary genomics, providing data for phylogenetic reconstruction based on genome rearrangement measures. Many pairwise rearrangement distances have been proposed, based on finding the minimum number of rearrangement events to transform one genome into the other, using some predefined operation. When more than two genomes are considered, we have the more challenging problem of rearrangement-based phylogeny reconstruction. Given a set of genomes and a distance notion, there are at least two natural ways to define the "target" genome. On the one hand, finding a genome that minimizes the sum of the distances from this to any other, called the median genome. Finding a genome that minimizes the maximum distance to any other, called the closest genome. Considering genomes as permutations, some distance metrics have been extensively studied. We investigate median and closest problems on permutations over the metrics: breakpoint, swap, block-interchange, short-block-move, and transposition. In biological matters some values are usually small, such as the solution value d or the number k of input permutations. For each of these metrics and parameters d or k, we analyze the closest and the median problems from the viewpoint of parameterized complexity. We obtain the following results: NP-hardness for finding the median/closest permutation for some metrics, even for k = 3; Polynomial kernels for the problems of finding the median permutation of all studied metrics, considering the target distance d as parameter; NP-hardness result for finding the closest permutation by short-block-moves; FPT algorithms and infeasibility of polynomial kernels for finding the closest permutation for some metrics parameterized by the target distance d.

cs.CC↗

arXiv2022

Simpler and efficient characterizations of tree t-spanners for graphs with few P4's and (k, l)-graphs

A tree $t$-spanner of a graph $G$ is a spanning tree $T$ in which the distance between any two adjacent vertices of $G$ is at most $t$. The smallest $t$ for which $G$ has a tree $t$-spanner is called tree stretch index. The $t$-admissibility problem aims to decide whether the tree stretch index is at most $t$. Regarding its optimization version, the smallest $t$ for which $G$ is $t$-admissible is the stretch index of $G$, denoted by $σ_T(G)$. Given a graph with $n$ vertices and $m$ edges, the recognition of $2$-admissible graphs can be done $O(n+m)$ time, whereas $t$-admissibility is NP-complete for $σ_T(G) \leq t$, $t \geq 4$ and deciding if $t = 3$ is an open problem, for more than 20 years. Since the structural knowledge of classes can be determinant to classify $3$-admissibility's complexity, in this paper we present simpler and faster algorithms to check $2$ and $3$-admissibility for families of graphs with few $P_4$'s and $(k,\ell)$-graphs. Regarding $(0,\ell)$-graphs, we present lower and upper bounds for the stretch index of these graphs and characterize graphs whose stretch indexes are equal to the proposed upper bound. Moreover, we prove that $t$-admissibility is NP-complete even for line graphs of subdivided graphs.

cs.DM↗

arXiv2020

Total tessellation cover and quantum walk

We propose the total staggered quantum walk model and the total tessellation cover of a graph. This model uses the concept of total tessellation cover to describe the motion of the walker who is allowed to hop both to vertices and edges of the graph, in contrast with previous models in which the walker hops either to vertices or edges. We establish bounds on $T_t(G)$, which is the smallest number of tessellations required in a total tessellation cover of $G$. We highlight two of these lower bounds $T_t(G) \geq ω(G)$ and $T_t(G)\geq is(G)+1$, where $ω(G)$ is the size of a maximum clique and $is(G)$ is the number of edges of a maximum induced star subgraph. Using these bounds, we define the good total tessellable graphs with either $T_t(G)=ω(G)$ or $T_t(G)=is(G)+1$. The $k$-total tessellability problem aims to decide whether a given graph $G$ has $T_t(G) \leq k$. We show that $k$-total tessellability is in $\mathcal{P}$ for good total tessellable graphs. We establish the $\mathcal{NP}$-completeness of the following problems when restricted to the following classes: ($is(G)+1$)-total tessellability for graphs with $ω(G) = 2$; $ω(G)$-total tessellability for graphs $G$ with $is(G)+1 = 3$; $k$-total tessellability for graphs $G$ with $\max\{ω(G), is(G)+1\}$ far from $k$; and $4$-total tessellability for graphs $G$ with $ω(G) = is(G)+1 = 4$. As a consequence, we establish hardness results for bipartite graphs, line graphs of triangle-free graphs, universal graphs, planar graphs, and $(2,1)$-chordal graphs.

cs.DM↗

arXiv2019

The Tessellation Cover Number of Good Tessellable Graphs

A tessellation of a graph is a partition of its vertices into vertex disjoint cliques. A tessellation cover of a graph is a set of tessellations that covers all of its edges, and the tessellation cover number, denoted by $T(G)$, is the size of a smallest tessellation cover. The \textsc{$t$-tessellability} problem aims to decide whether a graph $G$ has $T(G)\leq t$ and is $\mathcal{NP}$-complete for $t\geq 3$. Since the number of edges of a maximum induced star of $G$, denoted by $is(G)$, is a lower bound on $T(G)$, we define good tessellable graphs as the graphs~$G$ such that $T(G)=is(G)$. The \textsc{good tessellable recognition (gtr)} problem aims to decide whether $G$ is a good tessellable graph. We show that \textsc{gtr} is $\mathcal{NP}$-complete not only if $T(G)$ is known or $is(G)$ is fixed, but also when the gap between $T(G)$ and $is(G)$ is large. As a byproduct, we obtain graph classes that obey the corresponding computational complexity behaviors.

cs.CC↗

arXiv2017

Fast and Simple Jumbled Indexing for Binary RLE Strings

Important papers have appeared recently on the problem of indexing binary strings for jumbled pattern matching, and further lowering the time bounds in terms of the input size would now be a breakthrough with broad implications. We can still make progress on the problem, however, by considering other natural parameters. Badkobeh et al.\ (IPL, 2013) and Amir et al.\ (TCS, 2016) gave algorithms that index a binary string in $O (n + ρ^2 \log ρ)$ time, where $n$ is the length and $ρ$ is the number of runs, and Giaquinta and Grabowski (IPL, 2013) gave one that runs in $O (n + ρ^2)$ time. In this paper we propose a new and very simple algorithm that also runs in $O(n + ρ^2)$ time and can be extended either so that the index returns the position of a match (if there is one), or so that the algorithm uses only $O (n)$ bits of space.

cs.DS↗