Maise Dantas da Silva

Applying modular decomposition to parameterized bicluster editing

A graph G is said to be a cluster graph if G is a disjoint union of cliques (complete subgraphs), and a bicluster graph if G is a disjoint union of bicliques (complete bipartite subgraphs). In this work, we study the parameterized version of the NP-hard BICLUSTER GRAPH EDITING problem, which consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most k modifications are allowed in the edge set of any input graph (BICLUSTER(k) GRAPH EDITING problem), this problem is FPT, solvable in O(4 κ m) time by applying a search tree algorithm. It is shown an algorithm with O(4 κ + n + m) time, which uses a new strategy based on modular decomposition techniques. Furthermore, the same techniques lead to a new form of obtaining a problem kernel with O(κ 2 ) vertices for the CLUSTER(κ) GRAPH EDITING problem, in O(n+m) time. This problem consists of obtaining a cluster graph by modifying at most k edges in an input graph. A previous FPT algorithm for this problem was presented by Gramm et al. [11]. In their solution, a problem kernel with O(k 2 ) vertices and O(k 3 ) edges is built in O(n 3 ) time.

Parameterized Complexity of Flood-Filling Games on Trees

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. As for many colored graph problems, Flood-filling games have relevant interpretations in bioinformatics. The standard versions of Flood-It and Free-Flood-It are played on n ×m grids. In this paper we analyze the complexity of these games when played on trees. We prove that Flood-It remains NP-hard on trees whose leaves are at distance at most d = 2 from the pivot, and that Flood-It is in FPT when parameterized by the number of colors c in such trees (for any constant d). We also show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, keeping complexity features; this implies that Flood-It on trees inherits several complexity results already proved for RSCS, such as some interesting FPT and W[1]-hard cases. We introduce a new variant of Flood-It, called Multi-Flood-It, where each move of the game is played on various pivots. We also present a general framework for reducibility from Flood-It to Free-Flood-It, by defining a special graph operator ψ such that Flood-It played on a graph class \(\mathcal{F}\) is reducible to Free-Flood-It played on the image of \(\mathcal{F}\) under ψ. An interesting particular case occurs when \(\mathcal{F}\) is closed under ψ. Some NP-hard cases for Free-Flood-It on trees can be derived using this approach. We conclude by showing some results on parameterized complexity for Free-Flood-It played on pc-trees (phylogenetic colored trees). We prove that some results valid for Flood-It on pc-trees can be inherited by Free-Flood-It on pc-trees, using another type of reducibility framework.

Tractability and hardness of flood-filling games on trees

This work presents new results on flood-filling games, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. A flooding move consists of changing the color of the monochromatic component containing a vertex p (the pivot of the move). These games are originally played on grids; however, when played on trees, we have interesting applications and significant effects on problem complexity. In this paper, a complete mapping of the complexity of flood-filling games on trees is made, charting the consequences of single and aggregate parameterizations by: number of colors, number of moves, maximum distance from the pivot, number of occurrences of a color, number of leaves, and difference between number of moves and number of colors.We show that Flood-It on trees and Restricted Shortest Common Supersequence (RSCS) are analogous problems, in the sense that they can be translated from one to another, preserving complexity issues; this implies interesting FPT and W1]-hard cases to Flood-It. Restricting attention to phylogenetic colored trees (where each color occurs at most once in any root-leaf path, in order to model phylogenetic sequences), we also show some impressive NP-hard and FPT results for both games. In addition, we prove that Flood-It and Free-Flood-It remain NP-hard when played on 3-colored trees, which closes an open question posed by Fleischer and Woeginger. Finally, we present a general framework for reducibility from Flood-It to Free-Flood-It; some NP-hard cases for Free-Flood-It can be derived using this approach.

Revisiting the complexity of and/or graph solution

An and/or graph is an acyclic, edge-weighted directed graph containing a single source vertex such that every vertex v has a label f(v)∈{and,or}. A solution subgraph H of an and/or-graph must contain the source and obey the following rule: if an and-vertex (resp. or-vertex) is included in H then all (resp. one) of its out-edges must also be included in H. X-y graphs are defined as a generalization of and/or graphs: every vertex vi of an x-y graph has a label xi-yi, in such a way that if a vertex vi is included in a solution subgraph H of an x-y graph then xi of its yi out-edges must also be included in H. In this work, we analyze complexity aspects (both from the classical and the parameterized point of view) of finding solution subgraphs of minimum weight for and/or and x-y graphs.

Algorithms, kernels and lower bounds for the Flood-It game parameterized by the vertex cover number

Abstract Flood-It is a combinatorial problem on a colored graph whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a pivot vertex p . A flooding move consists of changing the color of the monochromatic component (maximal monochromatic connected subgraph) containing p . This problem generalizes a combinatorial game named alike which is played on m × n grids. It is known that Flood-It is NP-hard even for 3 × n grids and for instances with bounded number of colors, diameter, treewidth, or pathwidth. In [Fellows, Souza, Protti, Dantas da Silva, Tractability and hardness of flood-filling games on trees, Theoretical Computer Science, 576, 102-116, 2015] it is shown that Flood-It is W[1]-hard when played on trees with bounded number of colors, and the number of leaves is a single parameter. Contrasting with such results, in this work we show that Flood-It is fixed-parameter tractable when parameterized by either the vertex cover number or the neighborhood diversity. Additionally, we prove that Flood-It does not admit a polynomial kernel when the vertex cover number is a single parameter, unless coNP ⊆ NP ∕ poly . Finally, lower bounds based on the (Strong) Exponential Time Hypothesis as well as an upper bound for the required time to solve Flood-It are also provided.

A Survey on the Complexity of Flood-Filling Games

This survey is offered in honour of the special occasion of the birthday celebration of science and education pioneer Professor Juraj Hromkovic. In this survey, we review recent results on one-player flood-filling games on graphs, Flood-It and Free-Flood-It, in which the player aims to make the board monochromatic with a minimum number of flooding moves. As for many colored graph problems, flood-filling games have relevant interpretations in bioinformatics. The original versions of Flood-It and Free-Flood-It are played on \(n \times m\) grids, but several studies were devoted to analyzing the complexity of these games when the “board” (the graph) belongs to other graph classes. A complete mapping of the complexity of flood-filling games on trees is presented, charting the consequences of single and aggregate parameterizations. The Flood-It problem on trees and the Restricted Shortest Common Supersequence (RSCS) problem are analogous. Flood-It remains NP-hard when played on 3-colored trees. A general framework for reducibility from Flood-It to Free-Flood-It is revisited. The complexity behavior of these games when played on various kinds of graphs is surveyed, such as Cartesian products of cycles and paths, circular grids, split graphs, co-comparability graphs, and AT-free graphs. We review a recent investigation of the parameterized complexity of Flood-It when the size of a minimum vertex cover is the structural parameter. Some educational aspects of the game are also reviewed. Happy Birthday, Juraj!

The Flood-It game parameterized by the vertex cover number

Flood-It is a combinatorial problem on a colored graph whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a pivot vertex p. A flooding move consists of changing the color of the monochromatic component (maximal monochromatic connected subgraph) containing p. This problem generalizes a combinatorial game named alike which is played on m×n grids. It is known that Flood-It remains NP-hard even for 3×n grids and for instances with bounded number of colors, diameter, or treewidth. In contrast, in this work we show that Flood-It is fixed-parameter tractable when parameterized by the vertex cover number, and admits a polynomial kernelization when, besides the vertex cover number, the number of colors is an additional parameter.