FPT Algorithm Research Articles

Parameterized Study of Steiner Tree on Unit Disk Graphs

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset \(R\subseteq V(G)\) of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from \(V\setminus R\). The vertices of R are referred to as terminals and the vertices of \(V(G)\setminus R\) as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in \(n^{O(\sqrt{t+k})}\) time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time \(2^{O(k)}n^{O(1)}\). In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs Fomin (Discret. Comput. Geometry 62(4):879–911, 2019). We mention that the algorithmic results can be made to work for Steiner Tree problem on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree problem on disk graphs parameterized by k, is W[1]-hard.

Parameterized Complexity of Maximum Edge Colorable Subgraph

A graph H is p-edge colorable if there is a coloring \(\psi : E(H) \rightarrow \{1,2,\dots ,p\}\), such that for distinct \(uv, vw \in E(H)\), we have \(\psi (uv) \ne \psi (vw)\). The Maximum Edge-Colorable Subgraph problem takes as input a graph G and integers l and p, and the objective is to find a subgraph H of G and a p-edge-coloring of H, such that \(|E(H)| \ge l\). We study the above problem from the viewpoint of Parameterized Complexity. We obtain FPT algorithms when parameterized by: (1) the vertex cover number of G, by using Integer Linear Programming, and (2) l, a randomized algorithm via a reduction to Rainbow Matching, and a deterministic algorithm by using color coding, and divide and color. With respect to the parameters \(p+k\), where k is one of the following: (1) the solution size, l, (2) the vertex cover number of G, and (3) \(l - {\texttt {mm}}(G)\), where \({\texttt {mm}}(G)\) is the size of a maximum matching in G; we show that the (decision version of the) problem admits a kernel with \({\mathcal {O}}(k \cdot p)\) vertices. Furthermore, we show that there is no kernel of size \({\mathcal {O}}(k^{1-\epsilon } \cdot f(p))\), for any \(\epsilon > 0\) and computable function f, unless \({\textsf {NP}}\subseteq \textsf {coNP/poly}\).

Second-Order Finite Automata

Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each k and w one can count the number of distinct functions computable by ODDs of width at most w and length k in time h(|Σ|,w) ⋅ kO(1), for a suitable h:mathbb {N}times mathbb {N}rightarrow mathbb {N}. This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter w and in the length k of the ODDs.

Orienting Ordered Scaffolds: Complexity and Algorithms

Despite the recent progress in genome sequencing and assembly, many of the currently available assembled genomes come in a draft form. Such draft genomes consist of a large number of genomic fragments (scaffolds), whose order and/or orientation (i.e., strand) in the genome are unknown. There exist various scaffold assembly methods, which attempt to determine the order and orientation of scaffolds along the genome chromosomes. Some of these methods (e.g., based on FISH physical mapping, chromatin conformation capture, etc.) can infer the order of scaffolds, but not necessarily their orientation. This leads to a special case of the scaffold orientation problem (i.e., deducing the orientation of each scaffold) with a known order of the scaffolds. We address the problem of orientating ordered scaffolds as an optimization problem based on given weighted orientations of scaffolds and their pairs (e.g., coming from pair-end sequencing reads, long reads, or homologous relations). We formalize this problem using notion of a scaffold graph (i.e., a graph, where vertices correspond to the assembled contigs or scaffolds and edges represent connections between them). We prove that this problem is NP-hard, and present a polynomial-time algorithm for solving its special case, where orientation of each scaffold is imposed relatively to at most two other scaffolds. We further develop an FPT algorithm for the general case of the OOS problem.

Spy game: FPT-algorithm, hardness and graph products

In the (s,d)-spy game over a graph G, k guards and one spy occupy some vertices of G and, at each turn, the spy may move with speed s (along at most s edges) and each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy wins if she reaches a vertex at distance more than the surveilling distance d from every guard. This game was introduced by Cohen et al. (2016) [15] and is related to two well-studied games: Cops and Robber game and Eternal Dominating game. The guard number gns,d(G) is the minimum number of guards such that the guards have a winning strategy (of controlling the spy) in the graph G. In 2018, it was proved that deciding if the spy has a winning strategy is NP-hard for every speed s≥2 and distance d≥0. In this paper, we initiate the investigation of the guard number in grids and in graph products. We obtain a strict upper bound on the strong product of two general graphs and obtain examples with King grids that match this bound and other examples for which the guard number is smaller. We also obtain the exact value of the guard number in the lexicographical product of two general graphs for any distance d≥2. From the algorithmic point of view, we prove a positive result: if the number k of guards is fixed, the spy game is solvable in polynomial XP time O(n3k+2) for every speed s≥2 and distance d≥0. In other words, the spy game is XP when parameterized by the number of guards. This XP algorithm is used to obtain an FPT algorithm on the P4-fewness of the graph. As a negative result, we prove that the spy game is W[2]-hard even in bipartite graphs when parameterized by the number of guards, for every speed s≥2 and distance d≥0, extending the hardness result of Cohen et al. in 2018.

Tree diet: reducing the treewidth to unlock FPT algorithms in RNA bioinformatics

Hard graph problems are ubiquitous in Bioinformatics, inspiring the design of specialized Fixed-Parameter Tractable algorithms, many of which rely on a combination of tree-decomposition and dynamic programming. The time/space complexities of such approaches hinge critically on low values for the treewidth tw of the input graph. In order to extend their scope of applicability, we introduce the Tree-Diet problem, i.e. the removal of a minimal set of edges such that a given tree-decomposition can be slimmed down to a prescribed treewidth tw'. Our rationale is that the time gained thanks to a smaller treewidth in a parameterized algorithm compensates the extra post-processing needed to take deleted edges into account. Our core result is an FPT dynamic programming algorithm for Tree-Diet, using 2^{O(tw)}n time and space. We complement this result with parameterized complexity lower-bounds for stronger variants (e.g., NP-hardness when tw' or tw-tw' is constant). We propose a prototype implementation for our approach which we apply on difficult instances of selected RNA-based problems: RNA design, sequence-structure alignment, and search of pseudoknotted RNAs in genomes, revealing very encouraging results. This work paves the way for a wider adoption of tree-decomposition-based algorithms in Bioinformatics.

New FPT Algorithms for Finding the Temporal Hybridization Number for Sets of Phylogenetic Trees

We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5^kcdot ncdot m). We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in O((8k)^d5^ kcdot kcdot ncdot m) time. Lastly, we introduce an O(6^kk!cdot kcdot n^2) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

Star colouring of bounded degree graphs and regular graphs

A k-star colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that f(u)≠f(v) for every edge uv of G, and every bicoloured connected subgraph of G is a star. The star chromatic number of G, χs(G), is the least integer k such that G is k-star colourable. We prove that χs(G)≥⌈(d+4)/2⌉ for every d-regular graph G with d≥3. We reveal the structure and properties of even-degree regular graphs G that attain this lower bound. The structure of such graphs G is linked with a certain type of Eulerian orientations of G. Moreover, this structure can be expressed in the LC-VSP framework of Telle and Proskurowski (SIDMA, 1997), and hence can be tested by an FPT algorithm with the parameter either treewidth, cliquewidth, or rankwidth. We prove that for p≥2, a 2p-regular graph G is (p+2)-star colourable only if n:=|V(G)| is divisible by (p+1)(p+2). For each p≥2 and n divisible by (p+1)(p+2), we construct a 2p-regular Hamiltonian graph on n vertices which is (p+2)-star colourable.The problem k-Star Colourability takes a graph G as input and asks whether G is k-star colourable. We prove that 3-Star Colourability is NP-complete for planar bipartite graphs of maximum degree three and arbitrarily large girth. Besides, it is coNP-hard to test whether a bipartite graph of maximum degree eight has a unique 3-star colouring up to colour swaps. For k≥3, k-Star Colourability of bipartite graphs of maximum degree k is NP-complete, and does not even admit a 2o(n)-time algorithm unless ETH fails.

On the Parameterized Complexity of Maximum Degree Contraction Problem

In the Maximum Degree Contraction problem, the input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time $$n^{\mathcal {O}(k)}$$ . As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van’t Hof, and Paulusma studied the problem in the realm of parameterized complexity and proved, among other things, that it admits an FPT algorithm running in time $$(d + k)^{2k} \cdot n^{\mathcal {O}(1)} = 2^{\mathcal {O}(k \log (k+d) )} \cdot n^{\mathcal {O}(1)}$$ , and remains NP-hard for every constant $$d \ge 2$$ (Acta Informatica (2014)). We present a different FPT algorithm that runs in time $$2^{\mathcal {O}(dk)} \cdot n^{\mathcal {O}(1)}$$ . In particular, our algorithm runs in time $$2^{\mathcal {O}(k)} \cdot n^{\mathcal {O}(1)}$$ , for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by $$k + d$$ . We answer this question in the negative and prove that it does not admit a polynomial compression unless $$\textsf {NP}\subseteq \textsf {coNP}/poly$$ .

An Fpt Algorithm for Node-Disjoint Subtrees Problems Parameterized by Treewidth

An Fpt Algorithm for Node-Disjoint Subtrees Problems Parameterized by Treewidth

Twin-width I: Tractable FO Model Checking

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

Inserting an edge into a geometric embedding

The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e[14] is a helpful tool for designing heuristics that minimize edge crossings in topological drawings of general graphs. Unfortunately, not all such topological drawings are stretchable, i.e., there may not exist an equivalent straight-line drawing. That is, there is no planar straight-line drawing Γ of G such that in Γ+e the edge e crosses the same edges as in the topological drawing of G+e and it does so in the same order. This motivates the study of the computational complexity of the problem Geometric Edge Insertion: Given a combinatorially embedded graph G, compute a geometric embedding Γ of G that minimizes the crossings in Γ+e.We give a characterization of the stretchable topological drawings of G+e that also applies to the case where the outer face is fixed; this answers an open question of Eades et al. [8]. Algorithmically, we focus on the case where the outer face is not fixed. We show that Geometric Edge Insertion can be solved efficiently for graphs of maximum degree 5. For the general case, we show a (Δ−2)-approximation, where Δ is the maximum vertex degree of G and an FPT algorithm with respect to the minimum number of crossings. Finally, we consider the problem of testing whether there exists a solution of Geometric Edge Insertion that achieves the lower bound obtained by a topological insertion.

Matchings under distance constraints I.

This paper introduces the d-distance matching problem, in which we are given a bipartite graph G=(S,T;E) with S={s_1,dots ,s_n}, a weight function on the edges and an integer din mathbb Z_+. The goal is to find a maximum-weight subset Msubseteq E of the edges satisfying the following two conditions: (i) the degree of every node of S is at most one in M, (ii) if s_it,s_jtin M, then |j-i|ge d. This question arises naturally, for example, in various scheduling problems. We show that the problem is NP-complete in general and admits a simple 3-approximation. We give an FPT algorithm parameterized by d and also show that the case when the size of T is constant can be solved in polynomial time. From an approximability point of view, we show that the integrality gap of the natural integer programming model is at most 2-frac{1}{2d-1}, and give an LP-based approximation algorithm for the weighted case with the same guarantee. A combinatorial (2-frac{1}{d})-approximation algorithm is also presented. Several greedy approaches are considered, and a local search algorithm is described that achieves an approximation ratio of 3/2+epsilon for any constant epsilon >0 in the unweighted case. The novel approaches used in the analysis of the integrality gap and the approximation ratio of locally optimal solutions might be of independent combinatorial interest.

C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time O(r(n^2) + n^2) for general, possibly non-flat, instances.

Finding Temporal Paths Under Waiting Time Constraints

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration varDelta , referred to as varDelta -restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the distance to disjoint path of the underlying graph, which implies W[1]-hardness for many other parameters like feedback vertex number and pathwidth. A natural question is thus whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

New Algorithms for Mixed Dominating Set

A mixed dominating set is a collection of vertices and edges that dominates all vertices and edges of a graph. We study the complexity of exact and parameterized algorithms for \textsc{Mixed Dominating Set}, resolving some open questions. In particular, we settle the problem's complexity parameterized by treewidth and pathwidth by giving an algorithm running in time $O^*(5^{tw})$ (improving the current best $O^*(6^{tw})$), as well as a lower bound showing that our algorithm cannot be improved under the Strong Exponential Time Hypothesis (SETH), even if parameterized by pathwidth (improving a lower bound of $O^*((2 - \varepsilon)^{pw})$). Furthermore, by using a simple but so far overlooked observation on the structure of minimal solutions, we obtain branching algorithms which improve both the best known FPT algorithm for this problem, from $O^*(4.172^k)$ to $O^*(3.510^k)$, and the best known exponential-time exact algorithm, from $O^*(2^n)$ and exponential space, to $O^*(1.912^n)$ and polynomial space.

Parameterized complexity of fair feedback vertex set problem

Given a graph G=(V,E), a subset S⊆V(G) is said to be a feedback vertex set of G if G−S is a forest. In the Feedback Vertex Set (FVS) problem, we are given an undirected graph G, and a positive integer k, the question is whether there exists a feedback vertex set of size at most k. In this paper, we study three variants of the FVS problem: Unrestricted Fair FVS, Restricted Fair FVS, and Relaxed Fair FVS. In Unrestricted Fair FVS, we are given a graph G and a positive integer ℓ, the question is does there exist a feedback vertex set S⊆V(G) (of any size) such that for every vertex v∈V(G), v has at most ℓ neighbours in S. First, we study Unrestricted Fair FVS from different parameterizations such as treewidth, treedepth, and neighbourhood diversity and obtain several results (both tractability and intractability). Next, we study Restricted Fair FVS, where we are also given an integer k in the input and we demand the size of S to be at most k. This problem is trivially NP-complete; we show that Restricted Fair FVS when parameterized by the solution size k and the maximum degree Δ of the graph G, admits a kernel of size O(Δk). Finally, we study the Relaxed Fair FVS problem, where we want that the size of S is at most k and for every vertex v outside S, v has at most ℓ neighbours in S. We give an FPT algorithm for Relaxed Fair FVS problem running in time cknO(1), for a fixed constant c.

Metric Dimension Parameterized By Treewidth

A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time f(text {pw})n^{o(text {pw})} on n-vertex graphs of constant degree, with text {pw} the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter text {tl}+Delta, where text {tl} is the tree-length and Delta the maximum-degree of the input graph.

Finding Cuts of Bounded Degree: Complexity, FPT and Exact Algorithms, and Kernelization

A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce a natural generalization of this problem, which we call d -Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d -Cut on $$(2d+2)$$ -regular graphs and a polynomial algorithm for graphs of maximum degree at most $$d+2$$ . The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [DAM, 2020], is a polynomial kernel for d -Cut for every positive integer d, parameterized by the vertex deletion distance of the input graph to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time $$\mathcal {O}^*\!\left( 2^n\right)$$ .

An improved FPT algorithm for the flip distance problem

Given a set P of points in the Euclidean plane and two triangulations of P, the flip distance between these two triangulations is the minimum number of flips required to transform one triangulation into the other. The Parameterized Flip Distance problem is to decide if the flip distance between two given triangulations is equal to a given integer k. The previous best FPT algorithm runs in time O⁎(k⋅ck) (c≤2×1411), where each step has fourteen possible choices, and the length of the action sequence is bounded by 11k. By analyzing the underlying properties of the flip sequence, each step of our algorithm has only five possible choices. Based on an auxiliary graph G, we prove that the length of the action sequence for our algorithm is bounded by 2|G|. As a result, we present an FPT algorithm running in time O⁎(k⋅32k).