Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund and Husfeldt, ICALP 2014; Mari et al., SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing . In Min-Sum Cycle Packing , we try to find, in a weighted undirected graph, \( k \) vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed \( k \) , solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by \( k \) . The same results hold for the version of the problem where the task is to find \( k \) edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing , which is a special case of Min-Sum Cycle Packing that asks to find a packing of \( k \) shortest cycles in a graph. We prove this problem to be Fixed-Parameter Tractable (FPT) when parameterized by \( k \) on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Whether Min-Sum Cycle Packing is FPT on planar graphs, or Shortest Cycle Packing on general graphs, remains open.

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

The Euclidean Traveling Salesman Problem with Neighborhoods (ETSPN) is a well-studied problem in computational geometry and has a wealth of results. In this problem, given a set of geometric neighborhoods (or regions), the goal is to compute a shortest route that visits at least one point of each neighborhood. The problem is a generalization of the standard Euclidean TSP and hence is also NP-hard, even when the neighborhoods are disjoint unit disks or unit squares in the plane. A longstanding open problem on this topic is the existence of PTASes for ETSPN with unit disk (and unit square) neighborhoods. Prior to this work, the best-known approximation factor for unit disks is 6.75, and PTASes are only known for the special case where the unit disks/squares are of bounded depth, i.e., each point lies in at most a constant number of disks.

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Björklund, Husfeldt, ICALP 2014;

Abstract In this paper, we introduce a directed variant of the classical Bandwidthproblem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph $$\varvec{D}$$ D and an ordering $$\varvec{\sigma }$$ σ of its vertices, the digraph bandwidth of $$\varvec{\sigma }$$ σ with respect to $$\varvec{D}$$ D is equal to the maximum value of $$\varvec{\sigma (v)}-\varvec{\sigma (u)}$$ σ ( v ) - σ ( u ) over all arcs $$\varvec{(u,v)}$$ ( u , v ) of $$\varvec{D}$$ D going forward along $$\varvec{\sigma }$$ σ (that is, when $$\varvec{\sigma (u)} < \varvec{\sigma (v)}$$ σ ( u ) < σ ( v ) ). The Digraph Bandwidth problem takes as input a digraph $$\varvec{D}$$ D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidtheasily reduces to Digraph Bandwidth and thus, it immediately implies that Digraph Bandwidth is -hard. While an $$\varvec{\mathcal {O}}^{\star }\varvec{(n!)}$$ O ⋆ ( n ! ) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form $$\varvec{2}^{\varvec{\mathcal {O}(n)}}$$ 2 O ( n ) . In particular, we obtain the following results. Here, $$\varvec{n}$$ n and $$\varvec{m}$$ m denote the number of vertices and arcs of the input digraph $$\varvec{D}$$ D , respectively. Digraph Bandwidth can be solved in $$\varvec{\mathcal {O}}^\star (\varvec{3}^{\varvec{n}} \cdot \varvec{2}^{\varvec{m}})$$ O ⋆ ( 3

In the F-Deletion problem, where F is a fixed finite family of graphs, the input is a graph G and an integer k, and the goal is to determine if there exists a set of at most k vertices whose deletion results in a graph that does not contain any graph of F as a minor. The F-Deletion problem encapsulates a large class of natural and interesting graph problems like Vertex Cover, Feedback Vertex Set, Treewidth-η Deletion, Treedepth-η Deletion, Pathwidth-η Deletion, Outerplanar Deletion, Vertex Planarization and many more. We study the F-Deletion problem from the kernelization perspective. In a seminal work, Fomin et al. [FOCS 2012] gave a polynomial kernel for this problem when the family F contains at least one planar graph. The asymptotic growth of the size of the kernel is not uniform with respect to the family F: that is, the size of the kernel is k^{f(F)}, for some function f that depends only on F. Later Giannopoulou et al. [TALG 2017] showed that the non-uniformity in the kernel size bound is unavoidable as Treewidth-η Deletion cannot admit a kernel of size 𝒪(k^{(η+1)/2 - ε}), for any ε > 0, unless NP ⊆ coNP/poly. On the other hand it was also shown that Treedepth-η Deletion admits a uniform kernel of size f(F) ⋅ k⁶ depicting that there are subclasses of F where the asymptotic kernel sizes do not grow as a function of the family F. This work led to the question of determining classes of F where the problem admits uniform polynomial kernels. In this paper, we show that if all the graphs in F are connected and ℱ contains K_{2,p} (a bipartite graph with 2 vertices on one side and p vertices on the other), then the problem admits a uniform kernel of size f(F) ⋅ k^10. The graph K_{2,p} is one natural extension of the graph θ_p, where θ_p is a graph on two vertices and p parallel edges. The case when F contains θ_p has been studied earlier and serves as (the only) other example where the problem admits a uniform polynomial kernel.

In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of n points in ℝ2 called terminals and a parameter k, and the goal is to compute a Steiner tree that spans all the terminals and contains at most k points of ℝ2 as Steiner points such that the maximum edge-length of the Steiner tree is minimized, where the length of a tree edge is the Euclidean distance between its two endpoints. The problem is well-studied and is known to be NP-hard. In this paper, we give a kO(k)nO(1)-time algorithm for Euclidean Bottleneck Steiner Tree, which implies that the problem is fixed-parameter tractable (FPT). This settles an open question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that the ℓ1 and ℓ∞ variants of the problem are FPT. Our approach can be generalized to the problem with ℓp metric for any rational 1 ≤ ρ ≤ ∞, or even other metrics on ℝ2.

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set \(\mathcal{D}\) of \(n\) unit disks inducing a unit-disk graph \(G_{\mathcal{D}}\) and a number \(p\in[n]\) , one can partition \(\mathcal{D}\) into \(p\) subsets \(\mathcal{D}_{1},\dots,\mathcal{D}_{p}\) such that for every \(i\in[p]\) and every \(\mathcal{D}^{\prime}\subseteq\mathcal{D}_{i}\) , the graph obtained from \(G_{\mathcal{D}}\) by contracting all edges between the vertices in \(\mathcal{D}_{i}\backslash\mathcal{D}^{\prime}\) admits a tree decomposition in which each bag consists of \(O(p+|\mathcal{D}^{\prime}|)\) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved recently by Marx et al. [SODA ’22] and Bandyapadhyay et al. [SODA ’22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem for unit-disk graphs, resolving an open question in the work by Panolan et al. [SODA ’19]. On the algorithmic side, we obtain a new algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in \(2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}\) time, where \(k\) denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA ’22] which runs in \(2^{O(k^{27/28})}\cdot n^{O(1)}\) time. We also show that the problem cannot be solved in \(2^{o(\sqrt{k})}\cdot n^{O(1)}\) time assuming the Exponential Time Hypothesis, which implies that our algorithm is almost optimal.

We revisit a natural variant of geometric set cover, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set $S$ of points and a set $\mathcal{R}$ of geometric objects, and the goal is to find a subset $\mathcal{R}^*\subseteq\mathcal{R}$ to cover all points in $S$ such that the \textit{membership} of $S$ with respect to $\mathcal{R}^*$, denoted by $\mathsf{memb}(S,\mathcal{R}^*)$, is minimized, where $\mathsf{memb}(S,\mathcal{R}^*)=\max_{p\in S}|\{R\in\mathcal{R}^*: p\in R\}|$. We achieve the following two main results. * We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time $n^{O(\mathsf{opt})}$ where $\mathsf{opt}$ is the optimum of the problem (i.e., the minimum membership). * We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of $o(\log n)$ in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find $\mathcal{R}^*\subseteq\mathcal{R}$ to cover $S$ such that the ply of $\mathcal{R}^*$ is minimized, where the ply is defined as the maximum number of objects in $\mathcal{R}^*$ which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in $O(n^{12})$ time. We give a significantly simpler constant-approximation algorithm with near-linear running time.

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into $k$ clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a $(15+ε)$-approximation algorithm that runs in $2^{0(k^2\log k)}\cdot n^3$ time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + $ε$)-approximation with running time $2^{O(k\log(k/ε))}n^3$, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + $ε$)-approximation with running time $2^{O(k/ε^2 \cdot\log(k/ε))}dn^3$ and a $(1+ε)$-approximation with running time $2^{O(kd\log ((k/ε)))}n^{3}$ in the Euclidean space; and a (1 + $ε$)-approximation in the Euclidean space with running time $2^{O(k/ε^2 \cdot\log(k/ε))}dn^3$ if we are allowed to violate the capacities by (1 + $ε$)-factor. We complement this result by showing that there is no (1 + $ε$)-approximation algorithm running in time $f(k)\cdot n^{O(1)}$, if any capacity violation is not allowed.

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependencies on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k-means and k-median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the “best explanation” by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

We study two “above guarantee” versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least distG(s,t)+k. Bezáková et al. [7] proved that on undirected graphs the problem is fixed-parameter tractable (FPT). Our first main result establishes a connection between Longest Detour on directed graphs and 3- Disjoint Paths on directed graphs. Using these new insights, we design a 2O(k)⋅nO(1) time algorithm for the problem on directed planar graphs. Furthermore, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k. We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs.

k-means and k-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost proposed an elegant model of explainable k-means and k-median clustering in ICML 2020. In this model, a decision tree with k leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the ``best explanation'' by using a decision tree with k leaves? (2) For a given set of points, how to find a decision tree with k leaves minimizing the k-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on H-minor free graphs. In particular, we obtain the following results (where k is the solution-size parameter). time algorithms for Edge Bipartization and Odd Cycle Transversal; a time algorithm for Edge Multiway Cut and a time algorithm for Vertex Multiway Cut (with undeletable terminals), where r is the number of terminals to be separated; a time algorithm for Edge Multicut and a time algorithm for Vertex Multicut (with undeletable terminals), where r is the number of terminal pairs to be separated; a time algorithm for Group Feedback Edge Set and a time algorithm for Group Feedback Vertex Set, where g is the size of the group. In addition, our approach also gives time algorithms for all above problems with the exception of time for Edge/Vertex Multicut and time for Group Feedback Edge/Vertex Set. All of our FPT algorithms (the first four items above) are randomized, as they use known randomized kernelization algorithms as sub-routines. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an h-almost-embeddable graph for an arbitrary but fixed constant h. Our new decomposition theorem generalizes known Contraction Decomposition Theorem. Prior studies on this topic exhibited that the classes of planar graphs [Klein, SICOMP, 2008], graphs of bounded genus [Demaine, Hajiaghayi and Mohar, Combinatorica 2010] and H-minor free graphs [Demaine, Hajiaghayi and Kawarabayashi, STOC 2011] admit a Contraction Decomposition Theorem. In particular we show the following. Let G be a graph of bounded genus, or more generally, an h-almost-embeddable graph for an arbitrary but fixed constant h. Then for every p ∊ ℕ, there exist disjoint sets Z1, …, Zp ⊆ V(G) such that for every i ∊ {1, …, p} and every Z′ ⊆ Zi, the treewidth of G/(Zi\Z′) is upper bounded by O(p + |Z′|), where the constant hidden in O(·) depends on h. Here G/(Zi\Z′) denotes the graph obtained from G by contracting every edge with both endpoints in Zi\Z′. When Z′ = , this corresponds to classical Contraction Decomposition Theorem.

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bez\'akov\'a et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O (k)} n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path Above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path Above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path Above Diameter is solvable in polynomial time for each k\in{1,\dots, 4} and is NP-complete for every k\geq 5. The parameterized complexity of Longest Path Above Diameter on general directed graphs remains an interesting open problem.

Abstract We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most $$m-k$$ m - k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most $$m-k$$ m - k arcs. We show that (i) Directed Multiplicative Spanner admits a polynomial kernel of size $$\mathcal {O}(k^4t^5)$$ O ( k 4 t 5 ) and can be solved in randomized $$(4t)^k\cdot n^{\mathcal {O}(1)}$$ ( 4 t ) k · n O ( 1 ) time, (ii) the weighted variant of Directed Multiplicative Spanner can be solved in $$k^{2k}\cdot n^{\mathcal {O}(1)}$$ k 2 k · n O ( 1 ) time on directed acyclic graphs, (iii) Directed Additive Spanner is $${{\,\mathrm{\mathsf{W}}\,}}[1]$$ W [ 1 ] -hard when parameterized by k for every fixed $$t\ge 1$$ t ≥ 1 even when the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is $${{\,\mathrm{\mathsf{FPT}}\,}}$$ FPT when parameterized by t and k.

We consider a generalization of the fundamental k-means clustering for data with incomplete or corrupted entries. When data objects are represented by points in ℝd, a data point is said to be incomplete when some of its entries are missing or unspecified. An incomplete data point with at most Δ unspecified entries corresponds to an axis-parallel affine subspace of dimension at most Δ, called a Δ-point. Thus we seek a partition of n input Δ-points into k clusters minimizing the k-means objective. For Δ = 0, when all coordinates of each point are specified, this is the usual k-means clustering. We give an algorithm that finds an (1 + ∊)-approximate solution in time f(k, ∊, Δ) · n2 · d for some function f of k, ∊, and Δ only.

The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph G and a set of k pairs of terminals in G, it asks for the existence of k vertex-disjoint paths connecting each pair of terminals. The proof of Robertson and Seymour [JCTB 1995] of the existence of an n3 algorithm for any fixed k is one of the highlights of their Graph Minors project. In this paper, we focus on the version of the problem where all the paths are required to be shortest paths. This problem, called the disjoint shortest paths problem, was introduced by Eilam-Tzoreff [DAM 1998] where she proved that the case k = 2 admits a polynomial time algorithm. This problem has received some attention lately, especially since the proof of the existence of a polynomial time algorithm in the directed case when k = 2 by Bérczi and Kobayashi [ESA 2017]. However, the existence of a polynomial algorithm when k = 3 in the undirected version remained open since 1998. In this paper we show that for any fixed k, the disjoint shortest paths problem admits a polynomial time algorithm. In fact for any fixed C, the algorithm can be extended to treat the case where each path connecting the pair (s, t) has length at most d(s, t) + C.

Over the past few decades, the study of dense structures from the perspective of approximation algorithms has become a wide area of research. However, from the viewpoint of parameterized algorithm, this area is largely unexplored. In particular, properties of random samples have been successfully deployed to design approximation schemes for a number of fundamental problems on dense structures [Arora et al. FOCS 1995, Goldreich et al. FOCS 1996, Giotis and Guruswami SODA 2006, Karpinksi and Schudy STOC 2009]. In this paper, we fill this gap, and harness the power of random samples as well as structure theory to design kernelization as well as parameterized algorithms on dense structures. In particular, we obtain linear vertex kernels for Edge-Disjoint Paths, Edge Odd Cycle Transversal, Minimum Bisection, d-Way Cut, Multiway Cut and Multicut on everywhere dense graphs. In fact, these kernels are obtained by designing a polynomial-time algorithm when the corresponding parameter is at most Ω(n). Additionally, we obtain a cubic kernel for Vertex-Disjoint Paths on everywhere dense graphs. In addition to kernelization results, we obtain randomized subexponential-time parameterized algorithms for Edge Odd Cycle Transversal, Minimum Bisection, and d-Way Cut. Finally, we show how all of our results (as well as EPASes for these problems) can be de-randomized.

The line graph of a graph G is the graph L(G) whose vertex set is the edge set of G and there is an edge between e,f ∈ E(G) if e and f share an endpoint in G. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most k edges from the input graph G such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with O(k⁵) vertices. This answers an open question posed by Falk Hüffner at Workshop on Kernels (WorKer) in 2013.

In the past decade, the design of fault tolerant data structures for networks has become a central topic of research. Particular attention has been given to the construction of a subgraph H of a given digraph D with as fewest arcs/vertices as possible such that, after the failure of any set F of at most k ≥ 1 arcs, testing whether D-F has a certain property P is equivalent to testing whether H-F has that property. Here, reachability (or, more generally, distance preservation) is the most basic requirement to maintain to ensure that the network functions properly. Given a vertex s ∈ V(D), Baswana et al. [STOC'16] presented a construction of H with O(2^kn) arcs in time O(2^{k}nm) where n=|V(D)| and m= |E(D)| such that for any vertex v ∈ V(D): if there exists a path from s to v in D-F, then there also exists a path from s to v in H-F. Additionally, they gave a tight matching lower bound. While the question of the improvement of the dependency on k arises for special classes of digraphs, an arguably more basic research direction concerns the dependency on n (for reachability between a pair of vertices s,t ∈ V(D)) - which are the largest classes of digraphs where the dependency on n can be made sublinear, logarithmic or even constant? Already for the simple classes of directed paths and tournaments, Ω(n) arcs are mandatory. Nevertheless, we prove that "almost acyclicity" suffices to eliminate the dependency on n entirely for a broad class of dense digraphs called bounded independence digraphs. Also, the dependence in k is only a polynomial factor for this class of digraphs. In fact, our sparsification procedure extends to preserve parity-based reachability. Additionally, it finds notable applications in Kernelization: we prove that the classic Directed Feedback Arc Set (DFAS) problem as well as Directed Edge Odd Cycle Transversal (DEOCT) (which, in sharp contrast to DFAS, is W[1]-hard on general digraphs) admit polynomial kernels on bounded independence digraphs. In fact, for any p ∈ N, we can design a polynomial kernel for the problem of hitting all cycles of length ℓ where (ℓ mod p = 1). As a complementary result, we prove that DEOCT is NP-hard on tournaments by establishing a combinatorial identity between the minimum size of a feedback arc set and the minimum size of an edge odd cycle transversal. In passing, we also improve upon the running time of the sub-exponential FPT algorithm for DFAS in digraphs of bounded independence number given by Misra et at. [FSTTCS 2018], and give the first sub-exponential FPT algorithm for DEOCT in digraphs of bounded independence number.

For a fixed graph $H$, the $H$-free-editing problem asks whether we can modify a given graph $G$ by adding or deleting at most $k$ edges such that the resulting graph does not contain $H$ as an induced subgraph. The problem is known to be NP-complete for all fixed $H$ with at least $3$ vertices and it admits a $2^{O(k)}n^{O(1)}$ algorithm. Cai and Cai showed that the $H$-free-editing problem does not admit a polynomial kernel whenever $H$ or its complement is a path or a cycle with at least $4$ edges or a $3$-connected graph with at least $1$ edge missing. Their results suggest that if $H$ is not independent set or a clique, then $H$-free-editing admits polynomial kernels only for few small graphs $H$, unless $\textsf{coNP} \in \textsf{NP/poly}$. Therefore, resolving the kernelization of $H$-free-editing for small graphs $H$ plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs $H$ on $4$ vertices. Namely, we give the first polynomial kernel for paw-free editing with $O(k^{6})$vertices.

We initiate the parameterized complexity study of minimum t-spanner problems on directed graphs. For a positive integer t, a multiplicative t-spanner of a (directed) graph G is a spanning subgraph H such that the distance between any two vertices in H is at most t times the distance between these vertices in G, that is, H keeps the distances in G up to the distortion (or stretch) factor t. An additive t-spanner is defined as a spanning subgraph that keeps the distances up to the additive distortion parameter t, that is, the distances in H and G differ by at most t. The task of Directed Multiplicative Spanner is, given a directed graph G with m arcs and positive integers t and k, decide whether G has a multiplicative t-spanner with at most m-k arcs. Similarly, Directed Additive Spanner asks whether G has an additive t-spanner with at most m-k arcs. We show that - Directed Multiplicative Spanner admits a polynomial kernel of size 𝒪(k⁴t⁵) and can be solved in randomized (4t)^k⋅ n^𝒪(1) time, - Directed Additive Spanner is W[1]-hard when parameterized by k even if t = 1 and the input graphs are restricted to be directed acyclic graphs. The latter claim contrasts with the recent result of Kobayashi from STACS 2020 that the problem for undirected graphs is FPT when parameterized by t and k.

A proper edge coloring of a graph is adjacent vertex distinguishing if no two adjacent vertices see the same set of colors. Using a clever application of the local lemma, Hatami [ J. Combin. Theory Ser. B, 95 (2005), pp. 246--256] proved that every graph with maximum degree $\Delta$ and no isolated edge has an adjacent vertex distinguishing edge coloring with $\Delta + 300$ colors, provided $\Delta$ is large enough. We show that this bound can be reduced to $\Delta + 19$. This is motivated by the conjecture of Zhang, Liu, and Wang [ Appl. Math. Lett., 15 (2002), pp. 623--626] that $\Delta + 2$ colors are enough for $\Delta \geqslant 3$.

In this paper, we introduce a directed variant of the classical Bandwidth problem and study it from the view-point of moderately exponential time algorithms, both exactly and approximately. Motivated by the definitions of the directed variants of the classical Cutwidth and Pathwidth problems, we define Digraph Bandwidth as follows. Given a digraph D and an ordering sigma of its vertices, the digraph bandwidth of sigma with respect to D is equal to the maximum value of sigma(v)-sigma(u) over all arcs (u,v) of D going forward along sigma (that is, when sigma(u) < sigma (v)). The Digraph Bandwidth problem takes as input a digraph D and asks to output an ordering with the minimum digraph bandwidth. The undirected Bandwidth easily reduces to Digraph Bandwidth and thus, it immediately implies that Directed Bandwidth is {NP-hard}. While an O^*(n!) time algorithm for the problem is trivial, the goal of this paper is to design algorithms for Digraph Bandwidth which have running times of the form 2^O(n). In particular, we obtain the following results. Here, n and m denote the number of vertices and arcs of the input digraph D, respectively. - Digraph Bandwidth can be solved in O^*(3^n * 2^m) time. This result implies a 2^O(n) time algorithm on sparse graphs, such as graphs of bounded average degree. - Let G be the underlying undirected graph of the input digraph. If the treewidth of G is at most t, then Digraph Bandwidth can be solved in time O^*(2^(n + (t+2) log n)). This result implies a 2^(n+O(sqrt(n) log n)) algorithm for directed planar graphs and, in general, for the class of digraphs whose underlying undirected graph excludes some fixed graph H as a minor. - Digraph Bandwidth can be solved in min{O^*(4^n * b^n), O^*(4^n * 2^(b log b log n))} time, where b denotes the optimal digraph bandwidth of D. This allow us to deduce a 2^O(n) algorithm in many cases, for example when b <= n/(log^2n). - Finally, we give a (Single) Exponential Time Approximation Scheme for Digraph Bandwidth. In particular, we show that for any fixed real epsilon > 0, we can find an ordering whose digraph bandwidth is at most (1+epsilon) times the optimal digraph bandwidth, in time O^*(4^n * (ceil[4/epsilon])^n).