In the DELETION TO INDUCED MATCHING problem, we are given a graph $G$ on $n$ vertices, $m$ edges and a non-negative integer $k$ and asks whether there exists a set of vertices $S \subseteq V(G) $ such that $|S|\le k$ and the size of any connected component in $G-S$ is exactly 2. In this paper, we provide a fixed-parameter tractable (FPT) algorithm of running time $O^*(1.748^{k})$ for the DELETION TO INDUCED MATCHING problem using branch-and-reduce strategy and path decomposition. We also extend our work to the exact-exponential version of the problem.
We give an explicit construction of length-$n$ binary codes capable of correcting the deletion of two bits that have size $2^n/n^{4+o(1)}$. This matches up to lower order terms the existential result, based on an inefficient greedy choice of codewords, that guarantees such codes of size $Ω(2^n/n^4)$. Our construction is based on augmenting the classic Varshamov-Tenengolts construction of single deletion codes with additional check equations. We also give an explicit construction of binary codes of size $Ω(2^n/n^{3+o(1)})$ that can be list decoded from two deletions using lists of size two. Pre…
In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice's edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al…
A graph is said to be a Konig graph if the size of its maximum matching is equal to the size of its minimum vertex cover. The Konig Edge Deletion problem asks if in a given graph there exists a set of at most k edges whose deletion results in a Konig graph. While the vertex version of the problem (Konig vertex deletion) has been shown to be fixed-parameter tractable more than a decade ago, the fixed-parameter-tractability of the Konig Edge Deletion problem has been open since then, and has been conjectured to be W[1]-hard in several papers. In this paper, we settle the conjecture by proving it…
A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an approximate bottleneck matching in sublinear update time. In this work, we answer this question in the affirmative for points on a real line and for points in the plane with a bounded geometric spread. For a set $P$ of $n$ points on a line, we show that there exists a dynamic algorithm that maintains a bottleneck matching of $P$ and supports insertion and deletion in $O(\…
We study the parameterized and classical complexity of two related problems on undirected graphs $G=(V,E)$. In Strong Triadic Closure we aim to label the edges in $E$ as strong and weak such that at most~$k$ edges are weak and $G$ contains no induced $P_3$ with two strong edges. In Cluster Deletion, we aim to destroy all induced $P_3$s by a minimum number of edge deletions. We first show that Strong Triadic Closure admits a $4k$-vertex kernel. Then, we study parameterization by $\ell:=|E|-k$ and show that both problems are fixed-parameter tractable and unlikely to admit a polynomial kernel wit…
The core number of a vertex is a basic index depicting cohesiveness of a graph, and has been widely used in large-scale graph analytics. In this paper, we study the update of core numbers of vertices in dynamic graphs with edge insertions/deletions, which is known as the core maintenance problem. Different from previous approaches that just focus on the case of single-edge insertion/deletion and sequentially handle the edges when multiple edges are inserted/deleted, we investigate the parallelism in the core maintenance procedure. Specifically, we show that if the inserted/deleted edges consti…
We consider the $Π$-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties $Π$. Given an input graph $G$, this problem asks whether there is a subset of at most $k$ vertices whose removal ensures the resulting graph does not contain a graph from $Π$ as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property $Π$ by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for $Π$…
Vertex deletion problems ask whether it is possible to delete at most $k$ vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our resu…
Pattern matching is a fundamental process in almost every scientific domain. The problem involves finding the positions of a given pattern (usually of short length) in a reference stream of data (usually of large length). The matching can be an exact or as an approximate (inexact). Exact matching is to search for the pattern without allowing for mismatches (or insertions and deletions) of one or more characters in the pattern), while approximate matching is the opposite. For exact matching, several data structures that can be built in linear time and space are used and in practice nowadays. Fo…
We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass $2$-approximation streaming algorithm can be easily obtained with space $O(n \log n)$, where $n$ denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space $O(n^{3/2-δ})$ is granted, for every $δ> 0$. Specifically, for every $0 \le ε\le 1$, we show that in the one-pass turnstile streaming model, in order to comp…
In the {claw, diamond}-free edge deletion problem, we are given a graph $G$ and an integer $k>0$, the question is whether there are at most $k$ edges whose deletion results in a graph without claws and diamonds as induced graphs. Based on some refined observations, we propose a kernel of $O(k^3)$ vertices and $O(k^4)$ edges, significantly improving the previous kernel of $O(k^{12})$ vertices and $O(k^{24})$ edges. In addition, we derive an $O^*(3.792^k)$-time algorithm for the {claw, diamond}-free edge deletion problem.
The unit interval vertex deletion problem asks for a set of at most $k$ vertices whose deletion from an $n$-vertex graph makes it a unit interval graph. We develop an $O(k^4)$-vertex kernel for the problem, significantly improving the $O(k^{53})$-vertex kernel of Fomin, Saurabh, and Villanger [ESA'12; SIAM J. Discrete Math 27(2013)]. We introduce a novel way of organizing cliques of a unit interval graph. Our constructive proof for the correctness of our algorithm, using interval models, greatly simplifies the destructive proofs, based on forbidden induced subgraphs, for similar problems in li…
For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. We prove that $H$-free Edge Deletion is NP-complete if $H$ is a graph with at least two edges and $H$ has a component with maximum number of vertices which is a tree or a regular graph. Furthermore, we obtain that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cdot |G|^{O(1)}$, unless Exponential Time Hypothesis fails.
A directed graph $G$ is called a pumpkin if $G$ is a union of induced paths with a common start vertex $s$ and a common end vertex $t$, and the internal vertices of every two paths are disjoint. We give an algorithm that given a directed graph $G$ and an integer $k$, decides whether a pumpkin can be obtained from $G$ by deleting at most $k$ vertices. The algorithm runs in $O^*(2^k)$ time.
A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $l_1$ and $l_2$, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis. We analyze the structure of the so-called almost bipartite permutation graphs …
Two prevalent models in the data stream literature are the insertion-only and turnstile models. Unfortunately, many important streaming problems require a $Θ(\log(n))$ multiplicative factor more space for turnstile streams than for insertion-only streams. This complexity gap often arises because the underlying frequency vector $f$ is very close to $0$, after accounting for all insertions and deletions to items. Signal detection in such streams is difficult, given the large number of deletions. In this work, we propose an intermediate model which, given a parameter $α\geq 1$, lower bounds the n…
A graph $G$ is called a \emph{block graph} if each maximal $2$-connected component of $G$ is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph $G$ and a positive integer $k$ and the objective to check whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that the graph induced on $V(G)\setminus S$ is a block graph. In this paper we give an FPT algorithm with running time $4^kn^{O(1)}$ and a polynomial …
We consider the problem of maintaining an approximate maximum integral matching in a dynamic graph $G$, while the adversary makes changes to the edges of the graph. The goal is to maintain a $(1+ε)$-approximate maximum matching for constant $ε>0$, while minimizing the update time. In the fully dynamic setting, where both edge insertion and deletions are allowed, Gupta and Peng (see \cite{GP13}) gave an algorithm for this problem with an update time of $O(\sqrt{m}/ε^2)$. Motivated by the fact that the $O_ε(\sqrt{m})$ barrier is hard to overcome (see Henzinger, Krinninger, Nanongkai, and Saranur…
Let F be a fixed finite obstruction set of graphs and G be a graph revealed in an online fashion, node by node. The online Delayed F-Node-Deletion Problem (F-Edge-Deletion Problem}) is to keep G free of every H in F by deleting nodes (edges) until no induced subgraph isomorphic to any graph in F can be found in G. The task is to keep the number of deletions minimal. Advice complexity is a model in which an online algorithm has access to a binary tape of infinite length, on which an oracle can encode information to increase the performance of the algorithm. We are interested in the minimum numb…
For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. This brings us to the following natural parameterized questions: {\sc Vertex $(r,\ell)$-Partization} and {\sc Edge $(r,\ell)$-Partization}. An input to these problems consist of a graph $G$ and a positive integer $k$ and the objective is to decide whether there exists a set $S\subseteq V(G)$ ($S\subseteq E(G)$) such that the deletion of $S$ from $G$ results in an $(r,\ell)$-graph. These problems generalize well studied pro…
Let $d$-claw (or $d$-star) stand for $K_{1,d}$, the complete bipartite graph with 1 and $d\ge 1$ vertices on each part. The $d$-claw vertex deletion problem, $d$-CLAW-VD, asks for a given graph $G$ and an integer $k$ if one can delete at most $k$ vertices from $G$ such that the resulting graph has no $d$-claw as an induced subgraph. Thus, 1-CLAW-VD and 2-CLAW-VD are just the famous VERTEX COVER problem and the CLUSTER VERTEX DELETION problem, respectively. In this paper, we strengthen a hardness result in [M. Yannakakis, Node-Deletion Problems on Bipartite Graphs, SIAM J. Comput. (1981)], by s…
The study of parameterized streaming complexity on graph problems was initiated by Fafianie et al. (MFCS'14) and Chitnis et al. (SODA'15 and SODA'16). Simply put, the main goal is to design streaming algorithms for parameterized problems such that $O\left(f(k)\log^{O(1)}n\right)$ space is enough, where $f$ is an arbitrary computable function depending only on the parameter $k$. However, in the past few years, very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Verte…
\textsc{Densest $k$-Subgraph} is the problem to find a vertex subset $S$ of size $k$ such that the number of edges in the subgraph induced by $S$ is maximized. In this paper, we show that \textsc{Densest $k$-Subgraph} is fixed parameter tractable when parameterized by neighborhood diversity, block deletion number, distance-hereditary deletion number, and cograph deletion number, respectively. Furthermore, we give a $2$-approximation $2^{\tc(G)/2}n^{O(1)}$-time algorithm where $\tc(G)$ is the twin cover number of an input graph $G$.
For a set of graphs $\mathcal{H}$, the \textsc{$\mathcal{H}$-free Edge Deletion} problem asks to find whether there exist at most $k$ edges in the input graph whose deletion results in a graph without any induced copy of $H\in\mathcal{H}$. In \cite{cai1996fixed}, it is shown that the problem is fixed-parameter tractable if $\mathcal{H}$ is of finite cardinality. However, it is proved in \cite{cai2013incompressibility} that if $\mathcal{H}$ is a singleton set containing $H$, for a large class of $H$, there exists no polynomial kernel unless $coNP\subseteq NP/poly$. In this paper, we present a p…
In the fundamental Maximum Matching problem the task is to find a maximum cardinality set of pairwise disjoint edges in a given undirected graph. The fastest algorithm for this problem, due to Micali and Vazirani, runs in time $\mathcal{O}(\sqrt{n}m)$ and stands unbeaten since 1980. It is complemented by faster, often linear-time, algorithms for various special graph classes. Moreover, there are fast parameterized algorithms, e.g., time $\mathcal{O}(km\log n)$ relative to tree-width $k$, which outperform $\mathcal{O}(\sqrt{n}m)$ when the parameter is sufficiently small. We show that the Micali…
A diamond is a graph obtained by removing an edge from a complete graph on four vertices. A graph is diamond-free if it does not contain an induced diamond. The Diamond-free Edge Deletion problem asks whether there exist at most $k$ edges in the input graph $G$ whose deletion results in a diamond-free graph. For this problem, a polynomial kernel of $O(k^4$) vertices was found by Fellows et. al. (Discrete Optimization, 2011). In this paper, we give an improved kernel of $O(k^3)$ vertices for Diamond-free Edge Deletion. Further, we give an $O(k^2)$ vertex kernel for a related problem {Diamond,K_…
{\sc Directed Feedback Vertex Set (DFVS)} is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the {\sc ${\cal H}$-free SCC Deletion} problem. Here, one is given a digraph $D$, an integer $k$ and the objective is to decide whether there is a vertex set of size at most $k$ whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family ${\cal H}$ as (not necessarily induced) subgraphs. When ${\cal H}$ comprises only the digraph with a single…
Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly …
A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and an integer $k$, whether there is a set $S$ of at most $k$ vertices in $G$ such that $G-S$ is distance-hereditary. This problem is important due to its connection to the graph parameter rank-width that distance-hereditary graphs are exactly graphs of rank-width at most $1$. Eiben, Ganian, and Kwon (MFCS' 16) proved tha…
In the Cograph Deletion (resp., Cograph Editing) problem the input is a graph $G$ and an integer $k$, and the goal is to decide whether there is a set of edges of size at most $k$ whose removal from $G$ (resp., removal and addition to $G$) results in a graph that does not contain an induced path with four vertices. In this paper we give algorithms for Cograph Deletion and Cograph Editing whose running times are $O^*(2.303^k)$ and $O^*(4.329^k)$, respectively.
We study the problem of deleting the smallest set $S$ of vertices (resp. edges) from a given graph $G$ such that the induced subgraph (resp. subgraph) $G \setminus S$ belongs to some class $\mathcal{H}$. We consider the case where graphs in $\mathcal{H}$ have treewidth bounded by $t$, and give a general framework to obtain approximation algorithms for both vertex and edge-deletion settings from approximation algorithms for certain natural graph partitioning problems called $k$-Subset Vertex Separator and $k$-Subset Edge Separator, respectively. For the vertex deletion setting, our framework co…
We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. Very recently, extending the framework of Baswana, Gupta and Sen [FOCS 2011], Solomon [FOCS 2016] gave a randomized dynamic algorithm for this problem that has an approximation ratio of $2$ and an amortised update time of $O(1)$ with high probability. This algorithm requires the assump…
Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of lin…
We study the maximum matching problem in fully dynamic graphs: a graph is undergoing both edge insertions and deletions, and the goal is to efficiently maintain a large matching after each edge update. This problem has received considerable attention in recent years. The known algorithms naturally exhibit a trade-off between the quality of the matching maintained (i.e., the approximation ratio) and the time needed per update. While several interesting results have been obtained, the optimal behavior of this trade-off remains largely unclear. Our main contribution is a new approach to designing…
In 2005, Goddard, Hedetniemi, Hedetniemi and Laskar [Generalized subgraph-restricted matchings in graphs, Discrete Mathematics, 293 (2005) 129 - 138] asked the computational complexity of determining the maximum cardinality of a matching whose vertex set induces a disconnected graph. In this paper we answer this question. In fact, we consider the generalized problem of finding $c$-disconnected matchings; such matchings are ones whose vertex sets induce subgraphs with at least $c$ connected components. We show that, for every fixed $c \geq 2$, this problem is NP-complete even if we restrict the…
We consider the problem of estimating the weight of a maximum weighted matching of a weighted graph $G(V,E)$ whose edges are revealed in a streaming fashion. We develop a reduction from the maximum weighted matching problem to the maximum cardinality matching problem that only doubles the approximation factor of a streaming algorithm developed for the maximum cardinality matching problem. Our results hold for the insertion-only and the dynamic (i.e, insertion and deletion) edge-arrival streaming models. The previous best-known reduction is due to Bury and Schwiegelshohn (ESA 2015) who develop …
In this paper, we study the parameterized complexity and inapproximability of the {\sc Induced Matching} problem in hamiltonian bipartite graphs. We show that, given a hamiltonian cycle in a hamiltonian bipartite graph, the problem is W[1]-hard and cannot be solved in time $n^{o(k^{\frac{1}{2}})}$ unless W[1]=FPT, where $n$ is the number of vertices in the graph. In addition, we show that unless NP=P, the maximum induced matching in a hamiltonian graph cannot be approximated within a ratio of $n^{1-ε}$, where $n$ is the number of vertices in the graph. For a bipartite hamiltonian graph in $n$ …
We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size $k$ in time $f(k) \cdot n^{O(1)}$, or width parameterizations, finding arbitrarily large optimal solutions in time $f(w) \cdot n^{O(1)}$ for some width measure $w$ like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simult…
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity…
We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes O(log n) expected amortized time for each edge update where n is the number of vertices in the graph. While there is a trivial O(n) algorithm for edge update, the previous best known result for this problem for a graph with n vertices and m edges is O({(n+ m)}^{0.7072})which is sub-linear only for a sparse graph. For the related problem of maximum matching, Onak and Rubinfield designed a randomized data structure that achieves O(log^2 n) amortize…
In the (fully) dynamic set cover problem, we have a collection of $m$ sets from a universe of size $n$ that undergo element insertions and deletions; the goal is to maintain an approximate set cover of the universe after each update. We give an $O(f^2)$ update time algorithm for this problem that achieves an $f$-approximation, where $f$ is the maximum number of sets that an element belongs to; under the unique games conjecture, this approximation is best possible for any fixed $f$. This is the first algorithm for dynamic set cover with approximation ratio that {exactly} matches $f$ (as opposed…
We resolve the space complexity of linear sketches for approximating the maximum matching problem in dynamic graph streams where the stream may include both edge insertion and deletion. Specifically, we show that for any $ε> 0$, there exists a one-pass streaming algorithm, which only maintains a linear sketch of size $\tilde{O}(n^{2-3ε})$ bits and recovers an $n^ε$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. In contrast to the extensively studied insertion-only model, to the best of our knowledge, no non-trivial single-pass streaming…
An induced matching $M$ in a graph $G$ is dominating if every edge not in $M$ shares exactly one vertex with an edge in $M$. The dominating induced matching problem (also known as efficient edge domination) asks whether a graph $G$ contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solvable in polynomial time for claw-free graphs. In the present paper, we study this problem for graphs containing no long claw, i.e. no induced subgraph obtained from the claw by subdividing each …
In this paper, we study the parallel and the space complexity of the graph isomorphism problem (\GI{}) for several parameterizations. Let $\mathcal{H}=\{H_1,H_2,\cdots,H_l\}$ be a finite set of graphs where $|V(H_i)|\leq d$ for all $i$ and for some constant $d$. Let $\mathcal{G}$ be an $\mathcal{H}$-free graph class i.e., none of the graphs $G\in \mathcal{G}$ contain any $H \in \mathcal{H}$ as an induced subgraph. We show that \GI{} parameterized by vertex deletion distance to $\mathcal{G}$ is in a parameterized version of $\AC^1$, denoted $\PL$-$\AC^1$, provided the colored graph isomorphism …
