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Register a new userParameterized Complexity of Deletion to Scattered Graph Classes
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Ashwin Jacob
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Graph-modification problems, where we add/delete a small number of vertices/edges to make the given graph to belong to a simpler graph class, is a well-studied optimization problem in all algorithmic paradigms including classical, approximation and parameterized complexity. Specifically, graph-deletion problems, where one needs to delete at most $k$ vertices to place it in a given non-trivial hereditary (closed under induced subgraphs) graph class, captures several well-studied problems including {sc Vertex Cover}, {sc Feedback Vertex Set}, {sc Odd Cycle Transveral}, {sc Cluster Vertex Deletion}, and {sc Perfect Deletion}. Investigation into these problems in parameterized complexity has given rise to powerful tools and techniques. While a precise characterization of the graph classes for which the problem is {it fixed-parameter tractable} (FPT) is elusive, it has long been known that if the graph class is characterized by a {it finite} set of forbidden graphs, then the problem is FPT. In this paper, we initiate a study of a natural variation of the problem of deletion to {it scattered graph classes} where we need to delete at most $k$ vertices so that in the resulting graph, each connected component belongs to one of a constant number of graph classes. A simple hitting set based approach is no longer feasible even if each of the graph classes is characterized by finite forbidden sets. As our main result, we show that this problem is fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the finite classes is known to be FPT and the properties that a graph belongs to each of the classes is expressible in CMSO logic. When each graph class has a finite forbidden set, we give a faster FPT algorithm using the well-known techniques of iterative compression and important separators.
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{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 arc, then this problem is precisely DFVS. Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if ${cal H}$ only contains rooted graphs or if ${cal H}$ contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of G{o}ke et al. [CIAC 2019] for the {sc 1-Out-Regular Vertex Deletion} and {sc Bounded Size Strong Component Vertex Deletion} problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for {sc DFVS}, without using the heavy machinery of shadow removal as is done by G{o}ke et al. [CIAC 2019].
We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph $G$, a non-trivial hereditary property $Pi$ and an integer parameter $k$, the general problem $P(G,Pi,k)$ asks whether there exists $k$ vertices of $G$ that induce a subgraph satisfying property $Pi$. This problem, $P(G,Pi,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[1]-complete by Khot and Raman, if $Pi$ includes all trivial graphs but not all complete graphs and vice versa; and is fixed-parameter tractable (FPT), otherwise. As the problem is W[1]-complete on general graphs when $Pi$ includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes. Motivated by this line of research, we study the problem on graphs which also belong to a hereditary graph class and establish a framework which settles the parameterized complexity of the problem for various hereditary graph classes. In particular, we show that: $P(G,Pi,k)$ is solvable in polynomial time when the graph $G$ is co-bipartite and $Pi$ is the property of being planar, bipartite or triangle-free (or vice-versa). $P(G,Pi,k)$ is FPT when the graph $G$ is planar, bipartite or triangle-free and $Pi$ is the property of being planar, bipartite or triangle-free, or graph $G$ is co-bipartite and $Pi$ is the property of being co-bipartite. $P(G,Pi,k)$ is W[1]-complete when the graph $G$ is $C_4$-free, $K_{1,4}$-free or a unit disk graph and $Pi$ is the property of being either planar or bipartite.
We consider the problems of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the result of the modification satisfies some property definable in first-order logic. We establish a number of sufficient and necessary conditions on the quantification pattern of the first-order formula phi for the problem to be fixed-parameter tractable or to admit a polynomial kernel.
A cactus is a connected graph that does not contain $K_4 - e$ as a minor. Given a graph $G = (V, E)$ and integer $k ge 0$, Cactus Vertex Deletion (also known as Diamond Hitting Set) is the problem of deciding whether $G$ has a vertex set of size at most $k$ whose removal leaves a forest of cacti. The current best deterministic parameterized algorithm for this problem was due to Bonnet et al. [WG 2016], which runs in time $26^kn^{O(1)}$, where $n$ is the number of vertices of $G$. In this paper, we design a deterministic algorithm for Cactus Vertex Deletion, which runs in time $17.64^kn^{O(1)}$. As a straightforward application of our algorithm, we give a $17.64^kn^{O(1)}$-time algorithm for Even Cycle Transversal. The idea behind this improvement is to apply the measure and conquer analysis with a slightly elaborate measure of instances.
A graph is $d$-orientable if its edges can be oriented so that the maximum in-degree of the resulting digraph is at most $d$. $d$-orientability is a well-studied concept with close connections to fundamental graph-theoretic notions and applications as a load balancing problem. In this paper we consider the d-ORIENTABLE DELETION problem: given a graph $G=(V,E)$, delete the minimum number of vertices to make $G$ $d$-orientable. We contribute a number of results that improve the state of the art on this problem. Specifically: - We show that the problem is W[2]-hard and $log n$-inapproximable with respect to $k$, the number of deleted vertices. This closes the gap in the problems approximability. - We completely characterize the parameterized complexity of the problem on chordal graphs: it is FPT parameterized by $d+k$, but W-hard for each of the parameters $d,k$ separately. - We show that, under the SETH, for all $d,epsilon$, the problem does not admit a $(d+2-epsilon)^{tw}$, algorithm where $tw$ is the graphs treewidth, resolving as a special case an open problem on the complexity of PSEUDOFOREST DELETION. - We show that the problem is W-hard parameterized by the input graphs clique-width. Complementing this, we provide an algorithm running in time $d^{O(dcdot cw)}$, showing that the problem is FPT by $d+cw$, and improving the previously best known algorithm for this case.
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