Exponential Time Hypothesis Research Articles

Results on a Super Strong Exponential Time Hypothesis

All known SAT-solving paradigms (backtracking, local search, and the polynomial method) only yield a 2n(1−1/O(k)) time algorithm for solving k-SAT in the worst case, where the big-O constant is independent of k. For this reason, it has been hypothesized that k-SAT cannot be solved in worst-case 2n(1−f(k)/k) time, for any unbounded ƒ : ℕ → ℕ. This hypothesis has been called the “Super-Strong Exponential Time Hypothesis” (Super Strong ETH), modeled after the ETH and the Strong ETH. We prove two results concerning the Super-Strong ETH:1. It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the “critical threshold”, where the clause-to-variable ratio is 2k ln 2 −Θ(1). We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. In particular, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in 2n(1−Ω( log k)/k) time, with high probability (over the choice of the formula and the randomness of the algorithm). It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1998).2. The Unique k-SAT problem is the special case where there is at most one satisfying assignment. It is natural to hypothesize that the worst-case (exponential-time) complexity of Unique k-SAT is substantially less than that of k-SAT. Improving prior reductions, we show the time complexities of Unique k-SAT and k-SAT are very tightly related: if Unique k-SAT is in 2n(1−f(k)/k) time for an unbounded f, then k-SAT is in 2n(1−f(k)(1−ɛ)/k) time for every ɛ > 0. Thus, refuting Super Strong ETH in the unique solution case would refute Super Strong ETH in general.

Manipulating Districts to Win Elections: Fine-Grained Complexity

Gerrymandering is a practice of manipulating district boundaries and locations in order to achieve a political advantage for a particular party. Lewenberg, Lev, and Rosenschein [AAMAS 2017] initiated the algorithmic study of a geographically-based manipulation problem, where voters must vote at the ballot box closest to them. In this variant of gerrymandering, for a given set of possible locations of ballot boxes and known political preferences of n voters, the task is to identify locations for k boxes out of m possible locations to guarantee victory of a certain party in at least ℓ districts. Here integers k and ℓ are some selected parameter.It is known that the problem is NP-complete already for 4 political parties and prior to our work only heuristic algorithms for this problem were developed. We initiate the rigorous study of the gerrymandering problem from the perspectives of parameterized and fine-grained complexity and provide asymptotically matching lower and upper bounds on its computational complexity. We prove that the problem is W[1]-hard parameterized by k + n and that it does not admit an f(n,k) · mo(√k) algorithm for any function f of k and n only, unless the Exponential Time Hypothesis (ETH) fails. Our lower bounds hold already for 2 parties. On the other hand, we give an algorithm that solves the problem for a constant number of parties in time (m+n)O(√k).

Balancing Spreads of Influence in a Social Network

The personalization of our news consumption on social media has a tendency to reinforce our pre-existing beliefs instead of balancing our opinions. To tackle this issue, Garimella et al. (NIPS'17) modeled the spread of these viewpoints, also called campaigns, using the independent cascade model introduced by Kempe, Kleinberg and Tardos (KDD'03) and studied an optimization problem that aims to balance information exposure when two opposing campaigns propagate in a network. This paper investigates a natural generalization of this optimization problem in which μ different campaigns propagate in the network and we aim to maximize the expected number of nodes that are reached by at least ν or none of the campaigns, where μ ≥ ν ≥ 2. Following Garimella et al., despite this general setting, we also investigate a simplified one, in which campaigns propagate in a correlated manner. While for the simplified setting, we show that the problem can be approximated within a constant factor for any constant μ and ν, for the general setting, we give reductions leading to several approximation hardness results when ν ≥ 3. For instance, assuming the gap exponential time hypothesis to hold, we obtain that the problem cannot be approximated within a factor of n−g(n) for any g(n) = o(1) where n is the number of nodes in the network. We complement our hardness results with an Ω(n−1/2)-approximation algorithm for the general setting when ν = 3 and μ is arbitrary.

Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems

In this work, we focus on several completion problems for subclasses of chordal graphs: M INIMUM F ILL -I N , I NTERVAL C OMPLETION , P ROPER I NTERVAL C OMPLETION , T RIVIALLY P ERFECT C OMPLETION , and T HRESHOLD C OMPLETION . In these problems, the task is to add at most k edges to a given graph to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related C HAIN C OMPLETION problem: • Assuming the Exponential Time Hypothesis, none of these problems can be solved in time 2 O ( n 1/2 /log c n ) or 2 O ( k 1/4 /log c k )· n O (1) , for some integer c . • Assuming the non-existence of a subexponential-time approximation scheme for M IN B ISECTION on d -regular graphs, for some constant d , none of these problems can be solved in time 2 o ( n ) or 2 o √k) }· n O (1) . For all the aforementioned completion problems, apart from P ROPER I NTERVAL C OMPLETION , FPT algorithms with running time of the form 2 O (√ k log k ) · n O (1) are known. Thus, the second result proves that a significant improvement of any of these algorithms would lead to a surprising breakthrough in the design of approximation algorithms for M IN B ISECTION . To prove our results, we use a reduction methodology based on combining the classic approach of starting with a sparse instance of 3-S AT , prepared using the Sparsification Lemma, with the existence of almost linear-size Probabilistically Checkable Proofs. Apart from our main results, we also obtain lower bounds excluding the existence of subexponential algorithms for the O PTIMUM L INEAR A RRANGEMENT problem, as well as improved, yet still not tight, lower bounds for F EEDBACK A RC S ET IN T OURNAMENTS .

Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth

Many combinatorial problems can be solved in time $\mathcal{O}^*(c^{\mathrm{tw}})$ on graphs of treewidth $\mathrm{tw}$, for a problem-specific constant $c$. In several cases, matching upper and lower bounds on $c$ are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, $\rm{I{\small NDEPENDENT}~S{\small ET}}$ cannot be solved in $O^*((2-\varepsilon)^{\mathrm{ctw}})$ time, and $\rm{D{\small OMINATING}~S{\small ET}}$ cannot be solved in $O^*((3-\varepsilon)^{\mathrm{ctw}})$ time. By designing a new crossover gadget, we extend these lower bounds even to planar graphs of bounded cutwidth or treewidth. Hence planarity does not help when solving $\rm{I{\small NDEPENDENT}~S{\small ET}}$ or $\rm{D{\small OMINATING}~S{\small ET}}$ on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.

Graph Motif Problems Parameterized by Dual

Let $G=(V,E)$ be a vertex-colored graph, where $C$ is the set of colors used to color $V$. The ${\rm G{\small RAPH}~M{\small OTIF}}$ (or $\rm GM$) problem takes as input $G$, a multiset $M$ of colors built from $C$, and asks whether there is a subset $S\subseteq V$ such that (i) $G[S]$ is connected and (ii) the multiset of colors obtained from $S$ equals $M$. The ${\rm C{\small OLORFUL}~G{\small RAPH}~M{\small OTIF}}$ (or ${\rm CGM}$) problem is the special case of $\rm GM$ in which $M$ is a set, and the ${\rm L{\small IST-COLORED}~G{\small RAPH}~M{\small OTIF}}$ (or $\rm LGM$) problem is the extension of $\rm GM$ in which each vertex $v$ of $V$ may choose its color from a list $\mathcal L(v)\subseteq C$ of colors. We study the three problems $\rm GM$, $\rm CGM$, and $\rm LGM$, parameterized by the dual parameter $\ell:=|V|-|M|$. For general graphs, we show that, assuming the strong exponential time hypothesis, $\rm CGM$ has no $(2-\epsilon)^\ell\cdot |V|^{\mathcal O(1)}$-time algorithm, which implies that a previous algorithm, running in $\mathcal O(2^\ell\cdot |E|)$ time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that $\rm LGM$ is W[1]-hard with respect to $\ell$ even if we restrict ourselves to lists of at most two colors. Finally, we show that if the input graph is a tree, then $\rm GM$ can be solved in $\mathcal O(3^\ell\cdot |V|)$ time but admits no polynomial-size problem kernel, while $\rm CGM$ can be solved in $\mathcal O(\sqrt{2}^{\ell} + |V|)$ time and admits a polynomial-size problem kernel.

Counting Induced Subgraphs: A Topological Approach to #W[1]-hardness

We investigate the problem #{{mathsf {IndSub}}}(varPhi ) of counting all induced subgraphs of size k in a graph G that satisfy a given property varPhi .This continues the work of Jerrum and Meeks who proved the problem to be #{{mathrm {W[1]}}}-hard for some families of properties which include (dis)connectedness [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Using the recent framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], we discover that for monotone properties varPhi , the problem #{{mathsf {IndSub}}}(varPhi ) is hard for #{{mathrm {W[1]}}} if the reduced Euler characteristic of the associated simplicial (graph) complex of varPhi is non-zero. This observation links #{{mathsf {IndSub}}}(varPhi ) to Karp’s famous Evasiveness Conjecture, as every graph complex with non-vanishing reduced Euler characteristic is known to be evasive. Applying tools from the “topological approach to evasiveness” which was introduced in the seminal paper of Khan, Saks and Sturtevant [FOCS 83], we prove that #{{mathsf {IndSub}}}(varPhi ) is #{{mathrm {W[1]}}}-hard for every monotone property varPhi that does not hold on the Hamilton cycle as well as for some monotone properties that hold on the Hamilton cycle such as being triangle-free or not k-edge-connected for k > 2. Moreover, we show that for those properties #{{mathsf {IndSub}}}(varPhi ) can not be solved in time f(k)cdot n^{o(k)} for any computable function f unless the Exponential Time Hypothesis (ETH) fails. In the final part of the paper, we investigate non-monotone properties and prove that #{{mathsf {IndSub}}}(varPhi ) is #{{mathrm {W[1]}}}-hard if varPhi is any non-trivial modularity constraint on the number of edges with respect to some prime q or if varPhi enforces the presence of a fixed isolated subgraph.

The directed 2-linkage problem with length constraints

The weak 2-linkage problem for digraphs asks for a given digraph and vertices s1,s2,t1,t2 whether D contains a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs [8]. Recently it was shown [3] that if D is equipped with a weight function w on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in D. In this paper we consider the unit weight case and prove that for every pair of constants k1,k2, there is a polynomial algorithm which decides whether the input digraph D has a pair of arc-disjoint paths P1,P2 such that Pi is an (si,ti)-path of length no more than d(si,ti)+ki, for i=1,2, where d(si,ti) denotes the length of the shortest (si,ti)-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution P1,P2 to the weak 2-linkage problem where each path Pi has length at most d(si,ti)+clog1+ϵ⁡n for some constant c.

Minimum fill-in: Inapproximability and almost tight lower bounds

Given an n×n sparse symmetric matrix with m nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values, so called fill-ins. The minimum fill-in problem asks whether it is possible to perform the elimination with at most k fill-ins. We exclude the existence of polynomial time approximation schemes for this problem, assuming P≠NP, and the existence of 2O(n1−δ)-time approximation schemes for any positive δ, assuming the Exponential Time Hypothesis. We also give a 2O(k1/2−δ)⋅nO(1) parameterized lower bound. All these results come as corollaries of a new reduction from vertex cover to the minimum fill-in problem, which might be of its own interest: All previous reductions for similar problems start from some kind of graph layout problems, and hence have limited use in understanding their fine-grained complexity.

Tight Hardness Results for Consensus Problems on Circular Strings and Time Series

Consensus problems for strings and sequences appear in numerous application contexts, ranging among bioinformatics, data mining, and machine learning. Closing some gaps in the literature, we show that several fundamental problems in this context are NP-and W[1]-hard, and that the known (including some brute-force) algorithms are close to optimality assuming the Exponential Time Hypothesis. Among our main contributions is to settle the complexity status of computing a mean in dynamic time warping spaces which, as pointed out by Brill et al. [DMKD 2019], suffered from many unproven or false assumptions in the literature. We prove this problem to be NP-hard and additionally show that a recent dynamic programming algorithm is essentially optimal. In this context, we study a broad family of circular string alignment problems. This family also serves as a key for our hardness reductions, and it is of independent (practical) interest in molecular biology. In particular, we show tight hardness and running time lower bounds for Circular Consensus String; notably, the corresponding non-circular version is easily linear-time solvable.

An Optimal XP Algorithm for Hamiltonian Cycle on Graphs of Bounded Clique-Width

In this paper, we prove that, given a clique-width k-expression of an n-vertex graph, Hamiltonian Cycle can be solved in time $$n^{\mathcal {O}(k)}$$. This improves the naive algorithm that runs in time $$n^{\mathcal {O}(k^2)}$$ by Espelage et al. (Graph-theoretic concepts in computer science, vol 2204. Springer, Berlin, 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $$n^{o(k)}$$ (Fomin et al. in SIAM J Comput 43:1541–1563, 2014). We present a technique of representative sets using two-edge colored multigraphs on k vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on k vertices with at most n edges.

On the parameterized complexity of [1,j]-domination problems

For a graph G, a set D⊆V(G) is called a [1,j]-dominating set if every vertex in V(G)∖D has at least one and at most j neighbors in D. A set D⊆V(G) is called a [1,j]-total dominating set if every vertex in V(G) has at least one and at most j neighbors in D. In the [1,j]-(Total) Dominating Set problem we are given a graph G and a positive integer k. The objective is to test whether there exists a [1,j]-(total) dominating set of size at most k. The [1,j]-Dominating Set problem is known to be NP-complete, even for restricted classes of graphs such as chordal and planar graphs, but polynomial-time solvable on split graphs. The [1,2]-Total Dominating Set problem is known to be NP-complete, even for bipartite graphs. As both problems generalize the Dominating Set problem, both are W[1]-hard when parameterized by solution size. In this work, we study the aforementioned problems on various graph classes from the perspective of parameterized complexity and prove the following results:•[1,j]-Dominating Set parameterized by solution size is W[1]-hard on d-degenerate graphs for d=j+1.•[1,j]-Dominating Set parameterized by solution size is FPT on nowhere dense graphs.•The known algorithm for [1,j]-Dominating Set on split graphs is optimal under the Strong Exponential Time Hypothesis (SETH).•Assuming SETH, we provide a lower bound for the running time of any algorithm solving the [1,2]-Total Dominating Set problem parameterized by pathwidth.•Finally, we study another variant of Dominating Set, called Restrained Dominating Set, that asks if there is a dominating set D of G of size at most k such that no vertex outside of D has all of its neighbors in D. We prove this variant is W[1]-hard even on 3-degenerate graphs.

On approximability of minimum color-spanning ball in high dimensions

This paper presents a lower bound on the running time of any approximation scheme for Minimum Color-Spanning Ball (MCSB) problem in high dimensional spaces. This bound is based on the Exponential Time Hypothesis (ETH).

Temporal vertex cover with a sliding time window

Modern, inherently dynamic systems are usually characterized by a network structure which is subject to discrete changes over time. Given a static underlying graph, a temporal graph can be represented via an assignment of a set of integer time-labels to every edge, indicating the discrete time steps when this edge is active. While most of the recent theoretical research on temporal graphs focused on temporal paths and other “path-related” temporal notions, only few attempts have been made to investigate “non-path” temporal problems. In this paper we introduce and study two natural temporal extensions of the classical problem VERTEX COVER. We present a thorough investigation of the computational complexity and approximability of these two temporal covering problems. We provide strong hardness results, complemented by approximation and exact algorithms. Some of our algorithms are polynomial-time, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH) and other plausible complexity assumptions.

On the Parameterized Complexity of Approximating Dominating Set

We study the parameterized complexity of approximating the k -Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F ( k ) ⋅ k whenever the graph G has a dominating set of size k . When such an algorithm runs in time T ( k ) ⋅ poly ( n ) (i.e., FPT-time) for some computable function T , it is said to be an F ( k )- FPT-approximation algorithm for k -DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] ≠ FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T , F and every constant ε > 0: • Assuming W[1] ≠ FPT, there is no F ( k )- FPT-approximation algorithm for k -DomSet. • Assuming the Exponential Time Hypothesis (ETH), there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n o ( k ) time. • Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n k − ε time. • Assuming the k -SUM Hypothesis, for every integer k ≥ 3, there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n ⌈ k /2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] ≠ FPT and (log 1/4 &minus ε k )-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F ( k )-FPT-approximation algorithm for any function F was shown under Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n &delta k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well-studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

Computing the chromatic number using graph decompositions via matrix rank

Computing the smallest number q such that the vertices of a given graph can be properly q-colored, known as the chromatic number, is one of the oldest and most fundamental problems in combinatorial optimization. The q-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. For example, algorithms are known to solve the problem on graphs of treewidth tw in time O⁎(qtw), while a running time of O⁎((q−ε)tw) is impossible assuming the Strong Exponential Time Hypothesis (SETH). While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying q-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators.We present two algorithms for q-Coloring parameterized by cutwidth ctw: a deterministic one that runs in time O⁎(2ω⋅ctw), where ω is the square matrix multiplication exponent, and a randomized one with runtime O⁎(2ctw). In sharp contrast to earlier work, the running time is independent of q. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in O⁎((2−ε)ctw) time assuming SETH. Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an O⁎((⌊d/2⌋+1)pw) time randomized algorithm for q-Coloring on graphs of pathwidth pw and maximum degree d. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no O⁎((⌊d/2⌋+1−ε)pw)-time algorithm exists assuming SETH.

Sliding Window Temporal Graph Coloring

Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static graphs, which often stand in stark contrast to practice where data is inherently dynamic and subject to discrete changes over time. A temporal graph is a graph whose edges are assigned a set of integer time labels, indicating at which discrete time steps the edge is active. In this paper we present a natural temporal extension of the classical graph coloring problem. Given a temporal graph and a natural number ∆, we ask for a coloring sequence for each vertex such that (i) in every sliding time window of ∆ consecutive time steps, in which an edge is active, this edge is properly colored (i.e. its endpoints are assigned two different colors) at least once during that time window, and (ii) the total number of different colors is minimized. This sliding window temporal coloring problem abstractly captures many realistic graph coloring scenarios in which the underlying network changes over time, such as dynamically assigning communication channels to moving agents. We present a thorough investigation of the computational complexity of this temporal coloring problem. More specifically, we prove strong computational hardness results, complemented by efficient exact and approximation algorithms. Some of our algorithms are linear-time fixed-parameter tractable with respect to appropriate parameters, while others are asymptotically almost optimal under the Exponential Time Hypothesis (ETH).

Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics. When asking for projected counts we are interested in counting the number of extensions of a given argumentation framework while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming (DP). Our algorithms run in time double or triple exponential in the treewidth depending on the considered semantics. Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension.

Optimal Surveillance of Covert Networks by Minimizing Inverse Geodesic Length

The inverse geodesic length (IGL) is a well-known and widely used measure of network performance. It equals the sum of the inverse distances of all pairs of vertices. In network analysis, IGL of a network is often used to assess and evaluate how well heuristics perform in strengthening or weakening a network. We consider the edge-deletion problem MINIGLED. Formally, given a graph G, a budget k, and a target inverse geodesic length T, the question is whether there exists a subset of edges X with |X| ≤ ck, such that the inverse geodesic length of G − X is at most T.In this paper, we design algorithms and study the complexity of MINIGL-ED. We show that it is NP-complete and cannot be solved in subexponential time even when restricted to bipartite or split graphs assuming the Exponential Time Hypothesis. In terms of parameterized complexity, we consider the problem with respect to various parameters. We show that MINIGL-ED is fixed-parameter tractable for parameter T and vertex cover by modeling the problem as an integer quadratic program. We also provide FPT algorithms parameterized by twin cover and neighborhood diversity combined with the deletion budget k. On the negative side we show that MINIGL-ED is W[1]-hard for parameter tree-width.

Parameterized complexity of fair deletion problems

Edge deletion problems are those where the goal is to find a subset of edges such that after its removal the graph satisfies the given graph property. Typically, we want to minimize the number of elements removed. In fair deletion problems, the objective is changed, so the maximum number of deletions in a neighborhood of a single vertex is minimized.We study the parameterized complexity of fair deletion problems concerning the structural parameters such as the tree-width, the path-width, the tree-depth, the size of minimum feedback vertex set, the neighborhood diversity, and the size of minimum vertex cover of graph G.We prove the W[1]-hardness of the fair FO vertex-deletion problem with respect to the combined size of the tree-depth and the minimum feedback vertex set number. Moreover, we show that there is no algorithm for fair FO vertex-deletion problem running in time no(k3), where n is the size of the graph and k is the sum of the mentioned parameters, provided that the Exponential Time Hypothesis holds.On the other hand, we present an FPT algorithm for the fair MSO edge-deletion problem parameterized by the size of minimum vertex cover and an FPT algorithm for the fair MSO vertex-deletion problem parameterized by the neighborhood diversity.