In the $k$-Center problem, we are given a graph $G=(V,E)$ with positive edge weights and an integer $k$ and the goal is to select $k$ center vertices $C \subseteq V$ such that the maximum distance from any vertex to the closest center vertex is minimized. On general graphs, the problem is NP-hard and cannot be approximated within a factor less than $2$.
Typical applications of the $k$-Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Such networks are often characterized as graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that $k$-Center is W[1]-hard on planar graphs of constant doubling dimension when parameterized by the number of centers $k$, the highway dimension $hd$ and the pathwidth $pw$. We extend their result and show that even if we additionally parameterize by the skeleton dimension $κ$, the $k$-Center problem remains W[1]-hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for $k$-Center that has runtime $f(k,hd,pw,κ) \cdot \vert V \vert^{o(pw + κ+ \sqrt{k+hd})}$ for any computable function $f$.
Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.
The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parame…
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing $W[1]$-hardness proofs for these problems, since XNLP-hardness implies $W[t]$-hardness for all $t$. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle …
We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with $k$ edges are $\# W[1]$-hard when parameterized by $k$. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for $k$-forests implies $\# W[1]$-hardness of the problem of counting bases of a matroid when parameterized by rank or nullity, even if the matroid is restricted to be representable over a field of characteristic $2$. We c…
In this paper we study the hardness of the $k$-Center problem on inputs that model transportation networks. For the problem, a graph $G=(V,E)$ with edge lengths and an integer $k$ are given and a center set $C\subseteq V$ needs to be chosen such that $|C|\leq k$. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks. Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs.…
The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph $G$ from some class $K$ of "pattern graphs" can be embedded into a given graph $H$ (that is, is isomorphic to a subgraph of $H$) is fixed-parameter tractable if $K$ is a class of graphs of bounded tree width and $W[1]$-complete otherwise. Towards this conjecture, we prove that the embedding problem is $W[1]$-complete if $K$ is the class of all grids or the class of all walls.
We study the complexity of local search for the Boolean constraint satisfaction problem (CSP), in the following form: given a CSP instance, that is, a collection of constraints, and a solution to it, the question is whether there is a better (lighter, i.e., having strictly less Hamming weight) solution within a given distance from the initial solution. We classify the complexity, both classical and parameterized, of such problems by a Schaefer-style dichotomy result, that is, with a restricted set of allowed types of constraints. Our results show that there is a considerable amount of such pro…
The central conjecture of parameterized complexity states that FPT is not equal to W[1], and is generally regarded as the parameterized counterpart to P != NP. We revisit the issue of the plausibility of FPT != W[1], focusing on two aspects: the difficulty of proving the conjecture (assuming it holds), and how the relation between the two classes might differ from the one between P and NP. Regarding the first aspect, we give new evidence that separating FPT from W[1] would be considerably harder than doing the same for P and NP. Our main result regarding the relation between FPT and W[1] state…
We provide new (parameterized) computational hardness results for Interval Scheduling on Unrelated Machines. It is a classical scheduling problem motivated from just-in-time or lean manufacturing, where the goal is to complete jobs exactly at their deadline. We are given $n$ jobs and $m$ machines. Each job has a deadline, a weight, and a processing time that may be different on each machine. The goal is find a schedule that maximized the total weight of jobs completed exactly at their deadline. Note that this uniquely defines a processing time interval for each job on each machine. Interval Sc…
We investigate the parameterized complexity of several problems formalizing cluster identification in graphs. In other words we ask whether a graph contains a large enough and sufficiently connected subgraph. We study here three relaxations of CLIQUE: $s$-CLUB and $s$-CLIQUE, in which the relaxation is focused on the distances in respectively the cluster and the original graph, and $γ$-COMPLETE SUBGRAPH in which the relaxation is made on the minimal degree in the cluster. As these three problems are known to be NP-hard, we study here their parameterized complexities. We prove that $s$-CLUB and…
The Set Packing problem is, given a collection of sets $\mathcal{S}$ over a ground set $\mathcal{U}$, to find a maximum collection of sets that are pairwise disjoint. The problem is among the most fundamental NP-hard optimization problems that have been studied extensively in various computational regimes. The focus of this work is on parameterized complexity, Parameterized Set Packing (PSP): Given $r \in {\mathbb N}$, is there a collection $ \mathcal{S}' \subseteq \mathcal{S}: |\mathcal{S}'| = r$ such that the sets in $\mathcal{S}'$ are pairwise disjoint? Unfortunately, the problem is not fix…
During recent years the field of fine-grained complexity has bloomed to produce a plethora of results, with both applied and theoretical impact on the computer science community. The cornerstone of the framework is the notion of fine-grained reductions, which correlate the exact complexities of problems such that improvements in their running times or hardness results are carried over. We provide a parameterized viewpoint of these reductions (PFGR) in order to further analyze the structure of improvable problems and set the foundations of a unified methodology for extending algorithmic results…
We show that some natural problems that are XNLP-hard (which implies W[t]-hardness for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds (which is strongly NP-complete, as shown by Itai), Minimum Maximum Outdegree (for which W[1]-hardness for treewidth was proven by Szeider), and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set (for w…
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 $F(k) \cdot k$ whenever the graph $G$ has a dominating set of size $k$. When such an algorithm runs in time $T(k) \cdot poly(n)$ (i.e., FPT-time) for some computable function $T$, it is said to be an $F(k)$-FPT-approximation algorithm for $k$-DomSet. We prove the following for every computable functions $T, F$ and every constant $\varepsilon > 0$: $\bullet$ Assuming $W[1]\…
The generalized coloring numbers of Kierstead and Yang (Order 2003) offer an algorithmically-useful characterization of graph classes with bounded expansion. In this work, we consider the hardness and approximability of these parameters. First, we complete the work of Grohe et al. (WG 2015) by showing that computing the weak 2-coloring number is NP-hard. Our approach further establishes that determining if a graph has weak $r$-coloring number at most $k$ is para-NP-hard when parameterized by $k$ for all $r \geq 2$. We adapt this to determining if a graph has $r$-coloring number at most $k$ as …
We complement the recent algorithmic result that Feedback Vertex Set is XP-time solvable parameterized by the mim-width of a given branch decomposition of the input graph [3] by showing that the problem is W[1]-hard in this parameterization. The hardness holds even for linear mim-width, as well as for H-graphs, where the parameter is the number of edges in H. To obtain this result, we adapt a reduction due to Fomin, Golovach and Raymond [2], following the same line of reasoning but adding a new gadget.
The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Nu…
We study the 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph $G=(V,E)$, an alphabet set $Σ$ and, for each $\{u, v\}\in E$, a constraint $C_{uv} \subseteq Σ\timesΣ$, the goal is to find an assignment $σ: V \to Σ$ that satisfies as many constraints as possible, where a constraint $C_{uv}$ is satisfied if $(σ(u),σ(v))\in C_{uv}$. While the approximability of 2-CSPs is quite well understood when $|Σ|$ is constant, many problems are still open when $|Σ|$ becomes super constant. One such problem is whether it is hard to approximate 2-CSPs to …
Given a graph property $Φ$, we consider the problem $\mathtt{EdgeSub}(Φ)$, where the input is a pair of a graph $G$ and a positive integer $k$, and the task is to decide whether $G$ contains a $k$-edge subgraph that satisfies $Φ$. Specifically, we study the parameterized complexity of $\mathtt{EdgeSub}(Φ)$ and of its counting problem $\#\mathtt{EdgeSub}(Φ)$ with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties $Φ$: the decision problem $\mathtt{EdgeSub}(Φ)$ always admits an FPT algorithm and the counting problem $\#\mathtt{EdgeSub}(Φ)$ al…
University of Primorska（波莫瑞大学）；Kyushu University（九州大学）；Brandenburg University of Technology（勃兰登堡工业大学）；IAM, University of Primorska（波莫瑞大学应用数学研究所）；Osaka Metropolitan University（大阪公立大学）
Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses
我们提出了一系列几乎解决的不可逼近性结果的三个基本问题。在我们的系列中的第一个是最大独立集问题的次指数时间不可近似性，这是在参数化复杂性领域研究的一个问题。第二个问题是有界度二部图的最大诱导匹配问题的逼近困难。在我们的系列中的最后一个是严格的硬度近似的k-超图定价问题，一个基本的问题，从算法博弈论的领域。特别地，假设指数时间假设，我们的两个主要结果是：-对于任何大于某个常数的r，任何求解最大独立集问题的r-近似算法必须至少运行2^{n^{1-ε}/r^{1+ε}}时间。这几乎与2^{n/r}的上限相匹配（Cygan等人，2008年）。它还改进了参数化复杂性域中的一些困难结果（例如，Escoffier等人，2012和Chitnis等人，2013）-对于任何大于某个常数的k，k-超图定价问题不存在多项式时间min（k^{1-ε}，n^{1/2-ε}）-近似算法，其中n是输入图中的顶点数。这几乎匹配min（O（k），\tilde O（\sqrt{n}））的上界（Balcan and Blum，2007 and an algorithm in this paper）。我们注意到一个有趣的事实，与多项式时间算法的n^{1/2-ε}困难性相反，k-超图定价问题允许在准多项式时间中对任何δ>0进行n^δ逼近。这使得这个问题在一个罕见的近似类，其中近似阈值可以显着提高，允许算法运行在准多项式时间。
In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $d\ge2$, an $O^*(d^{\textrm{tw}})$-time algorithm, where $\textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - $d$…
In the $(k,h)$-SetCover problem, we are given a collection $\mathcal{S}$ of sets over a universe $U$, and the goal is to distinguish between the case that $\mathcal{S}$ contains $k$ sets which cover $U$, from the case that at least $h$ sets in $\mathcal{S}$ are needed to cover $U$. Lin (ICALP'19) recently showed a gap creating reduction from the $(k,k+1)$-SetCover problem on universe of size $O_k(\log |\mathcal{S}|)$ to the $\left(k,\sqrt[k]{\frac{\log|\mathcal{S}|}{\log\log |\mathcal{S}|}}\cdot k\right)$-SetCover problem on universe of size $|\mathcal{S}|$. In this paper, we prove a more scal…
A set $D \subseteq V$ of a graph $G = (V,E)$ is called an outer-connected dominating set of $G$ if every vertex $v$ not in $D$ is adjacent to at least one vertex in $D$, and the induced subgraph of $G$ on $V \setminus D$ is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality for the input graph $G$. Given a positive integer $k$ and a graph $G = (V, E)$, the Outer-connected Domination Decision problem is to decide whether $G$ has an outer-connected dominating set of cardinality at most $k$. The Outer-connected Domination …
We consider the problem of protecting and manipulating elections by recounting and changing ballots, respectively. Our setting involves a plurality-based election held across multiple districts, and the problem formulations are based on the model proposed recently by~[Elkind et al, IJCAI 2019]. It turns out that both of the manipulation and protection problems are NP-complete even in fairly simple settings. We study these problems from a parameterized perspective with the goal of establishing a more detailed complexity landscape. The parameters we consider include the number of voters, and the…
In this paper, we show how one may (efficiently) construct two types of extremal combinatorial objects whose existence was previously conjectural. (*) Panchromatic Graphs: For fixed integer k, a k-panchromatic graph is, roughly speaking, a balanced bipartite graph with one partition class equipartitioned into k colour classes in which the common neighbourhoods of panchromatic k-sets of vertices are much larger than those of k-sets that repeat a colour. The question of their existence was raised by Karthik and Manurangsi [Combinatorica 2020]. (*) Threshold Graphs: For fixed integer k, a k-thres…
A set S of vertices of a graph is a defensive alliance if, for each element of S, the majority of its neighbors is in S. The problem of finding a defensive alliance of minimum size in a given graph is NP-hard and there are polynomial-time algorithms if certain parameters are bounded by a fixed constant. In particular, fixed-parameter tractability results have been obtained for some structural parameters such as the vertex cover number. However, for the parameter treewidth, the question of whether the problem is FPT has remained open. This is unfortunate because treewidth is perhaps the most pr…
In the Binary Networked Public Goods game, every player needs to decide if she participates in a public project whose utility is shared equally by the community. We study the problem of deciding if there exists a pure strategy Nash equilibrium (PSNE) in such games. The problem is already known to be NP-complete. We provide fine-grained analysis of this problem under the lens of parameterized complexity theory. We consider various natural graph parameters and show either W[1]-hardness or exhibit an FPT algorithm. We finally exhibit some special graph classes, for example path, cycle, bi-clique,…
Given an $n*n$ sparse symmetric matrix with $m$ nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values. To maintain the matrix sparse, we would like to minimize the number $k$ of these changes, hence called the minimum fill-in problem. Agrawal et al.~[FOCS'90] developed the first approximation algorithm, based on early heuristics by George [SIAM J Numer Anal 10] and by Lipton et al.~[SIAM J Numer Anal 16]. The objective function they used is $m+k$, the number of nonzero elements after elimination. An approximation algorithm using $k$ as the objective function…
The centrality of a vertex v in a network intuitively captures how important v is for communication in the network. The task of improving the centrality of a vertex has many applications, as a higher centrality often implies a larger impact on the network or less transportation or administration cost. In this work we study the parameterized complexity of the NP-complete problems Closeness Improvement and Betweenness Improvement in which we ask to improve a given vertex' closeness or betweenness centrality by a given amount through adding a given number of edges to the network. Herein, the clos…
We consider the problem of controlling the spread of harmful items in networks, such as the contagion proliferation of diseases or the diffusion of fake news. We assume the linear threshold model of diffusion where each node has a threshold that measures the node resistance to the contagion. We study the parameterized complexity of the problem: Given a network, a set of initially contaminated nodes, and two integers $k$ and $\ell$, is it possible to limit the diffusion to at most $k$ other nodes of the network by immunizing at most $\ell$ nodes? We consider several parameters associated to the…
