# A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

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## Abstract

**:**

## 1. Introduction

- Accuracy: the algorithm should always compute the best possible (optimum) solution.
- Efficiency: the runtime of the algorithm should be polynomial in the input size n.

#### Organization of the Survey

## 2. Preliminaries

**Parameterized approximation algorithms.**We briefly specify the different types of algorithms we will consider. As already defined in the introduction, an FPT algorithm computes the optimum solution in $f(k){n}^{O(1)}$ time for some parameter k and computable function $f:\mathbb{N}\to \mathbb{N}$ on inputs of size n. The common choices of parameters are the standard parameters based on solution size, structural parameters, guarantee parameters, and dual parameters.

**Kernelization**. A further topic closely related to the FPT algorithms is kernelization. Here, the idea is that an instance is efficiently pre-processed by removing the “easy parts” so that only the NP-hard core of the instance remains. More concretely, a kernelization algorithm takes an instance I and a parameter k of some problem and computes a new instance ${I}^{\prime}$ with parameter ${k}^{\prime}$ of the same problem. The runtime of this algorithm is polynomial in the size of the input instance I and k, while the size of the output ${I}^{\prime}$ and ${k}^{\prime}$ is bounded as a function of the input parameter k. For optimization problems, it should also be the case that any optimum solution to ${I}^{\prime}$ can be converted to an optimum solution of I in polynomial time. The new instance ${I}^{\prime}$ is called the kernel of I (for parameter k). A fundamental result in fixed-parameter tractability is that an (optimization) problem parameterized by k is FPT if and only if it admits a kernelization algorithm for the same parameter [17]. However the size of the guaranteed kernel will in general be exponential (or worse) in the input parameter. Therefore, an interesting question is whether an NP-hard problem admits small kernels of polynomial size. This can be interpreted as meaning that the problem has a very efficient pre-processing algorithm, which can be used prior to solving the kernel. It also gives an additional dimension to the parameterized complexity landscape.

**Complexity-Theoretic Hypotheses**. We assume that the readers have basic knowledge of (classic) parameterized complexity theory, including the W-hierarchy, the exponential time hypothesis (ETH), and the strong exponential time hypothesis (SETH). The reader may choose to recapitulate these definitions by referring to [6] (Sections 13 and 14).

## 3. FPT Hardness of Approximation

#### 3.1. W[1]-Hardness of Gap Problems

#### 3.1.1. Parameterized Intractability of Biclique and Applications to Parameterized Inapproximability

**Theorem**

**1**

**.**Given a bipartite graph $G(L\dot{\cup}R,E)$ and $k\in \mathbb{N}$ as input, it is W[1]-hard to distinguish between the following two cases:

- Completeness: There are k vertices in L with at least ${n}^{\Theta (\frac{1}{k})}$ common neighbors in R;
- Soundness: Any k vertices in L have at most $(k+6)!$ common neighbors in R.

- Property 1: Any $k+1$ distinct subsets in $\mathcal{T}$ have intersection size at most ℓ;
- Property 2: Any k distinct subsets in $\mathcal{T}$ have intersection size at least h.

**Open Question 1**(Lower bound of One-Sided k-Biclique under ETH and SETH) Can the running time lower bound on One-Sided k-Biclique be improved to ${n}^{\Omega (k)}$ under ETH? Can it be improved to ${n}^{k-o(1)}$ under SETH?

**Inapproximability of $\mathit{k}$-Dominating Set via Gadget Composition.**We shall discuss about the inapproximability of k-Dominating Set in detail in the next subsubsection. We would like to simply highlight here how the above framework was used by Chen and Lin [25] and Lin [26] to obtain inapproximability results for k-Dominating Set.

**Even Set.**A recent success story of Theorem 1 is its application to resolve a long standing open problem called k-Minimum Distance Problem (also referred to as k-Even Set), where we are given as input a generator matrix $\mathbf{A}\in {\mathbb{F}}_{2}^{n\times m}$ of a binary linear code and an integer k, and the goal is to determine whether the code has distance at most k. Recall that the distance of a linear code is $\underset{\overrightarrow{0}\ne \mathbf{x}\in {\mathbb{F}}_{2}^{m}}{min}\phantom{\rule{4pt}{0ex}}{\parallel \mathbf{A}\mathbf{x}\parallel}_{0}$ where ${\parallel \xb7\parallel}_{0}$ denote the 0-norm (aka the Hamming norm).

**Theorem**

**2**

**.**For any $\gamma \ge 1$, given input $(\mathbf{A},k)\in {\mathbb{F}}^{n\times m}\times \mathbb{N}$, it is W[1]-hard (under randomized reductions) to distinguish between

- Completeness: Distance of the code generated by $\mathbf{A}$ is at most k, and,
- Soundness: Distance of the code generated by $\mathbf{A}$ is more than $\gamma \xb7k$.

**Open Question 2.**Is it W[1]-hard to decide k-Minimum Distance Problem over ${\mathbb{F}}_{p}$ with $p>2$, when p is fixed and is not part on the input?

**Shortest Vector Problem.**Theorem 1 (or more precisely the constant inapproximability of k-Linear Dependent Set stated above) was also used to resolve the complexity of the parameterized k-Shortest Vector Problem in lattices, where the input (in the ${\ell}_{p}$ norm) is an integer $k\in \mathbb{N}$ and a matrix $\mathbf{A}\in {\mathbb{Z}}^{n\times m}$ representing the basis of a lattice, and we want to determine whether the shortest (non-zero) vector in the lattice has length at most k, i.e., whether $\underset{\overrightarrow{0}\ne \mathbf{x}\in {\mathbb{Z}}^{m}}{\mathrm{min}}\phantom{\rule{4pt}{0ex}}{\parallel \mathbf{A}\mathbf{x}\parallel}_{p}\le k$. Again, k is the parameter of the problem. It should also be noted here that (as in [32]), we require the basis of the lattice to be integer valued, which is sometimes not enforced in literature (e.g., [33,34]). This is because, if $\mathbf{A}$ is allowed to be any matrix in ${\mathbb{R}}^{n\times m}$, then parameterization is meaningless because we can simply scale $\mathbf{A}$ down by a large multiplicative factor.

**Theorem**

**3**

**.**For any $p>1$, there exists a constant ${\gamma}_{p}>1$ such that given input $(\mathbf{A},k)\in {\mathbb{Z}}^{n\times m}\times \mathbb{N}$, it is W[1]-hard (under randomized reductions) to distinguish between

- Completeness: The ${\ell}_{p}$ norm of the shortest vector of the lattice generated by $\mathbf{A}$ is $\le k$, and,
- Soundness: The ${\ell}_{p}$ norm of the shortest vector of the lattice generated by $\mathbf{A}$ is $>{\gamma}_{p}\xb7k$.

**Open Question 3**(Approximation of k-Shortest Vector Problem in ${\ell}_{1}$ norm]). Is k-Shortest Vector Problem in the ${\ell}_{1}$ norm in FPT?

#### 3.1.2. Parameterized Inapproximability of Dominating Set

**Theorem**

**4**

**.**Let $F:\mathbb{N}\to \mathbb{N}$ be any computable function. Given an instance $(G,k)$ of k-Dominating Set as input, it is $\mathrm{W}\left[1\right]$-hard to distinguish between the following two cases:

- Completeness: G has a dominating set of size k.
- Soundness: Every dominating set of G is of size at least $F(k)\xb7k$.

**From $\mathit{k}$-Multicolor Clique to $\mathit{k}$-Gap CSP.**Starting from an instance of k-Multicolor Clique, say G on vertex set $V:={V}_{1}\dot{\cup}{V}_{2}\dot{\cup}\dots \dot{\cup}{V}_{k}$, we write down a set of constraints $\mathcal{P}$ on a variable set $X:=\{{x}_{i,j}\mid i,j\in \left[k\right],i\ne j\}$ as follows. For every $i,j\in \left[k\right]$, such that $i\ne j$, define ${E}_{i,j}$ to be the set of all edges in G whose end points are in ${V}_{i}$ and ${V}_{j}$. An assignment to variable ${x}_{i,j}$ is an element of ${E}_{i,j}$, i.e., a pair of vertices, one from ${V}_{i}$ and the other from ${V}_{j}$. Suppose that ${x}_{i,j}$ was assigned the edge $\{{v}_{i},{v}_{j}\}$, where ${v}_{i}\in {V}_{i}$ and ${v}_{j}\in {V}_{j}$. Then we define the assignment of ${x}_{i,j}^{i}$ to be ${v}_{i}$ and the assignment of ${x}_{i,j}^{j}$ to be ${v}_{j}$. We define $\mathcal{P}:=\{{P}_{1},\dots ,{P}_{k}\}$, where the constraint ${P}_{i}$ is defined to be satisfied if the assignment to all of ${x}_{1,i}^{i},{x}_{2,i}^{i},\dots ,{x}_{i-1,i}^{i},{x}_{i+1,i}^{i},\dots ,{x}_{k,i}^{i}$ are the same. We refer to the problem of determining if there is an assignment to the variables in X such that all the constraints are satisfied as the k-CSP problem. Notice that while this is a natural way to write k-Multicolor Clique as a CSP, where we have tried to check if all variables having a vertex in common, agree on its assignment, there is no gap yet in the k-CSP problem. In particular, if there was a clique of size k in G then there is an assignment to the variables of X (by assigning the edges of the clique in G to the corresponding variable in X) such that all the constraints in $\mathcal{P}$ are satisfied; however, if every clique in G is of size less than k then there every assignment to the variables of X may violate only one constraint in $\mathcal{P}$ (and not more).

**From**$\mathit{k}$-Gap CSP to gap $\mathit{k}$

**-Dominating Set.**In the second part, starting from the aforementioned instance of k-Gap CSP (after boosting the gap), we construct an instance H of k-Dominating Set. The construction is due to Feige [29,38] and it proceeds as follows. Let $\mathcal{F}$ be the set of all functions from ${\{0,1\}}^{tk}$ to $\left(\genfrac{}{}{0pt}{}{k}{2}\right)$, i.e., $\mathcal{F}:=\{f:{\{0,1\}}^{tk}\to \left(\genfrac{}{}{0pt}{}{k}{2}\right)\}$. The graph H is on vertex set $U=A\dot{\cup}B$, where $A={\mathcal{P}}^{*}\times \mathcal{F}$ and $B=E(G)$, i.e., B is simply the edge set of G. We introduce an edge between all pairs of vertices in B. We introduce an edge between $a:=(S:=({s}_{1},\dots ,{s}_{t})\in {\left[\ell \right]}^{t},f:{\{0,1\}}^{tk}\to \left(\genfrac{}{}{0pt}{}{k}{2}\right))\in A$ and $e:=({v}_{i},{v}_{j})\in E$ if and only if the following holds.

- (Completeness) If there is an assignment to X that satisfies all constraints in ${\mathcal{P}}^{*}$, then the corresponding $\left(\genfrac{}{}{0pt}{}{k}{2}\right)$ vertices in B dominate all vertices in the graph H.
- (Soundness) If each assignment can only satisfy $(1-\alpha )$ fraction of constraints in ${\mathcal{P}}^{*}$, then any dominating set of H has size at least $F\left(\left(\genfrac{}{}{0pt}{}{k}{2}\right)\right)\xb7\left(\genfrac{}{}{0pt}{}{k}{2}\right)$.

- Assuming the Exponential Time Hypothesis (ETH), there is no $F(k)$-approximation algorithm for k-Dominating Set that runs in $T(k)\xb7{n}^{o(k)}$ time.
- Assuming the Strong Exponential Time Hypothesis (SETH), for every integer $k\ge 2$, there is no $F(k)$-approximation algorithm for k-Dominating Set that runs in $T(k)\xb7{n}^{k-\epsilon}$ time.

**Open Question 4**(Tight inapproximability of k-Dominating Set) Is there a ${(logn)}^{1-o(1)}$ factor approximation algorithm for k-Dominating Set running in time ${n}^{k-0.1}$?

**Open Question 5**($\mathrm{W}\left[2\right]$-completeness of approximating k-Dominating Set) Can we base total inapproximability of k-Dominating Set on $\mathrm{W}\left[2\right]\ne \mathrm{FPT}$?

#### 3.1.3. Parameterized Inapproximability of Steiner Orientation by Gap Amplification

**Open Question 6.**Is it W[1]-hard or ETH-hard to approximatek-Cliqueto within a constant factor in FPT time?

#### 3.2. Hardness from Gap Hypotheses

**Hypothesis**

**1**

**Lemma**

**1.**

**Open Question 7.**Does PIH hold if we assume that k-Cliqueis FPT inapproximable to within any constant factor?

**Hypothesis**

**2**

**.**For some constants $\epsilon ,\delta >0$, there is no $O({2}^{\delta n})$-time algorithm that can, given a 3CNF formula, distinguish between the following two cases:

- (Completeness) the formula is satisfiable.
- (Soundness) any assignment violates more than ε fraction of the clauses.

#### 3.2.1. Strong Inapproximability of k-Clique

#### 3.2.2. Strong Inapproximability of Multicolored Densest k-Subgraph and Label Cover

**Theorem**

**6**

**.**Assuming Gap-ETH, there is no ${k}^{1-o(1)}$-approximation for Multicolored Densest k-Subgraph.

#### 3.2.3. Inapproximability of k-Biclique and Densest k-Subgraph

**Open Question 8.**Is there an $f(k)\xb7{N}^{o(k)}$-time algorithm that approximates k-Bicliqueto within a constant factor?

**Theorem**

**8**

**Open Question 9.**Is there an $o(k)$-FPT-approximation algorithm forDenestk-Subgraph?

## 4. Algorithms

#### 4.1. Packing Problems

#### 4.1.1. Independent Set

`Gap-ETH`, no $g(k)$-approximation can be computed in $f(k){n}^{O(1)}$ time [39] for any computable functions f and g, where k is the solution size. On the other hand, for planar graphs a PTAS exists [82]. Hence a natural question is how the problem behaves for graphs that are “close” to being planar.

**Theorem**

**9**

**Theorem**

**10**

**.**For the Independent Set problem a $(1+\epsilon )$-approximation can be computed in ${2}^{k}{n}^{O(1/\epsilon )}$ time for any $\epsilon >0$, where k is the size of a minimum planar deletion set.

**Theorem**

**11**

**.**For the Independent Set of Rectangles problem a $(1+\epsilon )$-approximation can be computed in ${k}^{O(k/{\epsilon}^{8})}{n}^{O(1/{\epsilon}^{8})}$ time for any $\epsilon >0$, where k is the size of the optimum solution, or in $f(\delta ,\epsilon ){n}^{O(1)}$ time for some computable function f and any $\epsilon >0$ and $0<\delta <1$, where δ is the shrinking factor. Moreover, a $(1+\epsilon )$-approximate kernel with ${k}^{O(1/{\epsilon}^{8})}$ rectangles can be computed in polynomial time.

#### 4.1.2. Vertex Coloring

**Theorem**

**12**

- a $7/3$-approximation can be computed in ${k}^{k}{n}^{O(1)}$ time, where k is the size of a minimum planar deletion set, and
- a 2-approximation can be computed in $f(k){n}^{O(1)}$ time for some function f, where k is the size of an excluded minor of the input graph.

**Definition**

**1.**

- the union of all bags is the vertex set V of G,
- for every edge $(u,v)$ of G, there is a node of T for which the associated bag contains u and v, and
- for every vertex u of G, all nodes of T for which the associated bags contain u, induce a connected subtree of T.

**Theorem**

**13**

**.**For the Defective Coloring problem, given a tree decomposition of width k of the input graph,

- a solution with the optimum number of colors where each color class induces a graph of maximum degree $(1+\epsilon )\Delta $ can be computed in ${(k/\epsilon )}^{O(k)}{n}^{O(1)}$ time for any $\epsilon >0$,
- a 2-approximation (of the optimum number of colors) can be computed in ${k}^{O(k)}{n}^{O(1)}$ time, but
- no $(3/2-\epsilon )$-approximation (of the optimum number of colors) can be computed in $f(k){n}^{O(1)}$ time for any $\epsilon >0$ and computable function f, unless FPT = W[1].

- Introduce(x): create a graph containing a singleton vertex labelled $x\in \{1,\dots ,\ell \}$.
- Union( ${G}_{1},{G}_{2}$): return the disjoint union of two vertex-labelled graphs ${G}_{1}$ and ${G}_{2}$.
- Join( $G,x,y$): add all edges connecting a vertex of label x with a vertex of label y to the vertex-labelled graph G.
- Rename( $G,x,y$): change the label of every vertex of G with label x to $y\in \{1,\dots ,\ell \}$.

**Theorem**

**14**

**.**For the Equitable Coloring problem, given a cliquewidth expression with ℓ labels for the input graph, a solution with optimum number of colors where the ratio between the sizes of any two color classes is at most $1+\epsilon $, can be computed in ${(k/\epsilon )}^{O(k\ell )}{n}^{O(1)}$ time [108,109] for any $\epsilon >0$, where k is the optimum number of colors.

**Theorem**

**15**

**.**For the Min Sum Edge Coloring problem a $(1+\epsilon )$-approximation can be computed in $f(k,\epsilon )n$ time for any $\epsilon >0$, where k is the treewidth of the input graph.

#### 4.1.3. Subgraph Packing

**Theorem**

**16**

**.**For the Vertex Cycle Packing problem, a $(1+\epsilon )$-approximate kernel of size ${k}^{O(1/(\epsilon log\epsilon ))}$ can be computed in polynomial time, where k is the solution size.

#### 4.1.4. Scheduling

**Theorem**

**17**

**.**For the Unsplittable Flow on a Path problem a $(1+\epsilon )$-approximation can be computed in ${2}^{O(klogk)}{n}^{g(\epsilon )}$ time for some computable function g and any $\epsilon >0$, where k is the solution size.

**Theorem**

**18**

**.**For the Flow Time Scheduling problem a $(1+\epsilon )$-approximation can be computed in ${(mk)}^{O(m{k}^{3}/\epsilon )}{n}^{O(1)}$ time in the preemtive setting, and in ${(mk/\epsilon )}^{O(m{k}^{5})}{n}^{O(1)}$ time in the non-preemtive setting, for any $\epsilon >0$, where m is the number of machines and k is an upper bound on every processing time and weight.

#### 4.2. Covering Problems

#### 4.2.1. Minimization Variants

**Set Cover, Dominating Set and Vertex Cover.**As discussed in detail in Section 3.1.2, Set Cover and equivalently Dominating Set are very hard to approximate in the general case. Hence, special cases where some constraints are placed on the set system are often considered. Arguably the most well-studied special case of Set Cover is the Vertex Cover problem, in which the set system is a graph. That is, we would like to find the smallest set of vertices such that every edge has at least one endpoint in the selected set (i.e., the edge is “covered”). Vertex Cover is well known to be FPT [126] and admit a linear-size kernel [127]. A generalization of Vertex Cover on d-uniform hypergraph, where the input is now a hypergraph and the goal is to find the smallest set of vertices such that every hyperedge contain at least one vertex from the set, is also often referred to as d-Hitting Set in the parameterized complexity community. However, we will mostly use the nomenclature Vertex Cover on d-uniform hypergraph because many algorithms generalizes well from Vertex Cover in graphs to hypergraphs. Indeed, branching algorithms for Vertex Cover on graphs can be easily generalized to hypergraphs, and hence the latter is also FPT. Polynomial-size kernels are also known for Vertex Cover on d-uniform hypergraphs [128].

**Connected Vertex Cover.**A popular variant of Vertex Cover that is the Connected Vertex Cover problem, for which the computed solution is required to induce a connected subgraph of the input. Just as Vertex Cover, the problem is FPT [129]. However, unlike Vertex Cover, Connected Vertex Cover does not admit a polynomial-time kernel [130], unless NP⊆coNP/poly. In spite of this, a PSAKS for Connected Vertex Cover exists:

**Theorem**

**19**

**.**For any $\epsilon >0$, an $(1+\epsilon )$-approximate kernel with ${k}^{O(1/\epsilon )}$ vertices can be computed in polynomial time.

**Connected Dominating Set**. Similar to Connected Vertex Cover, the Connected Dominating Set problem is the variant of Dominating Set for which the solution additionally needs to induce a connected subgraph of the input graph. When placing no restriction on the input graph, the problem is as hard to approximate as Dominating Set. However, for some special classes of graphs, PSAKS or bi-PSAKS [133] are known; these include graphs with bounded expansion, nowhere dense graphs, and d-biclique-free graphs [134].

**Covering Problems parameterized by Graph Width Parameters.**Several works in literature also study the approximability of variants of Vertex Cover and Dominating Set parameterized by graph widths [105,135]. These variants include:

- Power Vertex Cover (PVC). Here, along with the input graph, each edge has an integer demand and we have to assign (power) values to vertices, such that each edge has at least one endpoint with a value at least its demand. The goal is to minimize the total assigned power. Note that this is generalizes of Vertex Cover, where edges have unit demands.
- Capacitated Vertex Cover (CVC). The problem is similar to Vertex Cover, except that each vertex has a capacity which limits the number of edges that it can cover. Once again, Vertex Cover is a special case of CVC where each vertex’s capacity is ∞.
- Capacitated Dominating Set (CDS). Analogous to CVC, this is a generalization of Dominating Set where each vertex has a capacity and it can only cover/dominate at most that many other vertices.

**Packing-Covering Duality and Erdos-Pósa Property.**Given a set system $(V,\mathcal{C})$ where V is the universe and $\mathcal{C}=\{{C}_{1},\dots ,{C}_{m}\}$ is a collection of subsets of V, Hitting Set is the problem of computing the smallest $S\subseteq V$ that intersects every ${C}_{i}$, and Set Packing is the problem of computing the largest subcollection ${\mathcal{C}}^{\prime}\subseteq \mathcal{C}$ such that no two sets in ${\mathcal{C}}^{\prime}$ intersect. It can also be observed that the optimal value for Hitting Set is at least the optimal value for Set Packing, while the standard LP relaxations for them (covering LP and packing LP) have the same optimal value by strong duality. Studying the other direction of the inequality (often called the packing-covering duality) for natural families of set systems has been a central theme in combinatorial optimization. The gap between the covering optimum and packing optimum is large in general (e.g., Dominating Set/Independent Set), but can be small for some families of set systems (e.g., s-t Cut /s-t Disjoint Paths and Vertex Cover/Matching especially in bipartite graphs).

#### 4.2.2. Maximization Variants

**Max $\mathit{k}$-Coverage.**Recall that here we are given a set system and the goal is to select k subsets whose union is maximized. It is well known that the simple greedy algorithm yields an $\left(\frac{e}{e-1}\right)$-approximation [142]. Furthermore, Fiege shows, in his seminal work [29], show that this is tight: $\left(\frac{e}{e-1}-\epsilon \right)$-approximation is NP-hard for any constant $\epsilon >0$. In fact, recently it has been shown that this inapproximability applies also to the parameterized setting. Specifically, under Gap-ETH, $\left(\frac{e}{e-1}-\epsilon \right)$-approximation cannot be achieved in FPT time [143] or even $f(k)\xb7{n}^{o(k)}$ time [70]. In other words, the trivial algorithm is tight in terms of running time, the greedy algorithm is tight in terms of approximation ratio, and there is essentially no trade-off possible between these two extremes. We remark here that this hardness of approximation is also the basis of hardness for k-Median and k-Means [143] (see Section 4.3).

**Max $\mathit{k}$-Vertex Cover.**Another special case of Max k-Coverage is the restriction when each element belongs to at most d subsets in the system. This corresponds exactly to the maximization variant of the Vertex Cover problem on d-uniform hypergraph, which will refer to as Max k-Vertex Cover. Note here that, for such set systems, their VC-dimensions are also bounded by $logd+1$ and hence the aforementioned PAS of [144] applies here as well. Nonetheless, Max k-Vertex Cover admits a much simpler PAS (and even PSAKS) compared to Max k-Coverage parameterized by k and VC-dimension, as we will discuss more below.

**Theorem**

**20**

**.**For the Max k-Vertex Cover problem in d-uniform hypergraphs, a $(1+\epsilon )$-approximation can be computed in ${O}^{*}\left({(d/\epsilon )}^{k}\right)$ time for any $\epsilon >0$. Moreover, an $(1+\epsilon )$-approximate kernel with $O(dk/\epsilon )$ vertices can be computed in polynomial time.

#### 4.2.3. Other Related Problems

**Min $\mathit{k}$-Uncovered.**The first is the Min k-Uncovered problem, where the input is a set system and we would like to select k sets as to minimize the number of uncovered elements. When we are concerned with exact solutions, this is of course the Set Cover. However, the optimization version becomes quite different from Max k-Coverage. In particular, since it is hard to determine whether we can find k subsets that cover the whole universe, the problem is not approximable at all in the general case. However, if restrict ourselves to graphs and hypergraphs (for which we refer to the problem as Min k-Vertex Cover), it is possible to get a (randomized) PAS for the problem [146]:

**Theorem**

**21**

**.**For the Min k-Vertex Uncovered problem in d-uniform hypergraphs, a $(1+\epsilon )$-approximation can be computed in ${O}^{*}\left({(d/\epsilon )}^{k}\right)$ time for any $\epsilon >0$.

**Min**$\mathit{k}$-

**Coverage.**Another variant of the Set Cover problem studied is Min k-Coverage [150,151,152], where we would like to select k subsets that minimizes the number of covered elements. We stress here that this problem is not a relaxation of Set Cover but rather is much more closely related to graph expansion problems (see [151]).

#### 4.3. Clustering

- Objective function: Three well-studied objective functions arek-Median ( $cost(P,C):={\sum}_{p\in P}\rho (C,p)$).
- -
k-Means ( $cost(P,C):={\sum}_{p\in P}\rho {(C,p)}^{2}$).- -
k-Center ( $cost(P,C):={max}_{p\in P}\rho (C,p)$). - Metric space: The ambient metric space X can be
- -
- A general metric space explicitly given by the distance $\rho :X\times X\to {\mathbb{R}}^{+}\cup \{0\}$.
- -
- The Euclidean space ${\mathbb{R}}^{d}$ equipped with the ${\ell}_{2}$ distance.
- -
- Other structured metric spaces including metrics with bounded doubling dimension or bounded highway dimension.

#### 4.3.1. General Metric Space

**Theorem**

**22**

**.**For any $\epsilon >0$, there is an $(1+2/e+\epsilon )$-approximation algorithm for k-Median, and an $(1+8/e+\epsilon )$-approximation algorithm for k-Means, both running in time ${(O(klogk/{\epsilon}^{2}))}^{k}{n}^{O(1)}$.

`Gap-ETH`, for any $\epsilon >0$, any $(1+2/e-\epsilon )$-approximation algorithm fork-Median, and any $(1+8/e-\epsilon )$-approximation algorithm fork-Means, must run in time at least ${n}^{{k}^{g(\epsilon )}}$.

**Algorithm for $\mathit{k}$-Median**. We briefly describe ideas for the algorithm for k-Median in Theorem 22. The main technical tool that the algorithm uses is a coreset, which will be also frequently used for Euclidean subspaces in the next subsection.

`Gap-ETH`.)

**Constructing a coreset.**As discussed above, a coreset is a fundamental building block for optimal parameterized approximation algorithms for k-Median and k-Means for general metrics. We briefly describe the construction of Chen [160] that gives a coreset of cardinality $\tilde{O}({k}^{2}{log}^{2}n/{\epsilon}^{2})$ for k-Median. Similar ideas can be also used to obtain an EPAS for Euclidean spaces parameterized by k, though better specific constructions are known in Euclidean spaces.

#### 4.3.2. Euclidean Space

**Euclidean$\mathit{k}$-Median with parameter $\mathit{d}$.**The first PTAS for Euclidean k-Median in Euclidean spaces with fixed d appears in Arora et al. [166]. The techniques extend Arora’s previous PTAS for the Euclidean Travelling Salesman problem in Euclidean spaces [167], first proving that there exists a near-optimal solution that interacts with a quadtree (a geometric division of ${\mathbb{R}}^{d}$ into a hierarchy of square regions) in a restricted sense, and finally finding such a tour using dynamic programming. The running time is ${n}^{O(1/\epsilon )}$ for $d=2$ and ${n}^{{(logn/\epsilon )}^{d-2}}$ for $d>2$. Kolliopoulous and Rao [168] improved the running time to ${2}^{O({(log(1/\epsilon )/\epsilon )}^{d-1})}n{log}^{d+6}n$, which is an EPAS with parameter d.

**Euclidean$\mathit{k}$-Median and Euclidean $\mathit{k}$-Means with parameter $\mathit{k}$.**

**Euclidean$\mathit{k}$-Means with parameter $\mathit{d}$.**

**Other metrics and $\mathit{k}$-CENTER.**For the k-Center problem an EPAS exists when parametrizing by both k and the doubling dimension [184], and also for planar graphs there is an EPAS for parameter k, which is implied by the EPTAS of Fox-Epstein et al. [185] (cf. [184]).

**Capacitated clustering and other variants.**Another example where the parameterization by k helps is Capacitated k-Median, where each possible center $c\in F$ has a capacity ${u}_{c}\in \mathbb{N}$ and can be assigned at most ${u}_{c}$ points. It is not known whether there exists a constant-factor approximation algorithm, and known constant factor approximation algorithms either open $(1+\epsilon )k$ centers [188] or violate capacity constraints by an $(1+\epsilon )$ factor [189]. Adamczyk et al. [190] gave a $(7+\epsilon )$-approximation algorithm in $f(k,\epsilon ){n}^{O(1)}$ time, showing that a constant factor parameterized approximation algorithm is possible. The approximation ratio was soon improved to $(3+\epsilon )$ [191]. For Capacitated Euclidean k-Means, 192] also gave a $(69+\epsilon )$-approximation algorithm for in $f(k,\epsilon ){n}^{O(1)}$ time.

**Open Question 10.**DoesCapacitatedk-Medianadmit an$(1+2/e)$-approximation algorithm in FPT time with parameter k? DoCapacitated Euclideank-Means/k-Medianadmit an EPAS with parameter k or d?

#### 4.4. Network Design

**Undirected graphs.**A well-studied parameter for Steiner Tree is the number of terminals, for which the problem has been known to be FPT since the early 1970s due to the work of Dreyfus and Wagner [196]. Their algorithm is based on dynamic programming and runs in ${3}^{k}{n}^{O(1)}$ time if k is the number of terminals. Faster algorithms based on the same ideas with runtime ${(2+\delta )}^{k}{n}^{O(1)}$ for any constant $\delta >0$ exist [197] (here the degree of the polynomial depends on $\delta $). The unweighted Steiner Tree problem also admits a ${2}^{k}{n}^{O(1)}$ time algorithm [198] using a different technique based on subset convolution. Given any of these exact algorithms as a subroutine, a faster PAS can also be found [20] (cf. Section 4.7). On the other hand, no exact polynomial-sized kernel exists [130] for the Steiner Tree problem, unless NP⊆coNP/poly. Interestingly though, a PSAKS can be obtained [18].

- each full-component ${C}_{i}$ contains at most ${2}^{\lceil 1/\epsilon \rceil}$ terminals (leaves),
- the sum of the weights of the full-components is at most $1+\epsilon $ times the cost of T, and
- taking any collection of Steiner trees ${T}_{1},\dots ,{T}_{\ell}$, such that each tree ${T}_{i}$ connects the subset of terminals that forms the leaves of full-component ${C}_{i}$, the union ${\bigcup}_{i=1}^{\ell}{T}_{i}$ is a feasible solution to the input instance.

**Theorem**

**23**

**.**For the Steiner Tree problem a $(1+\epsilon )$-approximation can be computed in ${(2+\delta )}^{(1-\epsilon /2)}{n}^{O(1)}$ time for any constant $\delta >0$ (and in ${2}^{(1-\epsilon /2)}{n}^{O(1)}$ time in the unweighted case) for any $\epsilon >0$, where k is the number of terminals. Moreover, a $(1+\epsilon )$-approximate kernel of size ${(k/\epsilon )}^{O({2}^{1/\epsilon})}$ can be computed in polynomial time.

**Theorem**

**24**

**.**For the Steiner Tree problem a $(1+\epsilon )$-approximation can be computed in ${2}^{O({k}^{2}/{\epsilon}^{4})}{n}^{O(1)}$ time for any $\epsilon >0$, where k is the number of non-terminals in the optimum solution. Moreover, a $(1+\epsilon )$-approximate kernel of size ${(k/\epsilon )}^{{2}^{O(1/\epsilon )}}$ can be computed in polynomial time.

**Theorem**

**25**

**.**For any constant $\lambda >0$, there is an FPTAS for the SLSN ${}_{{\mathcal{C}}_{\lambda}}$ problem. For the SLSN ${}_{{\mathcal{C}}^{\star}}$ problem a $(1+\epsilon )$-approximation can be computed in ${4}^{k}{(n/\epsilon )}^{O(1)}$ time for any $\epsilon >0$, where k is the number of terminal pairs. Moreover, under

`Gap-ETH`no $(5/3-\epsilon )$-approximation for SLSN ${}_{\mathcal{C}}$ can be computed in $f(k){n}^{O(1)}$ time for any $\epsilon >0$ and computable function f, whenever $\mathcal{C}$ is a recursively enumerable class for which $\mathcal{C}\neg \subseteq {\mathcal{C}}^{\star}\cup {\mathcal{C}}_{\lambda}$ for every constant λ.

**Open Question 11.**Given some class of graphs$\mathcal{C}\neg \subseteq {\mathcal{C}}^{\star}\cup {\mathcal{C}}_{\lambda}$, which approximation factor${\alpha}_{\mathcal{C}}$can be obtained in FPT time for SLSN ${}_{\mathcal{C}}$ parameterized by the number of terminals?

**Theorem**

**26**

**.**For the DlTSP problem a $2.5$-approximation can be computed in $O(k!\xb7k)+{n}^{O(1)}$ time, if the number of vertices with deadlines is k. Moreover, no $(2-\epsilon )$-approximation can be computed in $f(k){n}^{O(1)}$ time for any $\epsilon >0$ and computable function f, unless P = NP.

**Low dimensional metrics.**Just as for clustering problems, another well-studied parameter in network design is the dimension of the underlying geometric space. A typical setting is when the input is assumed to be a set of points in some k-dimensional ${\ell}_{p}$-metric, where distances between points x and y are given by a function $\mathrm{dist}(x,y)=({\sum}_{i=1}^{k}|{x}_{i}-{y}_{i}{{|}^{p})}^{1/p}$. Two prominent examples are Euclidean metrics (where $p=2$) and Manhattan metrics (where $p=1$). The dimension k of the metric space has been studied as a parameter from the parameterized approximation point-of-view avant la lettre for quite a while. It was shown [206,207] that both Steiner Tree and TSP are paraNP-hard for this parameter (since they are NP-hard even if $k=2$), and that they are APX-hard in general metrics [201,208]. However, a PAS for Euclidean metrics both for the Steiner Tree and the TSP problems were shown to exist in the seminal work of Arora [167,209]. The techniques are similar to those used for clustering, and we refer to Section 4.3.2 for an overview.

**Theorem**

**27**

**.**For the Steiner Tree and TSP problems a $(1+\epsilon )$-approximation can be computed in ${k}^{O{(\sqrt{k}/\epsilon )}^{k-1}}{n}^{2}$ time for any $\epsilon >0$, if the input consists of n points in k-dimensional Euclidean space.

**Theorem**

**28**

**.**For the TSP problem a $(1+\epsilon )$-approximation can be computed in ${2}^{{(k/\epsilon )}^{O({k}^{2})}}n{log}^{2}n$ time for any $\epsilon >0$, if the input consists of n points with doubling dimension k.

**Open Question 12.**Is there a PAS forSteiner Treeparameterized by the doubling dimension? Is there a PAS for eitherSteiner Treeor TSP parameterized by the highway dimension?

**Directed Graphs.**When considering directed input graphs (asymmetric metrics), the Directed Steiner Tree problem takes as input a terminal set and a special terminal called the root. The task is to compute a directed tree of minimum weight that contains a path from each terminal to the root. In general no $f(k)$-approximation can be computed in FPT time for any computable function f, when the parameter k is the number of Steiner vertices in the optimum solution [202]. A notable special case is the unweighted Directed Steiner Tree problem, which for this parameter admits a PAS. The techniques here are the same as those used to obtain Theorem 24 for the undirected case. However, in contrast to the undirected case which admits a PSAKS, no polynomial-sized $(2-\epsilon )$-approximate kernelization exists for Directed Steiner Tree [202], unless NP⊆coNP/poly. It is an intriguing question whether a 2-approximate kernel exists.

**Open Questions 13.**Is there a polynomial-sized 2-approximate kernel for the unweightedDirected Steiner Treeproblem parameterized by the number of Steiner vertices in the optimum solution?

**Theorem**

**29**

**.**For the Strongly Connected Steiner Subgraph problem a 2-approximation can be computed in ${(2+\delta )}^{k}{n}^{O(1)}$ time for any constant $\delta >0$, where k is the number of terminals. Moreover, under

`Gap-ETH`no $(2-\epsilon )$-approximation can be computed in $f(k){n}^{O(1)}$ time for any $\epsilon >0$ and computable function f.

`Gap-ETH`. Both a PAS and a PSAKS exist [50] for the special case when the input graph is planar and bidirected, i.e., for every directed edge $uv$ the reverse edge $vu$ exists and has the same cost.

**Theorem**

**30**

**.**For the Directed Steiner Network problem on planar bidirected graphs a $(1+\epsilon )$-approximation can be computed in $max\{{2}^{{k}^{{2}^{O(1/\epsilon )}}},{n}^{{2}^{O(1/\epsilon )}}\}$ time for any $\epsilon >0$, where k is the number of terminals. Moreover, a $(1+\epsilon )$-approximate kernel of size ${(k/\epsilon )}^{{2}^{O(1/\epsilon )}}$ can be computed in polynomial time.

#### 4.5. Cut Problems

#### 4.5.1. Multicut

**Undirected Multicut.**Undirected Multicut admits an $O(logk)$-approximation algorithm [223] in polynomial time, and is NP-hard to approximate within any constant factor assuming the Unique Games Conjecture [224]. Directed Multiway Cut admits an $1.2965$-approximation algorithm [225] in polynomial time, and is NP-hard to approximate within a factor $1.20016$ [226]. Undirected Multicut (and thus Directed Multiway Cut) admits an exact algorithm parametrized by $\mathrm{O}\mathrm{PT}$ [219,220].

**Directed Multicut.**Generally, Directed Multicut is a much harder computational task than Undirected Multicut in terms of both approximation and parameterized algorithms. Directed Multicut admits a $min(k,\tilde{O}({n}^{11/23}))$-approximation algorithm [229]. It is NP-hard to approximate within a factor $k-\epsilon $ for any $\epsilon >0$ for fixed k [230] under the Unique Games Conjecture, or ${2}^{\Omega ({log}^{1-\epsilon}n)}$ for any $\epsilon >0$ [231] for general k. Directed Multiway Cut admits an 2-approximation algorithm [232], which is tight even when $k=2$ [230]. Parameterizing by Opt, Directed Multicut is FPT for $k=2$, but Directed Multicut is W[1]-hard even when $k=4$ [46]. Directed Multiway Cut on the other hand is in FPT [221].

**Open Question 14.**What is the best approximation ratio (as a function of k) achieved by a parameterized algorithm (with parameter$\mathrm{O}\mathrm{PT}$)? Will it be close to$O(1)$or$\Omega (k)$?

#### 4.5.2. Minimum Bisection and Balanced Separator

#### 4.5.3. k-Cut

#### 4.6. $\mathcal{F}$-Deletition Problems

- Choose a graph width parameter (e.g., treewidth, pathwidth, cliquewidth, rankwidth, etc.) and $k\in \mathbb{N}$. Let $\mathcal{F}$ be the set of graphs G with the chosen width parameter at most k. The parameter of $\mathcal{F}$-Deletion is k.
- Choose a notion of subgraph (e.g., subgraph, induced subgraph, minor, etc.) and a finite family of forbidden graphs $\mathcal{H}$. Let $\mathcal{F}$ be the set of graphs G that do not have any graph in $\mathcal{H}$ as the chosen notion of subgraph. The parameter of $\mathcal{F}$-Deletion is $|\mathcal{H}|:={\sum}_{H\in \mathcal{H}}|V(H)|$.

#### 4.6.1. Treewidth and Planar Minor Deletion

**Open Question 15.**DoesWeighted Treewidthk-Deletionadmit an$f(k)$-approximation algorithm with parameter k for some functionf? DoesTreewidthk-Deletionadmit a c-approximation algorithm with parameter k for some universal constant c?

**Algorithms for Treewidth**$\mathit{k}$

**-Deletion.**Here we present high-level ideals of [255,260] for Treewidth k-Deletion and Weighted Treewidth k-Deletion respectively. These two algorithms share the following two important ingredients:

- Graphs with bounded treewidth admit good separators.
- There are good approximation algorithms to find such separators.

**Weighted Treewidth**$\mathit{k}$

**-Deletion.**Agrawal et al. [260] achieves an $O({log}^{1.5}n)$-approximation for Weighted Treewidth k-Deletion in time ${n}^{O(k)}$. It would be interesting to see whether the running time can be made FPT with parameter k.

**Treewidth**$\mathit{k}$

**-Deletion.**Gupta et al. [255] give an $O(logk)$-approximation algorithm that runs in time $f(k)\xb7{n}^{O(1)}$ for the unweighted version of Treewidth k-Deletion. The main structure of this algorithm is bottom-up iterative refinement. The algorithm maintains a feasible solution $S\subseteq V$ (we can start with $S=V$), and iteratively uses S to obtain another feasible solution ${S}^{\prime}$. If the new solution is not smaller (i.e., $|{S}^{\prime}|\ge |S|$), then $|S|\le O(logk)\xb7\mathrm{O}\mathrm{PT}$.

**Lemma**

**2**

#### 4.6.2. Subgraph Deletion

#### 4.6.3. Other Deletion Problems

**Chordal graphs.**A graph is chordal if it does not have an induced cycle of length $\ge 4$. Chordal graphs form a subclass of perfect graphs that have been actively studied. Initially motivated by efficient kernels, approximation algorithms for Chordal Deletion have been developed recently. The current best results are a $\mathrm{poly}(\mathrm{O}\mathrm{PT})$-approximation [140,266] and a $O({log}^{2}n)$-approximation [260].

**Edge versions.**While this subsection focused on the vertex deletion problem, there are some results on the edge deletion, edge addition, and edge modification versions. (Edge modification allows both addition and deletion.) Cao and Sandeep [267] studied Minimum Fill-In, whose goal is to add the minimum number of edges to make a graph chordal. They gave new inapproximability results implying improved time lower bounds for parameterized algorithms. Giannopoulou et al. [268] gave $O(1)$-approximation algorithms for Planar $\mathcal{H}$-Immersion Deletion parameterized by $\mathcal{H}$. Bliznets et al. [269] considered H-free edge modification for a forbidden induced subgraph H and give an almost complete characterization on its approximability depending on H.

**Directed graphs.**There is also a large body of work on parameterized algorithms for vertex deletion problems in directed graphs. While many of the known problems (including Directed Feedback Vertex Set [270]) admit an exact FPT algorithm, Lokshtanov et al. [48] studied Directed Odd Cycle Transversal, and proved that it is W[1]-hard and is unlikely to admit an PAS under the Parameterized Inapproximability Hypothesis (or Gap-ETH). They complemented the result by showing a 2-approximation algorithm running in time $f(\mathrm{O}\mathrm{PT}){n}^{O(1)}$.

#### 4.7. Faster Algorithms and Smaller Kernels via Approximation

**Theorem**

**31**

**.**Let $\delta >0$ be such that there exists an ${O}^{*}({\delta}^{k})$-time algorithm for Vertex Cover (e.g., $\delta =1.2738$). Then, for any $\epsilon >0$, there is an $(1+\epsilon )$-approximation for Vertex Cover that runs in ${O}^{*}({\delta}^{(1-\epsilon )k})$ time.

**Open Question 16.**LetLet $\delta >0$ be the smallest (known) constant such that an ${O}^{*}({\delta}^{k})$-time exact algorithm exists for Vertex Cover. Is there an algorithm that, for any $\epsilon >0$, runs in time $f(1/\epsilon )\xb7{O}^{*}({\lambda}^{k})$ for some constant $\lambda <\delta $?

**Open Question 17.**Is there an algorithm that runs in ${2}^{o(k)}{n}^{O(1)}$ time and achieves an approximation ratio of $(2-\rho )$ for some absolute constant $\rho >0$?

## 5. Future Directions

#### 5.1. Approximation Factors

- A parameterized approximation scheme (PAS) exists, i.e., for any constant $\epsilon >0$ a $(1+\epsilon )$-approximation can be computed in $f(k){n}^{O(1)}$ time for some parameter k. These are currently the most prevalent types of results in the literature. To just mention one example, the Steiner Tree problem is APX-hard, but admits a PAS [202] when parameterized by the number of non-terminals (so-called Steiner vertices) in the optimum solution (cf. Section 4.4).
- A lower bound excluding any non-trivial approximation factor exists. For example, under
`ETH`the Dominating Set problem has no $g(k)$-approximation in $f(k){n}^{o(k)}$ time [35] for any functions g and f, where k is the size of the largest dominating set. - A polynomial-time approximation algorithm can achieve a similar approximation ratio, i.e., the parameterization is not very helpful. For instance, for the k-Center problem [286] a 2-approximation can be computed in polynomial time [287], but even when parameterizing by k no $(2-\epsilon )$-approximation is possible [187] for any $\epsilon >0$, under standard complexity assumptions. A similar situation holds for Max k-Coverage, which we discussed in Section 4.2.2.
- Constant or logarithmic approximation ratios can be shown, and which beat any approximation ratio obtainable in polynomial time. For instance, Strongly Connected Steiner Subgraph problem: under standard complexity assumptions, for this problem no polynomial-time $O({log}^{2-\epsilon}n)$-approximation algorithm exists [214], and there is no FPT algorithm parameterized by the number k of terminals [213]. However it is not hard to compute a 2-approximation in ${2}^{O(k)}{n}^{O(1)}$ time [215], and no $(2-\epsilon )$-approximation algorithm with runtime $f(k){n}^{O(1)}$ exists [50] under
`Gap-ETH`, for any function f and any $\epsilon >0$ (cf. Section 4.4).

#### 5.2. Parameterized Running Times

`ETH`, it is even possible to provide lower bounds on the runtimes obtainable by any FPT or XP algorithm. Similar to approximation algorithms, this has lead to the discovery of a spectrum of tractability (cf. [6]): starting from slightly sub-exponential ${2}^{O(\sqrt{k})}{n}^{O(1)}$ time, through single exponential ${2}^{O(k)}{n}^{O(1)}$ time, to double exponential ${2}^{{2}^{O(k)}}{n}^{O(1)}$ time for FPT algorithms with matching asymptotic lower bounds under

`ETH`(e.g., for the Planar Vertex Cover, Vertex Cover, and Edge Clique Cover problems, respectively, each parameterized by the solution size). For XP algorithms, asymptotically tight runtime bounds of the form ${n}^{O(\sqrt{k})}$ and ${n}^{O(k)}$ can be obtained under

`ETH`(e.g., for the Clique problem parameterized by the solution size, and the Planar Bidirected Steiner Network problem parameterized by the number of terminals [50], respectively). Finally, problems that are NP-hard when the given parameter is constant do not even allow XP algorithms unless P = NP (e.g., the Graph Colouring problem where the parameter is the number of colours).

- An approximation is possible in $f(k){n}^{O(1)}$ time for some function f. Most current results are only concerned with the existence of an algorithm with this type of runtime, i.e., they do not provide any evidence that the obtained runtime is best possible, or try to optimize it. The only lower bounds known exclude certain types of approximation schemes when a hardness result for the parameterization by the solution size exists. For instance, it is known that if some problem does not admit a ${2}^{o(k)}{n}^{O(1)}$ time algorithm for this parameter k then it also does not admit an EPTAS with runtime ${2}^{o(1/\epsilon )}{n}^{O(1)}$ (cf. [5,8]).
- A certain approximation ratio cannot be obtained in $f(k){n}^{O(1)}$ time for any function f. For example, it is known that while a 2-approximation for the Strongly Connected Steiner Subgraph problem can be computed in ${2}^{O(k)}{n}^{O(1)}$ time [215], where k is the number of terminals, no $(2-\epsilon )$-approximation can be computed in $f(k){n}^{O(1)}$ time [50] for any function f, under
`Gap-ETH`(cf. Section 4.4).

#### 5.3. Kernel Sizes

- A polynomial-sized approximate kernelization scheme (PSAKS) exists, i.e., for any $\epsilon >0$ there is a $(1+\epsilon )$-approximate kernelization algorithm that computes a $(1+\epsilon )$-approximate kernel of size polynomial in the parameter k. For example, the Steiner Tree problem admits a PSAKS for both the parameterization in the number of terminals [18] and in the number of Steiner vertices in the optimum [202], even though neither of these two parameters admits a polynomial-sized (exact) kernel.

#### 5.4. Completeness in Hardness of Approximation

## Author Contributions

## Funding

## Conflicts of Interest

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Feldmann, A.E.; Karthik C. S.; Lee, E.; Manurangsi, P. A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms. *Algorithms* **2020**, *13*, 146.
https://doi.org/10.3390/a13060146

**AMA Style**

Feldmann AE, Karthik C. S., Lee E, Manurangsi P. A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms. *Algorithms*. 2020; 13(6):146.
https://doi.org/10.3390/a13060146

**Chicago/Turabian Style**

Feldmann, Andreas Emil, Karthik C. S., Euiwoong Lee, and Pasin Manurangsi. 2020. "A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms" *Algorithms* 13, no. 6: 146.
https://doi.org/10.3390/a13060146