arXiv subjects

Karthik C. S.

Publications and source records attributed to Karthik C. S..

At least 37 records · Page 2Linked to original sources

arXiv2023

Clustering Categorical Data: Soft Rounding k-modes

Over the last three decades, researchers have intensively explored various clustering tools for categorical data analysis. Despite the proposal of various clustering algorithms, the classical k-modes algorithm remains a popular choice for unsupervised learning of categorical data. Surprisingly, our first insight is that in a natural generative block model, the k-modes algorithm performs poorly for a large range of parameters. We remedy this issue by proposing a soft rounding variant of the k-modes algorithm (SoftModes) and theoretically prove that our variant addresses the drawbacks of the k-modes algorithm in the generative model. Finally, we empirically verify that SoftModes performs well on both synthetic and real-world datasets.

cs.LG↗

arXiv2023

On Complexity of 1-Center in Various Metrics

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=ω(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=Ω(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+ε)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

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arXiv2023

Can You Solve Closest String Faster than Exhaustive Search?

We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set $X \subseteq Σ^d$ of $n$ strings, find the string $x^*$ minimizing the radius of the smallest Hamming ball around $x^*$ that encloses all the strings in $X$. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: $\bullet$ In the continuous Closest String problem, the goal is to find the solution string $x^*$ anywhere in $Σ^d$. For binary strings, the exhaustive search algorithm runs in time $O(2^d poly(nd))$ and we prove that it cannot be improved to time $O(2^{(1-ε) d} poly(nd))$, for any $ε> 0$, unless the Strong Exponential Time Hypothesis fails. $\bullet$ In the discrete Closest String problem, $x^*$ is required to be in the input set $X$. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time $n^{2 \pm o(1)}$ whenever the dimension is $ω(\log n) < d < n^{o(1)}$. We complement this known hardness result with new algorithms, proving essentially that whenever $d$ falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-$d$ regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in $X$.

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arXiv2023

Deterministic Replacement Path Covering

In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph $G$, a vertex pair $(s,t) \in V(G)\times V(G)$, and a set of edge faults $F \subseteq E(G)$, a replacement path $P(s,t,F)$ is an $s$-$t$ shortest path in $G \setminus F$. For integer parameters $L,f$, a replacement path covering (RPC) is a collection of subgraphs of $G$, denoted by $\textit{G}_{L,f}=\{G_1,\ldots, G_r \}$, such that for every set $F$ of at most $f$ faults (i.e., $|F|\le f$) and every replacement path $P(s,t,F)$ of at most $L$ edges, there exists a subgraph $G_i\in \textit{G}_{L,f}$ that contains all the edges of $P$ and does not contain any of the edges of $F$. The covering value of the RPC $\textit{G}_{L,f}$ is then defined to be the number of subgraphs in $\textit{G}_{L,f}$. We present efficient deterministic constructions of $(L,f)$-RPCs whose covering values almost match the randomized ones, for a wide range of parameters. Our time and value bounds improve considerably over the previous construction of Parter (DISC 2019). We also provide an almost matching lower bound for the value of these coverings. A key application of our above deterministic constructions is the derandomization of the algebraic construction of the distance sensitivity oracle by Weimann and Yuster (FOCS 2010). The preprocessing and query time of the our deterministic algorithm nearly match the randomized bounds. This resolves the open problem of Alon, Chechik and Cohen (ICALP 2019).

cs.DS↗

arXiv2022

On Communication Complexity of Fixed Point Computation

Brouwer's fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from $[0,1]^n$ to $[0,1]^n$, and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of $2^{Ω(n)}$ for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. $\bullet$ Each player is given a function from $[0,1]^n$ to $[0,1]^{n/2}$, and their goal is to find an approximate fixed point of the concatenation of the functions. $\bullet$ Each player is given a function from $[0,1]^n$ to $[0,1]^{n}$, and their goal is to find an approximate fixed point of the interpolation of the functions. We show a randomized communication complexity lower bound of $2^{Ω(n)}$ for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the two-player communication model: A triangulation $T$ of the $d$-simplex is publicly known and one player is given a set $S_A\subset T$ and a coloring function from $S_A$ to $\{0,\ldots ,d/2\}$, and the other player is given a set $S_B\subset T$ and a coloring function from $S_B$ to $\{d/2+1,\ldots ,d\}$, such that $S_A\dot\cup S_B=T$, and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of $|T|^{Ω(1)}$ for the aforementioned problem as well (when $d$ is large).

cs.CC↗

arXiv2022

Obtaining Approximately Optimal and Diverse Solutions via Dispersion

There has been a long-standing interest in computing diverse solutions to optimization problems. Motivated by reallocation of governmental institutions in Sweden, in 1995 J. Krarup posed the problem of finding $k$ edge-disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer $k$, an approximation factor $c$, and an instance $I$ of an optimization problem, we aim to obtain a set of $k$ solutions to $I$ that a) are all $c$ approximately-optimal for $I$ and b) maximize the diversity of the $k$ solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. We show that, given any metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, this problem can be solved by combining ideas from dispersion and multicriteria optimization. We first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to be maximized (minimized) subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem with a little overhead.

cs.DS↗

arXiv2021

Johnson Coverage Hypothesis: Inapproximability of k-means and k-median in L_p metrics

K-median and k-means are the two most popular objectives for clustering algorithms. Despite intensive effort, a good understanding of the approximability of these objectives, particularly in $\ell_p$-metrics, remains a major open problem. In this paper, we significantly improve upon the hardness of approximation factors known in literature for these objectives in $\ell_p$-metrics. We introduce a new hypothesis called the Johnson Coverage Hypothesis (JCH), which roughly asserts that the well-studied max k-coverage problem on set systems is hard to approximate to a factor greater than 1-1/e, even when the membership graph of the set system is a subgraph of the Johnson graph. We then show that together with generalizations of the embedding techniques introduced by Cohen-Addad and Karthik (FOCS '19), JCH implies hardness of approximation results for k-median and k-means in $\ell_p$-metrics for factors which are close to the ones obtained for general metrics. In particular, assuming JCH we show that it is hard to approximate the k-means objective: $\bullet$ Discrete case: To a factor of 3.94 in the $\ell_1$-metric and to a factor of 1.73 in the $\ell_2$-metric; this improves upon the previous factor of 1.56 and 1.17 respectively, obtained under UGC. $\bullet$ Continuous case: To a factor of 2.10 in the $\ell_1$-metric and to a factor of 1.36 in the $\ell_2$-metric; this improves upon the previous factor of 1.07 in the $\ell_2$-metric obtained under UGC. We also obtain similar improvements under JCH for the k-median objective. Additionally, we prove a weak version of JCH using the work of Dinur et al. (SICOMP '05) on Hypergraph Vertex Cover, and recover all the results stated above of Cohen-Addad and Karthik (FOCS '19) to (nearly) the same inapproximability factors but now under the standard NP$\neq$P assumption (instead of UGC).

cs.CC↗

arXiv2021

Applications of Random Algebraic Constructions to Hardness of Approximation

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-threshold graph is, roughly speaking, a balanced bipartite graph in which the common neighbourhoods of k-sets of vertices on one side are much larger than those of (k+1)-sets. The question of their existence was raised by Lin [JACM 2018]. As applications of our constructions, we show the following conditional time lower bounds on the parameterized set intersection problem where, given a collection of n sets over universe [n] and a parameter k, the goal is to find k sets with the largest intersection. (*) Assuming ETH, for any computable function F, no $n^{o(k)}$-time algorithm can approximate the parameterized set intersection problem up to factor F(k). This improves considerably on the previously best-known result under ETH due to Lin [JACM 2018], who ruled out any $n^{o(\sqrt{k})}$ time approximation algorithm for this problem. (*) Assuming SETH, for every $\varepsilon>0$ and any computable function F, no $n^{k-\varepsilon}$-time algorithm can approximate the parameterized set intersection problem up to factor F(k). No result of comparable strength was previously known under SETH, even for solving this problem exactly.

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arXiv2020

On Approximability of Clustering Problems Without Candidate Centers

The k-means objective is arguably the most widely-used cost function for modeling clustering tasks in a metric space. In practice and historically, k-means is thought of in a continuous setting, namely where the centers can be located anywhere in the metric space. For example, the popular Lloyd's heuristic locates a center at the mean of each cluster. Despite persistent efforts on understanding the approximability of k-means, and other classic clustering problems such as k-median and k-minsum, our knowledge of the hardness of approximation factors of these problems remains quite poor. In this paper, we significantly improve upon the hardness of approximation factors known in the literature for these objectives. We show that if the input lies in a general metric space, it is NP-hard to approximate: $\bullet$ Continuous k-median to a factor of $2-o(1)$; this improves upon the previous inapproximability factor of 1.36 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ Continuous k-means to a factor of $4- o(1)$; this improves upon the previous inapproximability factor of 2.10 shown by Guha and Khuller (J. Algorithms '99). $\bullet$ k-minsum to a factor of $1.415$; this improves upon the APX-hardness shown by Guruswami and Indyk (SODA '03). Our results shed new and perhaps counter-intuitive light on the differences between clustering problems in the continuous setting versus the discrete setting (where the candidate centers are given as part of the input).

cs.CC↗

arXiv2020

On Hardness of Approximation of Parameterized Set Cover and Label Cover: Threshold Graphs from Error Correcting Codes

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 scalable version of his result: given any error correcting code $C$ over alphabet $[q]$, rate $ρ$, and relative distance $δ$, we use $C$ to create a reduction from the $(k,k+1)$-SetCover problem on universe $U$ to the $\left(k,\sqrt[2k]{\frac{2}{1-δ}}\right)$-SetCover problem on universe of size $\frac{\log|\mathcal{S}|}ρ\cdot|U|^{q^k}$. Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code $C$. We use the same threshold graphs mentioned above to prove inapproximability results, under W[1]$\neq$FPT and ETH, for the $k$-MaxCover problem introduced by Chalermsook et al. (SICOMP'20). Our inapproximaiblity results match the bounds obtained by Karthik et al. (JACM'19), although their proof framework is very different, and involves generalization of the distributed PCP framework. Prior to this work, it was not clear how to adopt the proof strategy of Lin to prove inapproximability results for $k$-MaxCover.

cs.CC↗

arXiv2020

On Efficient Low Distortion Ultrametric Embedding

A classic problem in unsupervised learning and data analysis is to find simpler and easy-to-visualize representations of the data that preserve its essential properties. A widely-used method to preserve the underlying hierarchical structure of the data while reducing its complexity is to find an embedding of the data into a tree or an ultrametric. The most popular algorithms for this task are the classic linkage algorithms (single, average, or complete). However, these methods on a data set of $n$ points in $Ω(\log n)$ dimensions exhibit a quite prohibitive running time of $Θ(n^2)$. In this paper, we provide a new algorithm which takes as input a set of points $P$ in $\mathbb{R}^d$, and for every $c\ge 1$, runs in time $n^{1+\fracρ{c^2}}$ (for some universal constant $ρ>1$) to output an ultrametric $Δ$ such that for any two points $u,v$ in $P$, we have $Δ(u,v)$ is within a multiplicative factor of $5c$ to the distance between $u$ and $v$ in the "best" ultrametric representation of $P$. Here, the best ultrametric is the ultrametric $\tildeΔ$ that minimizes the maximum distance distortion with respect to the $\ell_2$ distance, namely that minimizes $\underset{u,v \in P}{\max}\ \frac{\tildeΔ(u,v)}{\|u-v\|_2}$. We complement the above result by showing that under popular complexity theoretic assumptions, for every constant $\varepsilon>0$, no algorithm with running time $n^{2-\varepsilon}$ can distinguish between inputs in $\ell_\infty$-metric that admit isometric embedding and those that incur a distortion of $\frac{3}{2}$. Finally, we present empirical evaluation on classic machine learning datasets and show that the output of our algorithm is comparable to the output of the linkage algorithms while achieving a much faster running time.

cs.DS↗

arXiv2020

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

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.

cs.DS↗

arXiv2019

Parameterized Intractability of Even Set and Shortest Vector Problem

The $k$-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over $\mathbb F_2$, which can be stated as follows: given a generator matrix $\mathbf A$ and an integer $k$, determine whether the code generated by $\mathbf A$ has distance at most $k$, or in other words, whether there is a nonzero vector $\mathbf{x}$ such that $\mathbf A \mathbf{x}$ has at most $k$ nonzero coordinates. The question of whether $k$-Even Set is fixed parameter tractable (FPT) parameterized by the distance $k$ has been repeatedly raised in literature; in fact, it is one of the few remaining open questions from the seminal book of Downey and Fellows (1999). In this work, we show that $k$-Even Set is W[1]-hard under randomized reductions. We also consider the parameterized $k$-Shortest Vector Problem (SVP), in which we are given a lattice whose basis vectors are integral and an integer $k$, and the goal is to determine whether the norm of the shortest vector (in the $\ell_p$ norm for some fixed $p$) is at most $k$. Similar to $k$-Even Set, understanding the complexity of this problem is also a long-standing open question in the field of Parameterized Complexity. We show that, for any $p > 1$, $k$-SVP is W[1]-hard to approximate (under randomized reductions) to some constant factor.

cs.CC↗

arXiv2019

Hardness Amplification of Optimization Problems

In this paper, we prove a general hardness amplification scheme for optimization problems based on the technique of direct products. We say that an optimization problem $Π$ is direct product feasible if it is possible to efficiently aggregate any $k$ instances of $Π$ and form one large instance of $Π$ such that given an optimal feasible solution to the larger instance, we can efficiently find optimal feasible solutions to all the $k$ smaller instances. Given a direct product feasible optimization problem $Π$, our hardness amplification theorem may be informally stated as follows: If there is a distribution $\mathcal{D}$ over instances of $Π$ of size $n$ such that every randomized algorithm running in time $t(n)$ fails to solve $Π$ on $\frac{1}{α(n)}$ fraction of inputs sampled from $\mathcal{D}$, then, assuming some relationships on $α(n)$ and $t(n)$, there is a distribution $\mathcal{D}'$ over instances of $Π$ of size $O(n\cdot α(n))$ such that every randomized algorithm running in time $\frac{t(n)}{poly(α(n))}$ fails to solve $Π$ on $\frac{99}{100}$ fraction of inputs sampled from $\mathcal{D}'$. As a consequence of the above theorem, we show hardness amplification of problems in various classes such as NP-hard problems like Max-Clique, Knapsack, and Max-SAT, problems in P such as Longest Common Subsequence, Edit Distance, Matrix Multiplication, and even problems in TFNP such as Factoring and computing Nash equilibrium.

cs.CC↗

arXiv2019

Towards a General Direct Product Testing Theorem

The Direct Product encoding of a string $a\in \{0,1\}^n$ on an underlying domain $V\subseteq \binom{n}{k}$, is a function DP$_V(a)$ which gets as input a set $S\in V$ and outputs $a$ restricted to $S$. In the Direct Product Testing Problem, we are given a function $F:V\to \{0,1\}^k$, and our goal is to test whether $F$ is close to a direct product encoding, i.e., whether there exists some $a\in \{0,1\}^n$ such that on most sets $S$, we have $F(S)=$DP$_V(a)(S)$. A natural test is as follows: select a pair $(S,S')\in V$ according to some underlying distribution over $V\times V$, query $F$ on this pair, and check for consistency on their intersection. Note that the above distribution may be viewed as a weighted graph over the vertex set $V$ and is referred to as a test graph. The testability of direct products was studied over various specific domains and test graphs (for example see Dinur-Steurer [CCC'14]; Dinur-Kaufman [FOCS'17]). In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem? Towards this goal we introduce the notion of coordinate expansion of a test graph. Roughly speaking a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion then it admits a direct product testing theorem. Additionally, for every $k$ and $n$ we provide a direct product domain $V\subseteq \binom{n}{k}$ of size $n$, called the Sliding Window domain for which we prove direct product testability.

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arXiv2018

On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic

Given a set of $n$ points in $\mathbb R^d$, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the $\ell_p$-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when $d=ω(\log n)$ was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], David-Karthik-Laekhanukit [SoCG'18]). In this paper, we show that for every $p\in\mathbb R_{\ge 1}\cup\{0\}$, under the Strong Exponential Time Hypothesis (SETH), for every $\varepsilon>0$, the following holds: $\bullet$ No algorithm running in time $O(n^{2-\varepsilon})$ can solve the Closest Pair problem in $d=(\log n)^{Ω_{\varepsilon}(1)}$ dimensions in the $\ell_p$-metric. $\bullet$ There exists $δ= δ(\varepsilon)>0$ and $c = c(\varepsilon)\ge 1$ such that no algorithm running in time $O(n^{1.5-\varepsilon})$ can approximate Closest Pair problem to a factor of $(1+δ)$ in $d\ge c\log n$ dimensions in the $\ell_p$-metric. At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on $n$ vertices with $n^{2-\varepsilon}$ edges whose vertices can be realized as points in a $(\log n)^{Ω_\varepsilon(1)}$-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].

cs.CG↗

arXiv2018

On the Complexity of Closest Pair via Polar-Pair of Point-Sets

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $Δ$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $Δ$ is called the sphericity of $G$, and if the collection of spheres are non-overlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.

cs.CG↗

arXiv2018

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 $F(k) \cdot k$ 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 $W[1]\neq FPT$, there is no $F(k)$-FPT-approximation algorithm for $k$-DomSet. $\bullet$ Assuming the Exponential Time Hypothesis (ETH), there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{o(k)}$ time. $\bullet$ Assuming the Strong Exponential Time Hypothesis (SETH), for every integer $k \geq 2$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{k - \varepsilon}$ time. $\bullet$ Assuming the $k$-Sum Hypothesis, for every integer $k \geq 3$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{\lceil k/2 \rceil - \varepsilon}$ time. 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.

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