In the classical survivable network design problem (SNDP), we are given an undirected graph G=(V,E) with costs on edges and a connectivity requirement k(s,t) for each pair of vertices. The goal is to find a minimum-cost subgraph H⊆ G such that every pair (s,t) is connected by k(s,t) edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS’98, Combinatorica’01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an O(k 3 log n )-approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS’09, Theory Comput.’12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This article concerns the generalization of EC-SNDP into the subset-to-subset setting, namely Group EC-SNDP. We develop a framework, which yields the first non-trivial (true) approximation algorithm for Group EC-SNDP. Previously, only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15], and a true approximation algorithm is known only for the single-source variant with connectivity requirement k(S,T) ∈ { 0,1,2} [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor. Comput. Sci.’12]. On the negative side, in terms of the number of connectivity demands q , we give an Ω (q /log q )-hardness result for large k , complementing the previous inapproximability results, e.g., hardness in terms of k : k 1/5-ɛ -hardness [Cheriyan et al., SODA’12; Laekhanukit, SODA’14; Chalermsook et al., SODA’15; Manurangsi, IPL’19]; hardness in terms of n : 2 log 1-ɛ n -hardness [Chalermsook et al., SODA’15].

In the Directed Steiner Tree (DST) problem, we are given a directed graph \(G=(V,E)\) on \(n\) vertices with edge-costs \(c\in{\mathbb{R}}_{\geq 0}^{E}\) , a root vertex \(r\in V\) , and a set \(K\subseteq V\setminus\{r\}\) of \(k\) terminals. The goal is to find a minimum-cost subgraph of \(G\) that contains a path from \(r\) to every terminal \(t\in K\) . DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of \(O(k^{\epsilon})\) for any constant \(\epsilon \gt 0\) , and the best quasi-polynomial-time approximation ratio of \(O\left(\frac{\log^{2}k}{\log\log k}\right)\) . Toward understanding this gap, we study the integrality gap of the standard flow linear programming relaxation for the problem. We show that the linear program (LP) has an integrality gap of \(\Omega(n^{0.0418})\) . Previously, the integrality gap of the LP is only known to be \(\Omega\left(\frac{\log^{2}n}{\log\log n}\right)\) [Halperin et al., SODA’03 & SIAM J. Comput.] and \(\Omega(\sqrt{k})\) [Zosin-Khuller, SODA’02] in some instance with \(\sqrt{k}=O\left(\frac{\log n}{\log\log n}\right)\) . Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in \(n\) , the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

In the classical survivable network design problem (SNDP), we are given an undirected graph $G-(V,E)$ with costs on edges and a connectivity requirement $k(5,t)$ for each pair of vertices. The goal is to find a minimum-cost subgraph $H\sqsubseteq G$ such that every pair $(s,t)$ are connected by $k(s,t)$ edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS’98, Combinatorica’01] gives a 2-approximation algorithm for EC-SNDP, and a decade later, an $O(k^{3}\log n)-$ approximation algorithm for VC-SNDP, where k is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS’09, Theory Comput’12]. While there is a rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This paper concerns the generalization of SNDP into the subset-to-subset setting, namely Group EC-SNDR We develop the framework, which yields the first non-trivial (true) approximation algorithm for Group. EC-SNDE Previously only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA’15l, and a true approximation algorithm is known only for the single-source variant with connectivity requirement $k(S,T)\in\{0,1,2\}$ [Gupta, Krishnaswamy, and Ravi, SODA’10; Khandekar, Kortsarz, and Nutov, FSTTCS’09 and Theor Comput. Sci’12].

In the k-outconnected directed Steiner tree problem (k-DST), we are given an n-vertex directed graph G = (V,E) with edge costs, a connectivity requirement k, a root r ∈ V and a set of terminals T ⊆ V. The goal is to find a minimum-cost subgraph H ⊆ G that has k edge-disjoint paths from the root vertex r to every terminal t ∈ T. The problem is NP-hard, and inapproximability results are known in several parameters, e.g., hardness in terms of n: log^{2-ε}n-hardness for k = 1 [Halperin and Krauthgamer, STOC'03], 2^{log^{1-ε}n}-hardness for general case [Cheriyan, Laekhanukit, Naves and Vetta, SODA'12], hardness in terms of k [Cheriyan et al., SODA'12; Laekhanukit, SODA'14; Manurangsi, IPL'19] and hardness in terms of |T| [Laekhanukit, SODA'14]. In this paper, we show the approximation hardness of k-DST for various parameters. - Ω(|T|/log |T|)-approximation hardness, which holds under the standard complexity assumption NP≠ ZPP. The inapproximability ratio is tightened to Ω(|T|) under the Strongish Planted Clique Hypothesis [Manurangsi, Rubinstein and Schramm, ITCS 2021]. The latter hardness result matches the approximation ratio of |T| obtained by a trivial approximation algorithm, thus closing the long-standing open problem. - Ω(2^{k/2} / k)-approximation hardness for the general case of k-DST under the assumption NP≠ZPP. This is the first hardness result known for survivable network design problems with an inapproximability ratio exponential in k. - Ω((k/L)^{L/4})-approximation hardness for k-DST on L-layered graphs for L ≤ O(log n). This almost matches the approximation ratio of O(k^{L-1}⋅ L ⋅ log |T|) achieved in O(n^L)-time due to Laekhanukit [ICALP'16]. We further extend our hardness results in terms of |T| to the undirected cases of k-DST, namely the single-source k-vertex-connected Steiner tree and the k-edge-connected group Steiner tree problems. Thus, we obtain Ω(|T|/log |T|) and Ω(|T|) approximation hardness for both problems under the assumption NP≠ ZPP and the Strongish Planted Clique Hypothesis, respectively. This again matches the upper bound obtained by trivial algorithms.

In the Directed Steiner Tree (DST) problem, we are given a directed graph G = (V, E) on n vertices with edge-costs , a root vertex r, and a set K of k terminals. The goal is to find a minimum-cost subgraph of G that contains a path from r to every terminal t ∊ k. DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of O(k∊) for any constant ∊ > 0, and the best quasi-polynomial-time approximation ratio of . Towards understanding this gap, we study the integrality gap of the standard flow LP relaxation for the problem. We show that the LP has an integrality gap polynomial in n. Previously, the integrality gap LP is only known to be [Halperin et al., SODA'03 & SIAM J. Comput.] and [Zosin-Khuller, SODA'02] in some instance with . Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in n, the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $$G=(V, E)$$ with edge costs $$c \in {\mathbb {R}}_{\ge 0}^E$$ , a root $$r \in V$$ and k terminals $$K\subseteq V$$ , we need to output the minimum-cost arborescence in G that contains an $$r \rightarrow t$$ path for every $$t \in K$$ . Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time $$O(\log ^2k/\log \log k)$$ -approximation Algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound $$d_v$$ on each vertex $$v \in V$$ , and we require that every vertex v in the output tree has at most $$d_v$$ children. We give a quasi-polynomial time $$(O(\log n \log k), O(\log ^2 n))$$ -bicriteria approximation: The Algorithm produces a solution with cost at most $$O(\log n\log k)$$ times the cost of the optimum solution that violates the degree constraints by at most a factor of $$O(\log ^2n)$$ . This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of $$O(\log ^2n)$$ is an $$O(\log n)$$ -factor away from the approximation lower bound of $$\Omega (\log n)$$ from the set-cover hardness. The hardness result holds even on the special case of the Degree-Bounded Group Steiner Tree problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an $$(O(\log n\log k), O(\log n))$$ -bicriteria approximation Algorithm for DB-GST-T.

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether (1 + ∊)-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter c, find a smaller graph, which we call connectivity-c mimicking network, which preserves connectivity among k terminals exactly up to the value of c. We show that connectivity-c mimicking networks with O(kc4) edges exist and can be found in time m(c log n)O(c). We also give a separate algorithm that constructs such graphs with k · O(c)2c edges in time mcO(c) logO(1) n. These results lead to the first data structures for answering fully dynamic offline c-edge-connectivity queries for c ≥ 4 in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

In the classic Directed Steiner Tree problem (DST), we are given an edge-weighted directed graph G = (V,E) with n nodes, a specified root node r ∈ V, and k terminals X ⊆ V-{r}. The goal is to find the cheapest F ⊆ E such that r can reach any terminal using only edges in F. Designing approximation algorithms for DST is quite challenging, to date the best approximation guarantee of a polynomial-time algorithm for DST is O(k^ε) for any constant ε > 0 [Charikar et al., 1999]. For network design problems like DST, one often relies on natural cut-based linear programming (LP) relaxations to design approximation algorithms. In general, the integrality gap of such an LP for DST is known to have a polynomial integrality gap lower bound [Zosin and Khuller, 2002; Li and Laekhanukit, 2021]. So particular interest has been invested in special cases or in strengthenings of this LP. In this work, we show the integrality gap is only O(log k) for instances of DST where no Steiner node has both an edge from another Steiner node and an edge to another Steiner node, i.e. the longest path using only Steiner nodes has length at most 1. This generalizes the well-studied case of quasi-bipartite DST where no edge has both endpoints being Steiner nodes. Our result is also optimal in the sense that the integrality gap can be as bad as poly(n) even if the longest path with only Steiner nodes has length 2.

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable (FPT) algorithms. The questions, which have been asked several times, are whether there is a nontrivial FPT-approximation algorithm for the Maximum Clique $({\sf Clique})$ and Minimum Dominating Set $({\sf DomSet})$ problems parameterized by the size of the optimal solution. In particular, letting ${\sf OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t({\sf OPT}){\operatorname{poly}}(N)$ time and outputs a solution of size $f({\sf OPT})$ for any computable functions $t$ and $f$ that are independent of $N$ (for ${\sf Clique}$, we want $f({\sf OPT})=\omega(1)$)? In this paper, we show that both ${\sf Clique}$ and ${\sf DomSet}$ admit no nontrivial FPT-approximation algorithm, i.e., there is no $o({\sf OPT})$-FPT-approximation algorithm for ${\sf Clique}$ and no $f({\sf OPT})$-FPT-approximation algorithm for ${\sf DomSet}$ for any function $f$. In fact, our results imply something even stronger: The best way to solve ${\sf Clique}$ and ${\sf DomSet}$, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis [I. Dinur. ECCC, TR16-128, 2016; P. Manurangsi and P. Raghavendra, preprint, arXiv:1607.02986, 2016], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3 \sf SAT formula and one which is not even $(1 - \varepsilon)$-satisfiable for some constant $\varepsilon > 0$. Besides ${\sf Clique}$ and ${\sf DomSet}$, we also rule out nontrivial FPT-approximation for the Maximum Biclique problem, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs, and we rule out the $k^{o(1)}$-FPT-approximation algorithm for the Densest $k$-Subgraph problem.

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA’16 8 J.ACM’17, Bodwin-Williams SODA’16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O ( n 3/5 + ε )-approximation for distance preservers and pairwise spanners (for arbitrary constant ε > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O (log n )-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O (1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O ( n 3/5 + ε )-approximation for the Directed Steiner Forest problem (for arbitrary constant ε > 0) when all edges have uniform costs, improving the previous best O ( n 2/3 + ε )-approximation due to Berman et al. [ICALP’11] (which holds for general edge costs).

In the Directed Steiner Tree (DST) problem we are given an n-vertex directed edge-weighted graph, a root r , and a collection of k terminal nodes. Our goal is to find a minimum-cost subgraph that contains a directed path from r to every terminal. We present an O(log^2 k /log log k )-approximation algorithm for DST that runs in quasi-polynomial-time, i.e., in time n^polylog(k). By making standard complexity assumptions, we show the matching lower bound of Omega(log^2 k/loglogk) for the class of quasi-polynomial time algorithms, meaning that our approximation ratio is asymptotically the best possible. This is the first improvement on the DST problem since the classical quasi-polynomial-time O (log^3 k ) approximation algorithm by Charikar et al. [SODA’98J. Algorithms’99]. (The paper erroneously claims an O (log^2 k ) approximation due to a mistake in prior work.)

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer $$r>1$$ , and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of (Throughout, $${\tilde{O}}$$ and $$O^*$$ omit $$\hbox {polyloglog} (r)$$ and factors polynomial in the input size, respectively.) The best known time bounds for all problems were $$O^*(2^{n/r})$$ (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by $$\exp (n^{1-o(1)}/r^{1+o(1)})$$ lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking $$O^*(2^{n/r})$$ bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016).

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

In the directed Steiner tree (DST) problem, we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost subgraph that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial time, i.e., in time $n^{{poly}\log (k)}$. By assuming the projection game conjecture and ${NP}\not\subseteq{\bigcap}_{0<\epsilon<1}{ZPTIME}(2^{n^\epsilon})$ and adjusting the parameters in the hardness result of [Halperin and Krauthgamer, Polylogarithmic inapproximability, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 585--594], we show the matching lower bound of $\Omega(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial time algorithms, meaning that our approximation ratio is asymptotically the best possible. Our algorithm is proceeded by reducing DST to an intermediate problem, namely, the group Steiner tree on trees with dependency constraint problem, which we approximate using the framework developed by [Rothvoß, Directed Steiner Tree and the Lasserre Hierarchy, preprint, arxiv:1111.5473, 2011] and [Friggstad et al., Linear programming hierarchies suffice for directed Steiner tree, in Proceedings of the 17th Annual Conference on Integer Programming and Combinatorial Optimization, 2014, pp. 285--296].

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ 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 $\Delta$ is called the sphericity of $G$, and if the collection of spheres are nonoverlapping, 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.

In the Group Steiner Tree problem (GST), we are given a (vertex or edge)-weighted graph $G=(V,E)$ on $n$ vertices, a root vertex $r$ and a collection of groups $\{S_i\}_{i\in[h]}: S_i\subseteq V(G)$. The goal is to find a min-cost subgraph $H$ that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group $S_i$ has a demand $k_i\in[k],k\in\mathbb N$, and we wish to find a min-cost $H\subseteq G$ such that, for each group $S_i$, there is a vertex in $S_i$ connected to the root via $k_i$ (vertex or edge) disjoint paths. While GST admits $O(\log^2 n\log h)$ approximation, its high connectivity variants are Label-Cover hard, and for the vertex-weighted version, the hardness holds even when $k=2$. Previously, positive results were known only for the edge-weighted version when $k=2$ [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where the disjoint paths may end at different vertices in a group [Chalermsook et al., SODA 2015]. Our main result is an $O(\log n\log h)$ approximation for Restricted Group SNDP that runs in time $n^{f(k, w)}$, where $w$ is the treewidth of $G$. This nearly matches the lower bound when $k$ and $w$ are constant. The key to achieving this result is a non-trivial extension of the framework in [Chalermsook et al., SODA 2017], which embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.

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.

We consider questions that arise from the intersection between the areas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [1], [2], [3]) are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting OPT be the optimum and N be the size of the input, is there an algorithm that runs in t(OPT) poly(N) time and outputs a solution of size f(OPT), for any functions t and f that are independent of N (for Clique, we want f(OPT) = ω(1))? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(OPT)-FPT-approximation algorithm for Clique and no f(OPT)-FPT-approximation algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (GapETH) [4], [5], which states that no 2o(n)-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - ε)-satisfiable for some constant ε > 0. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be W[1]-hard [6], [7], [8]. Additionally, we rule out ko(1)-FPT-approximation algorithm for Densest k-Subgraph although this ratio does not yet match the trivial O(k)-approximation algorithm. To the best of our knowledge, prior results only rule out constant factor approximation for Clique [9], [10] and log1/4+ε(OPT) approximation for DomSet for any constant ε > 0 [11]. Our result on Clique significantly improves on [9], [10]. However, our result on DomSet is incomparable to [11] since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption.

The Group Steiner Tree (GST) problem is a classical problem in combinatorial optimization and theoretical computer science. In the Edge-Weighted Group Steiner Tree (EW-GST) problem, we are given an undirected graph G = (V, E) on n vertices with edge costs c : E → ℝ≥0, a source vertex s and a collection of subsets of vertices, called groups, S1,…, Sk ⊆ V. The goal is to find a minimum-cost tree H ⊆ G that connects s to some vertex from each group Si, for all i = 1, 2,…, k. The Node-Weighted Group Steiner Tree (NW-GST) problem has the same setting, but the costs are associated with nodes. The goal is to find a minimum- cost node set X ⊆ V such that G[X] connects every group to the source. When G is a tree, both EW-GST and NW-GST admit a polynomial-time O (log n log k) approximation algorithm due to the seminal result of [Garg et al., SODA'98 and J. Algorithm]. The matching hardness of log2 −∊ n is known even for tree instances of EW-GST and NW-GST [Halperin and Krauthgamer STOC'03]. In general graphs, most of polynomial-time approximation algorithms for EW- GST reduce the problem to a tree instance using the metric- tree embedding, incurring a loss of O(log n) on the approximation factor [Bartal, FOCS'96; Fakcharoenphol et al., FOCS'03 and JCSS]. This yields an approximation ratio of O(log n log k) for EW-GST. Using metric-tree embedding, this factor cannot be improved: The loss of O(log n) is necessary on some input graphs (e.g., grids and expanders). There are alternative approaches that avoid metric-tree embedding, e.g., the algorithm of [Chekuri and Pal, FOCS'05], which gives a tight approximation ratio, but none of which achieves polylogarithmic approximation in polynomial-time. This state of the art shows a clear lack of understanding of GST in general graphs beyond the metric-tree embedding technique. For NW-GST (for which the metric-tree embedding does not apply), not even a polynomial-time polyloga- rithmic approximation algorithm is known. In this paper, we present O(log n log k) approximation algorithms that run in time nÕ(tw(G)2 ‘for both NW-GST and EW-GST1, where tw(G) denotes the treewidth of graph G. The key to both results is a different type of “tree- embedding” that produces a tree of much bigger size, but does not cause any loss on the approximation factor. Our embedding is inspired by dynamic programming, a technique which is typically not applicable to Group Steiner problems.

It was recently found that there are very close connectionsbetween the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA ‘16, Bodwin-Williams SODA ‘16]. We study these problemsfrom an optimization point of view, where ratherthan studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O(n3/5+∊)-approximation for distance preservers and pairwisespanners (for arbitrary constant ∊ > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O(logn)-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O(1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O(n3/5+∊)- approximation for the Directed Steiner Forest problem (for arbitrary constant ∊ > 0) when all edges have uniform costs, improving the previous best O(n2/3+∊)- approximation due to Berman et al. [ICALP ‘11] (whichholds for general edge costs).

Real-word networks are often prone to failures. A reliable network needs to cope with this situation and must provide a backup communication channel. This motivates the study of survivable network design, which has been a focus of research for a few decades. To date, survivable network design problems on undirected graphs are well-understood. For example, there is a 2 approximation in the case of edge failures [Jain, FOCS'98/Combinatorica'01]. The problems on directed graphs, in contrast, have seen very little progress. Most techniques for the undirected case like primal-dual and iterative rounding methods do not seem to extend to the directed case. Almost no non-trivial approximation algorithm is known even for a simple case where we wish to design a network that tolerates a single failure.

Directed Steiner problems are fundamental problems in Combinatorial Optimization and Theoretical Computer Science. An important problem in this genre is the k-edge connected directed Steiner tree (k-DST) problem. In this problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H subseteq G that connects r to each terminal t in T by k edge-disjoint r, t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k=1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14], by Laekhanukit [SODA'14] and in a survey by Kortsarz and Nutov [Handbook of Approximation Algorithms and Metaheuristics], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in a special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k=2 and h=2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D*k^{D-1}*log(n))-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r, t-paths, each of length at most D, for every terminal t in T. For the case k=1 (DST), our algorithm yields an approximation ratio of O(D*log(h)), thus implying an O(log^3(h))-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Our algorithm is based on an LP-formulation that allows us to embed a solution to a tree-instance of GST, which does not preserve connectivity. We show, however, that one can randomly extract a solution of k-DST from the tree-instance of GST. Our algorithm is almost tight when k and D are constants since the case that k=1 and D=3 is NP-hard to approximate to within a factor of O(log(h)), and our algorithm archives the same approximation ratio for this special case. We also remark that the k^{1/4-epsilon}-hardness instance of k-DST is a DAG with 6 layers, and our algorithm gives O(k^5*log(n))-approximation for this special case. Consequently, as our algorithm works for general graphs, we obtain an O(D*k^{D-1}*log(n))-approximation algorithm for a D-shallow instance of the k edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k edgedisjoint paths.

In the Set Connectivity problem, we are given an n-node edge-weighted undirected graph and a collection of h set pairs (Si, Ti), where Si and Ti are subsets of nodes. The goal is to compute a min-cost subgraph H so that, for each set pair (Si, Ti), there exists at least one path in H between some node in Si and some node in Ti. In this paper, we initiate the study of the Survivable Set Connectivity problem (SSC), i.e., the generalization of Set Connectivity where we are additionally given an integer requirement ki ≥ 1 for each set pair (Si, Ti), and we want to find a min-cost subgraph H so that there are at least ki edge-disjoint paths in H between Si and Ti. We achieve the following main results: We show that there is no poly-logarithmic approximation for SSC unless NP has a quasi-polynomial time algorithm. This result is based on a reduction from the Minimum Label Cover problem, and the result holds even for the special case where Si = {r} for all i, i.e., for the high-connectivity variant of the classical Group Steiner Tree problem. More precisely, we prove an approximability lower bound of 2log1−ε n for SSC, for any constant ∊ > 0, which is almost polynomial on n. A technical novelty of our proof is the first use of a padding scheme technique for an edge-connectivity problem on undirected graphs. (Prior to our results, the applications of this technique only pertain to either node-connectivity problems or problems on directed graphs). We present a bicriteria approximation algorithm for SSC that computes a solution H of cost at most poly-logarithmically larger than the optimal cost and provides a connectivity at least Ω(ki/log n) for each set pair (Si,Ti). The main algorithmic idea is to solve a standard LP relaxation to the problem, and then embed the resulting fractional capacities into a tree via Räcke's cut-based tree embeddings. Based on that, we generate a random collection of Group Steiner Tree-like fractional solutions, which can then be handled by the rounding scheme of [Garg, Konjevod and Ravi – SODA'98]. The prior work on Set Connectivity and Group Steiner Tree used Bartal's distance-based tree embeddings which do not seem to generalize to the k-connectivity versions of these problems. Finally, we remark an interesting contrast demonstrated by our results: While our hardness result almost rules out “polynomial” approximation ratios, relaxing connectivity constraints allows us to obtain “poly-logarithmic” bounds. This naturally suggests that relaxing connectivity requirements might be a proper way in getting big improvements, even beyond (non-bicriteria) lower bounds, for other connectivity problems, especially those whose approximability lower bounds are derived from Minimum Label Cover.

The study of graph products is a major research topic and typically concerns the term f(G * H), e.g., to show that f(G * H) = f(G)f(H). In this paper, we study graph products in a non-standard form f(R[G * H]) where R is a “reduction”, a transformation of any graph into an instance of an intended optimization problem. We resolve some open problems as applications. The first problem is minimum consistent deterministic finite automaton (DFA). We show a tight n1-ϵapproximation hardness, improving the n1/14-ϵhardness of [Pitt and Warmuth, STOC 1989 and JACM 1993], where n is the sample size. (In fact, we also give improved hardnesses for the case of acyclic DFA and NFA.) Due to Board and Pitt [Theoretical Computer Science 1992], this implies the hardness of properly learning DFAs assuming NP ≠ RP (the weakest possible assumption). This affirmatively answers an open problem raised 25 years ago in the paper of Pitt and Warmuth and the survey of Pitt [All 1989]. Prior to our results, this hardness only follows from the stronger hardness of improperly learning DFAs, which requires stronger assumptions, i.e., either a cryptographic or an average case complexity assumption [Kearns and Valiant STOC 1989 and J. ACM 1994; Daniely et al. STOC 2014]. The second problem is edge-disjoint paths (EDP) on directed acyclic graphs (DAGs). This problem admits an O(√n)-approximation algorithm [Chekuri, Khanna, and Shepherd, Theory of Computing 2006] and a matching Ω(√n) integrality gap, but so far only an n1/26-ϵhardness factor is known [Chuzhoy et al., STOC 2007]. (n denotes the number of vertices.) Our techniques give a tight n1/2-ϵhardness for EDP on DAGs, thus resolving its approximability status. As by-products of our techniques: (i) We give a tight hardness of packing vertex-disjoint k-cycles for large k, complimenting [Guruswami and Lee, ECCC 2014] and matching [Krivelevich et al., SODA 2005 and ACM Transactions on Algorithms 2007]. (ii) We give an alternative (and perhaps simpler) proof for the hardness of properly learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004 and J. Comput.Syst. Sci. 2008]. Our new concept reduces the task of proving hardnesses to merely analyzing graph product inequalities, which are often as simple as textbook exercises. This concept was inspired by, and can be viewed as a generalization of, the graph product subadditivity technique we previously introduced in SODA 2013. This more general concept might be useful in proving other hardness results as well.

Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important task in PCPs literature as it would imply a smaller PCP with the same or stronger soundness. While this is a basic question in PCPs community, the connection between the parameters of PCPs and hardness of approximations is sometimes obscure to approximation algorithm community. In this paper, we investigate the connection between the parameters of 2P1R and the hardness of approximating the class of so-called connectivity problems, which includes as subclasses the survivable network design and (multi)cut problems. Based on recent development on 2P1R by Chan (STOC 2013) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form kσ, for some (very) small constant σ > 0, to hardness results of the form kc for some explicit constant c, where k is a connectivity parameter. In addition, we show how to convert these hardness into hardness results of the form Dc, where D is the number of demand pairs (or the number of terminals). Our results are as follows.

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques, we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω( k ϵ ) hardness bound for the rooted k -connectivity problem in undirected graphs. As a consequence, we obtain an Ω( k ϵ ) hardness bound for the undirected subset k -connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k -connectivity problem.

We consider the problem of finding semi-matching in bipartite graphs, which is also extensively studied under various names in the scheduling literature. We give faster algorithms for both weighted and unweighted cases. For the weighted case, we give an O ( nm log n )-time algorithm, where n is the number of vertices and m is the number of edges, by exploiting the geometric structure of the problem. This improves the classical O ( n 3 )-time algorithms by Horn [1973] and Bruno et al. [1974b]. For the unweighted case, the bound can be improved even further. We give a simple divide-and-conquer algorithm that runs in O (√ nm log n ) time, improving two previous O ( nm )-time algorithms by Abraham [2003] and Harvey et al. [2003, 2006]. We also extend this algorithm to solve the Balanced Edge Cover problem in O (√ nm log n ) time, improving the previous O ( nm )-time algorithm by Harada et al. [2008].

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the bipartite induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: For any r larger than some constant, any r-approximation algorithm for the independent set problem must run in at least 2n1-ε/r1+εtime. This nearly matches the upper bound of 2n/r[23]. It also improves some hardness results in the domain of parameterized complexity (e.g., [26], [19]). For any k larger than some constant, there is no polynomial time min{k1-ε, n1/2-ε} time min -approximation algorithm for the k-hypergraph pricing problem , where n is the number of vertices in an input graph. This almost matches the upper bound of min{O(k), Õ(√n) } min (by Balcan and Blum [3] and an algorithm in this paper). We note an interesting fact that, in contrast to n1/2-εhardness for polynomial-time algorithms, the k-hypergraph pricing problem admits nδapproximation for any δ > 0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time. The proofs of our hardness results rely on unexpectedly tight connections between the three problems. First, we establish a connection between the first and second problems by proving a new graph-theoretic property related to an induced matching number of dispersers. Then, we show that the n1/2-εhardness of the last problem follows from nearly tight subexponential time inapproximability of the first problem, illustrating a rare application of the second type of inapproximability result to the first one. Finally, to prove the subexponential-time inapproximability of the first problem, we construct a new PCP with several properties; it is sparse and has nearly-linear size, large degree, and small free-bit complexity. Our PCP requires no ground-breaking ideas but rather a very careful assembly of the existing ingredients in the PCP literature.

Graph product is a fundamental tool with rich applications in both graph theory and theoretical computer science. It is usually studied in the form f(G * H) where G and H are graphs, * is a graph product and f is a graph property. For example, if f is the independence number and * is the disjunctive product, then the product is known to be multiplicative: f(G * H) = f(G)f(H). In this paper, we study graph products in the following non-standard form: f((G⊕H)*J) where G, H and J are graphs, ⊕ and * are two different graph products and f is a graph property. We show that if f is the induced and semi-induced matching number, then for some products ⊕ and *, it is subadditive in the sense that f((G ⊕ H) * J) ≤ f(G * J) + f(H * J). Moreover, when f is the poset dimension number, it is almost subadditive. As applications of this result (we only need J = K2 here), we obtain tight hardness of approximation for various problems in discrete mathematics and computer science: bipartite induced and semi-induced matching (a.k.a. maximum expanding sequences), poset dimension, maximum feasible subsystem with 0/1 coefficients, unit-demand min-buying and single-minded pricing, donation center location, boxicity, cubicity threshold dimension and independent packing.

In the minimum-cost $k\hbox{-}(S,T)$ connected digraph (abbreviated as $k\hbox{-}(S,T)$ connectivity) problem we are given a positive integer $k$, a directed graph $G=(V,E)$ with nonnegative costs on the edges, and two subsets $S,T$ of $V$; the goal is to find a subset of edges $\widehat{E}$ of minimum cost such that the subgraph $(V,\widehat{E})$ has $k$ edge-disjoint directed paths from each vertex in $S$ to each vertex in $T$. Most of our results focus on a specialized version of the problem that we call the standard version, where every edge of positive cost has its tail in $S$ and its head in $T$. This version of the problem captures NP-hard problems such as the minimum-cost $k$-vertex connected spanning subgraph problem. We give an approximation algorithm with a guarantee of $O((\log{k})(\log{n}))$ for the standard version of the $k\hbox{-}(S,T)$ connectivity problem, where $n$ denotes the number of vertices. For $k=1$, we give a simple 2-approximation algorithm that generalizes a well-known 2-approximation algorithm for the minimum-cost strongly connected spanning subgraph problem. For $k=2$, we give a 3-approximation algorithm; this matches the best approximation guarantee known for the special case of the minimum-cost $2$-vertex connected spanning subgraph problem. Besides the standard version, we study another version that is intermediate between the standard version and the problem in its full generality. In the relaxed version of the $(S,T)$ connectivity problem, each edge of positive cost has its head in $T$ but there is no restriction on the tail. We study the relaxed version with the connectivity parameter $k$ fixed at one and observe that this version is at least as hard to approximate as the directed Steiner tree problem. We match this by giving an algorithm that achieves an approximation guarantee of $\alpha(n)+1$ for the relaxed $(S,T)$ connectivity problem, where $\alpha(n)$ denotes the best approximation guarantee available for the directed Steiner tree problem. The key to the analysis is a structural result that decomposes any feasible solution into a set of so-called junction trees that are disjoint on the vertices of $T$. Our algorithm and analysis specialize to the case when the input digraph is acyclic on $T$, meaning that there exists no dicycle that contains two distinct vertices of $T$. In this setting, we show that the relaxed $(S,T)$ connectivity problem is at least as hard to approximate as the set covering problem, and we prove that our algorithm achieves a matching approximation guarantee of $O(\log{|S|})$.

The Directed Steiner Tree (DST) problem is a cornerstone problem in network design. We focus on the generalization of the problem with higher connectivity requirements. The problem with one root and two sinks is APX-hard. The problem with one root and many sinks is as hard to approximate as the directed Steiner forest problem, and the latter is well known to be as hard to approximate as the label cover problem. Utilizing previous techniques (due to others), we strengthen these results and extend them to undirected graphs. Specifically, we give an Ω(k∊) hardness bound for the rooted k-connectivity problem in undirected graphs; this addresses a recent open question of Khanna. As a consequence, we also obtain the Ω(k∊) hardness of the undirected subset k-connectivity problem. Additionally, we give a result on the integrality ratio of the natural linear programming relaxation of the directed rooted k-connectivity problem.

We present an $O(\log^2{k})$-approximation algorithm for the problem of finding a $k$-vertex connected spanning subgraph of minimum cost, where $n$ is the number of vertices in an input graph, and $k$ is a connectivity requirement. Our algorithm is the first that achieves a polylogarithmic approximation ratio for all values of $k$ and $n$, and it works for both directed and undirected graphs. As in previous works, we use the Frank--Tardos algorithm for finding $k$-outconnected subgraphs as a subroutine. However, with our structural lemmas, we are able to show that we need only partial solutions returned by the Frank--Tardos algorithm; thus, we can avoid paying the whole cost of an optimal solution every time the algorithm is applied.

We present an O(log n• log k)-approximation algorithm for the problem of finding k-vertex connected spanning subgraph of minimum cost, where n is the number of vertices in the input graph, and k is the connectivity requirement. Our algorithm works for both directed and undirected graphs. The best known approximation guarantees for these problems are O(ln k• min{√k,n/n-k ln k}) by Kortsarz and Nutov, and O(ln{k}) in the case of undirected graphs where n≥ 6k2 by Cheriyan, Vempala, and Vetta. Our algorithm is the first that has a polylogarithmic guarantee for all values of k. Combining our algorithm with the algorithm of Kortsarz and Nutov in case of small k, e.g., k