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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 $H\subseteq V$ 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 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-SNDP. We develop the 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 $k(S,T)\in\{0,1,2\}$ [Gupta, Krishnaswamy, and Ravi, SODA'10; Khandekar, Kortsarz, and Nutov, FSTTCS'09 and Theor. Comput. Sci.'12]., Comment: 22 pages, 2 figures

Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints. The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation. Keywords: 2-node connected graphs, approximation algorithms, connectivity augmentation, greedy algorithm, network design, partition relaxation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and a parameter $\alpha>1$, and the goal is to design an $\alpha$-approximation algorithm with the fastest possible running time. We show the following results: - An $r$-approximation for maximum independent set in $O^*(\exp(\tilde O(n/r \log^2 r+r\log^2r)))$ time, - An $r$-approximation for chromatic number in $O^*(\exp(\tilde{O}(n/r \log r+r\log^2r)))$ time, - A $(2-1/r)$-approximation for minimum vertex cover in $O^*(\exp(n/r^{\Omega(r)}))$ time, and - A $(k-1/r)$-approximation for minimum $k$-hypergraph vertex cover in $O^*(\exp(n/(kr)^{\Omega(kr)}))$ time. (Throughout, $\tilde O$ and $O^*$ omit $\mathrm{polyloglog}(r)$ and factors polynomial in the input size, respectively.) The best known time bounds for all problems were $O^*(2^{n/r})$ [Bourgeois et al. 2009, 2011 & Cygan et al. 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., 2013 & Laekhanukit, 2014 (Ph.D. Thesis)]. Our results show that the naturally-looking $O^*(2^{n/r})$ bounds are not tight for all these problems. The key to these algorithmic results is a sparsification procedure, allowing the use of better approximation algorithms for bounded degree graphs. For obtaining 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, 2016]. It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture [Dinur 2016 & Manurangsi and Raghavendra, 2016]., Comment: 13 pages

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 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., Comment: The paper was previously titled, "The Curse of Medium Dimension for Geometric Problems in Almost Every Norm"

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), 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 $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=\omega(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is 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 (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - \epsilon)$-satisfiable for some constant $\epsilon > 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm., Comment: 43 pages. To appear in FOCS'17

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-Godwin SODA '16, Godwin-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 $O(n^{3/5 + \epsilon})$-approximation for distance preservers and pairwise spanners (for arbitrary constant $\epsilon > 0$). 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 $O(n^{3/5 + \epsilon})$-approximation for the Directed Steiner Forest problem (for arbitrary constant $\epsilon > 0$) when all edges have uniform costs, improving the previous best $O(n^{2/3 + \epsilon})$-approximation due to Berman et al.~[ICALP '11] (which holds for general edge costs)., Comment: 33 pages

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 n^{1 \eps} approximation hardness improving the n^{1/14 \eps} hardness 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 \neq 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(\sqrt{n}) approximation algorithm [Chekuri Khanna and Shepherd Theory of Computing 2006] and a matching \Omega(\sqrt{n}) integrality gap but so far only an n^{1/26 \eps} hardness factor is known [Chuzhoy et al. STOC 2007]. (n denotes the number of vertices.) Our techniques give a tight n^{1/2 \eps} hardness 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 sometime 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 (ECCC 2011) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form $k^\sigma$, for some (very) small constant $\sigma>0$, to hardness results of the form $k^c$ 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 $D^{c'}$, where $D$ is the number of demand pairs (or the number of terminals). Thus, we give improved hardness results of k^{1/2-\epsilon} and k^{1/10-\epsilon} for the root $k$-connectivity problem on directed and undirected graphs, k^{1/6-\epsilon} for the vertex-connectivity survivable network design problem on undirected graphs, and k^{1/6-\epsilon} for the vertex-connectivity $k$-route cut problem on undirected graphs.

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: $f((G\oplus H)*J)$ where $G$, $H$ and $J$ are graphs, $\oplus$ 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 $\oplus$ and *, it is subadditive in the sense that $f((G\oplus H)*J)\leq f(G*J)+f(H*J)$. Moreover, when $f$ is the poset dimension number, it is almost subadditive. As applications of this result (we only need $J=K_2$ 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., Comment: Preliminary version is published in SODA 2013

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 maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum 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 maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-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), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >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., Comment: The full version of FOCS 2013

We examine the effectiveness of packet routing in this model for the broad class next-hop preferences with filtering. Here each node v has a filtering list D(v) consisting of nodes it does not want its packets to route through. Acceptable paths (those that avoid nodes in the filtering list) are ranked according to the next-hop, that is, the neighbour of v that the path begins with. On the negative side, we present a strong inapproximability result. For filtering lists of cardinality at most one, given a network in which an equilibrium is guaranteed to exist, it is NP-hard to approximate the maximum number of packets that can be routed to within a factor of O(n^{1-\epsilon}), for any constant \epsilon >0. On the positive side, we give algorithms to show that in two fundamental cases every packet will eventually route with probability one. The first case is when each node's filtering list contains only itself, that is, D(v)={v}. Moreover, with positive probability every packet will be routed before the control plane reaches an equilibrium. The second case is when all the filtering lists are empty, that is, $\mathcal{D}(v)=\emptyset$. Thus, with probability one packets will route even when the nodes don't care if their packets cycle! Furthermore, with probability one every packet will route even when the control plane has em no equilibrium at all., Comment: ESA 2012

We consider the problem of finding \textit{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 case. 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 $O(n^3)$ algorithms by Horn [Operations Research 1973] and Bruno, Coffman and Sethi [Communications of the ACM 1974]. For the unweighted case, the bound could be improved even further. We give a simple divide-and-conquer algorithm which runs in $O(\sqrt{n}m\log n)$ time, improving two previous $O(nm)$-time algorithms by Abraham [MSc thesis, University of Glasgow 2003] and Harvey, Ladner, Lov\'asz and Tamir [WADS 2003 and Journal of Algorithms 2006]. We also extend this algorithm to solve the \textit{Balance Edge Cover} problem in $O(\sqrt{n}m\log n)$ time, improving the previous $O(nm)$-time algorithm by Harada, Ono, Sadakane and Yamashita [ISAAC 2008]., Comment: ICALP 2010

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.