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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

openaire: EC/H2020/759557/EU//ALGOCom We study the computational complexity of the non-dominated sorting problem (NDS): Given a set P of n points in Rm, for each point p ∈ P, compute ℓ, the length of longest domination chain p1 > p2 > ··· > pℓ = p, where x dominates y (denoted as x > y) if x is not larger than y in every coordinate. A special case of NDS, which we label as NDS1, is to find all the non-dominated points in P. NDS has emerged as a critical component for multi-objective optimization problems (MOPs). For m ≤ 3, Θ(n log n)-time is known. For a fixed small m > 3, the best bound is O(n logm-2 n log log n). For larger m, the best result is an O(mn2)-time algorithm. We show that the O(mn2) running time is nearly optimal by proving an almost matching conditional lower bound: for any ∈ > 0, and ω(log n) ≤ m ≤ (log n)o(1), there is no O(mn2-ϵ)-time algorithm for NDS or NDS1 unless a popular conjecture in fine-grained complexity theory is false. To complete our results, we present an algorithm for NDS with an expected running time O(mn + n2/m + n log2 n) on uniform random inputs.

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....

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

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.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Group Steiner Tree on Trees with Dependency Constraint (GSTTD) problem. Compared to the classic Group Steiner Tree on Trees problem, in GSTTD we are additionally given some dependency constraints among the nodes in the output tree that must be satisfied. The GSTTD instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky’s heigh-reduction theorem [Algorithmica’97] used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve GSTTD instances, which is inspired by the framework developed by [Rothvob, Preprint’11; Friggstad et al., IPCO’14]. We consider a Sherali-Adams lifting of a proper LP relaxation of GSTTD. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. The limited height of the tree and small number of labels on root-to-leaf paths guarantee that a small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves. We believe that our basic strategy of combining label-based reductions with a round-and-condition type of LP-rounding over hierarchies might find applications to other related problems.

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

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. In this paper, we study a survivable network design problem on directed graphs, 2-Connected Directed Steiner Tree (2-DST): given an n-vertex weighted directed graph, a root r, and a set of h terminals S, find a min-cost subgraph H that has two edge/vertex disjoint paths from r to any te S. 2-DST is a natural generalization of the classical Directed Steiner Tree problem (DST), where we have an additional requirement that the network must tolerate one failure. No non-trivial approximation is known for 2-DST. This was left as an open problem by Feldman et al., [SODA'09; JCSS] and has then been studied by Cheriyan et al. [SODA'12; TALG] and Laekhanukit [SODA'14]. However, no positive result was known except for the special case of a D-shallow instance [Laekhanukit, ICALP'16]. We present an O(D3logD#183; h2/D#183; logn) approximation algorithm for 2-DST that runs in time O(nO(D)), for any De[log2h]. This implies a polynomial-time O(helogn) approximation for any constant e>0, and a poly-logarithmic approximation running in quasi-polynomial time. We remark that this is essentially the best-known even for the classical DST, and the latter problem is O(log2-en)-hard to approximate [Halperin and Krauthgamer, STOC'03]. As a by product, we obtain an algorithm with the same approximation guarantee for the 2-Connected Directed Steiner Subgraph problem, where the goal is to find a min-cost subgraph such that every pair of terminals are 2-edge/vertex connected. Our approximation algorithm is based on a careful combination of several techniques. In more detail, we decompose an optimal solution into two (possibly not edge disjoint) divergent trees that induces two edge disjoint paths from the root to any given terminal. These divergent trees are then embedded into a shallow tree by means of Zelikovsky's height reduction theorem. On the latter tree we solve a 2-Connected Group Steiner Tree problem and then map back this solution to the original graph. Crucially, our tree embedding is achieved via a probabilistic mapping guided by an LP: This is the main technical novelty of our approach, and might be useful for future work.

The minimum cost subset k-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph G=(V,E) with costs on edges and a set of terminals T⊆V. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by k openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset k-connected subgraph problem which improves on the previous best approximation guarantee of O(k2logk) by Nutov (ACM Trans. Algorithms 9(1):1, 2012). Our approximation guarantee, ?(|T|), depends upon the number of terminals: $$\alpha(|T|) = \begin{cases} O(k \log^2 k) & \mbox{if } 2k\le |T| k. This suggests that the hardest instances of the problem are when |T|?k.

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-appro...

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

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.

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.