### Browsing by Author "Nanongkai, Danupon"

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Item Approximating k-Edge-Connected Spanning Subgraphs via a Near-Linear Time LP Solver(Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022-07-01) Chalermsook, Parinya; Huang, Chien Chung; Nanongkai, Danupon; Saranurak, Thatchaphol; Sukprasert, Pattara; Yingchareonthawornchai, Sorrachai; Computer Science Professors; École normale supérieure; University of Copenhagen; University of Michigan, Ann Arbor; Northwestern University; Department of Computer Science; Bojanczyk, Mikolaj; Merelli, Emanuela; Woodruff, David P.In the k-edge-connected spanning subgraph (kECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to k link failures: Given an n-node m-edge graph with a cost function on the edges, our goal is to compute a minimum-cost k-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the “uniform case” of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of 2ECSS. The fastest 2-approximation algorithm is however rather slow, taking O(mnk) time [Khuller, Vishkin, STOC'92]. A faster time complexity of O(n2) can be obtained, but with a higher approximation guarantee of (2k − 1) [Gabow, Goemans, Williamson, IPCO'93]. Our main contribution is an algorithm that (1 + ε)-approximates the optimal fractional solution in Õ(m/ε2) time (independent of k), which can be turned into a (2+ ε) approximation algorithm that runs in time (Equation presented) for (integral) kECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.Item Breaking quadratic time for small vertex connectivity and an approximation scheme(2019-06-23) Nanongkai, Danupon; Saranurak, Thatchaphol; Yingchareonthawornchai, Sorrachai; KTH Royal Institute of Technology; University of Chicago; Department of Computer Science; Charikar, Moses; Cohen, EdithVertex connectivity a classic extensively-studied problem. Given an integer k, its goal is to decide if an n-node m-edge graph can be disconnected by removing k vertices. Although a linear-time algorithm was postulated since 1974 [Aho, Hopcroft and Ullman], and despite its sibling problem of edge connectivity being resolved over two decades ago [Karger STOC’96], so far no vertex connectivity algorithms are faster than O(n2) time even for k = 4 and m = O(n). In the simplest case where m = O(n) and k = O(1), the O(n2) bound dates five decades back to [Kleitman IEEE Trans. Circuit Theory’69]. For higher m, O(m) time is known for k ≤ 3 [Tarjan FOCS’71; Hopcroft, Tarjan SICOMP’73], the first O(n2) time is from [Kanevsky, Ramachandran, FOCS’87] for k = 4 and from [Nagamochi, Ibaraki, Algorithmica’92] for k = O(1). For general k and m, the best bound is Õ (min(kn2, nω + nkω )) [Henzinger, Rao, Gabow FOCS’96; Linial, Lovász, Wigderson FOCS’86] where Õ hides polylogarithmic terms and ω < 2.38 is the matrix multiplication exponent. In this paper, we present a randomized Monte Carlo algorithm with Õ (m + k7/3n4/3) time for any k = O(n). This gives the first subquadratic time bound for any 4 ≤ k ≤ o(n2/7) (subquadratic time refers to O(m) + o(n2) time.) and improves all above classic bounds for all k ≤ n0.44. We also present a new randomized Monte Carlo (1 + ϵ)-approximation algorithm that is strictly faster than the previous Henzinger’s 2-approximation algorithm [J. Algorithms’97] and all previous exact algorithms. The story is the same for the directed case, where our exact Õ (min(km2/3n, km4/3))-time for any k = O(n) and (1 + ϵ)-approximation algorithms improve all previous exact bounds. Additionally, our algorithm is the first approximation algorithm on directed graphs. The key to our results is to avoid computing single-source connectivity, which was needed by all previous exact algorithms and is not known to admit o(n2) time. Instead, we design the first local algorithm for computing vertex connectivity; without reading the whole graph, our algorithm can find a separator of size at most k or certify that there is no separator of size at most k “near” a given seed node.Item From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more(Society for Industrial and Applied Mathematics Publications, 2020) Chalermsook, Parinya; Cygan, Marek; Kortsarz, Guy; Laekhanukit, Bundit; Manurangsi, Pasin; Nanongkai, Danupon; Trevisan, Luca; Department of Computer Science; Computer Science - Algorithms and Theoretical Computer Science (TCS); Professorship Chalermsook Parinya; University of Warsaw; Rutgers University-Camden; Shanghai University of Finance and Economics; University of California, Berkeley; KTH Royal Institute of TechnologyWe 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 (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 computable functions t and f that are independent of N (for Clique, we want f (OPT) = omega(1))? In this paper, we show that both Clique and DomSet admit no nontrivial 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. 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 [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 SAT formula and one which is not even (1 - epsilon)-satisfiable for some constant epsilon > 0. Besides Clique and 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.Item New Tools and Connections for Exponential-Time Approximation(2018) Bansal, Nikhil; Chalermsook, Parinya; Laekhanukit, Bundit; Nanongkai, Danupon; Nederlof, Jesper; Department of Computer Science; Professorship Chalermsook Parinya; Eindhoven University of Technology; Shanghai University of Finance and Economics; KTH Royal Institute of TechnologyIn 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 of1.r for maximum independent set in O∗(exp (O~ (n/ rlog 2r+ rlog 2r))) time,2.r for chromatic number in O∗(exp (O~ (n/ rlog r+ rlog 2r))) time,3.(2 - 1 / r) for minimum vertex cover in O∗(exp (n/ rΩ ( r ))) time, and4.(k- 1 / r) for minimum k-hypergraph vertex cover in O∗(exp (n/ (kr) Ω ( k r ))) time. (Throughout, O~ and O∗ omit 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 (n1 - o ( 1 )/ r1 + 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).Item Vertex Connectivity in Poly-Logarithmic Max-Flows(Association for Computing Machinery, 2021-06-15) Li, Jason; Nanongkai, Danupon; Panigrahi, Debmalya; Saranurak, Thatchaphol; Yingchareonthawornchai, Sorrachai; Department of Computer Science; Khuller, Samir; Williams, Virginia Vassilevska; Professorship Chalermsook Parinya; Computer Science - Algorithms and Theoretical Computer Science (TCS); Carnegie Mellon University; University of Copenhagen; Duke University; University of MichiganThe vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph, or leaves only a singleton vertex. In this paper, we give a reduction from the vertex connectivity problem to a set of maxflow instances. Using this reduction, we can solve vertex connectivity in (mα) time for any α ≥ 1, if there is a mα-time maxflow algorithm. Using the current best maxflow algorithm that runs in m4/3+o(1) time (Kathuria, Liu and Sidford, FOCS 2020), this yields a m4/3+o(1)-time vertex connectivity algorithm. This is the first improvement in the running time of the vertex connectivity problem in over 20 years, the previous best being an Õ(mn)-time algorithm due to Henzinger, Rao, and Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time was known before our work, even if we assume an (m)-time maxflow algorithm. Our new technique is robust enough to also improve the best Õ(mn)-time bound for directed vertex connectivity to mn1−1/12+o(1) time