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1. Approximation Algorithm for Norm Multiway Cut

   https://doi.org/10.4230/LIPIcs.ESA.2023.32  

   Carlson, Charlie; Jafarov, Jafar; Makarychev, Konstantin; Makarychev, Yury; Shan, Liren (August 2023, Proceedings of the European Symposium on Algorithms)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph G into k parts so as to separate k given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced l_p-norm Multiway Cut, a generalization of the problem, in which the goal is to minimize the l_p norm of the edge boundaries of k parts. We provide an O(log^{1/2} n log^{1/2 + 1/p} k) approximation algorithm for this problem, improving upon the approximation guarantee of O(log^{3/2} n log^{1/2} k) due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an O(log^{1/2} n log^{7/2} k) approximation algorithm with a weaker oracle and an O(log^{1/2} n log^{5/2} k) approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no n^{1/4-ε} approximation algorithm for every ε > 0 assuming the Hypergraph Dense-vs-Random Conjecture. 

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2. A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

   https://doi.org/10.1109/FOCS46700.2020.00111  

   Chuzhoy, Julia; Gao, Yu; Li, Jason; Nanongkai, Danupon; Peng, Richard; Saranurak, Thatchaphol (November 2020, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We consider the classical Minimum Balanced Cut problem: given a graph $$G$$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $$n$$-vertex $$m$$-edge graph $$G$$ and any parameter $$1\leq r\leq O(\log n)$$, computes a $$(\log m)^{r^2}$$-approximation for Minimum Balanced Cut on $$G$$, in time $$O\left ( m^{1+O(1/r)+o(1)}\cdot (\log m)^{O(r^2)}\right )$$. In particular, we obtain a $$(\log m)^{1/\epsilon}$$-approximation in time $$m^{1+O(1/\sqrt{\epsilon})}$$ for any constant $$\epsilon$$, and a $$(\log m)^{f(m)}$$-approximation in time $$m^{1+o(1)}$$, for any slowly growing function $$m$$. We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $$G$$ that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an $$n$$-vertex graph is $$n^{o(1)}$$, thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in $$n$$ factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs. 

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3. On Approximating Degree-Bounded Network Design Problems

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.39  

   Xiangyu Guo, Guy Kortsarz (October 2020, Leibniz international proceedings in informatics)

   null (Ed.)

   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 ∈ ℝ_{≥ 0}^E, a root r ∈ V and k terminals K ⊆ V, we need to output a minimum-cost arborescence in G that contains an rrightarrow t path for every t ∈ K. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time O(log²k/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 ∈ 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(log² n))-bicriteria approximation: The algorithm produces a solution with cost at most O(log nlog k) times the cost of the optimum solution that violates the degree constraints by at most a factor of O(log²n). This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of O(log²n) is an O(log n)-factor away from the approximation lower bound of Ω(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 nlog k), O(log n))-bicriteria approximation algorithm for DB-GST-T. 

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4. Ultrasparse Ultrasparsifiers and Faster Laplacian System Solvers

   https://doi.org/10.1145/3593809  

   Jambulapati, Arun; Sidford, Aaron (February 2024, ACM Transactions on Algorithms)

   In this paper we provide anO(mloglog^O(1)nlog (1/ϵ))-expected time algorithm for solving Laplacian systems onn-nodem-edge graphs, improving upon the previous best expected runtime of\(O(m \sqrt {\log n} \mathrm{log log}^{O(1)} n \log (1/\epsilon)) \)achieved by (Cohen, Kyng, Miller, Pachocki, Peng, Rao, Xu 2014). To obtain this result we provide efficient constructions of low spectral stretch graph approximations with improved stretch and sparsity bounds. As motivation for this work, we show that for every set of vectors in\(\mathbb {R}^d \)(not just those induced by graphs) and all integerk> 1 there exist an ultra-sparsifier withd− 1 +O(d/k) re-weighted vectors of relative condition number at mostk². For smallk, this improves upon the previous best known multiplicative factor of\(k \cdot \tilde{O}(\log d) \), which is only known for the graph case. Additionally, in the graph case we employ our low-stretch subgraph construction to obtainn− 1 +O(n/k)-edge ultrasparsifiers of relative condition numberk^{1 +o(1)}fork=ω(log ^δn) for anyδ> 0: this improves upon the previous work fork=o(exp (log ^{1/2 −δ}n)). 

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5. Node-weighted Network Design in Planar and Minor-closed Families of Graphs

   https://doi.org/10.1145/3447959  

   Chekuri, Chandra; Ene, Alina; Vakilian, Ali (June 2021, ACM Transactions on Algorithms)

   null (Ed.)

   We consider node-weighted survivable network design (SNDP) in planar graphs and minor-closed families of graphs. The input consists of a node-weighted undirected graph G = ( V , E ) and integer connectivity requirements r ( uv ) for each unordered pair of nodes uv . The goal is to find a minimum weighted subgraph H of G such that H contains r ( uv ) disjoint paths between u and v for each node pair uv . Three versions of the problem are edge-connectivity SNDP (EC-SNDP), element-connectivity SNDP (Elem-SNDP), and vertex-connectivity SNDP (VC-SNDP), depending on whether the paths are required to be edge, element, or vertex disjoint, respectively. Our main result is an O ( k )-approximation algorithm for EC-SNDP and Elem-SNDP when the input graph is planar or more generally if it belongs to a proper minor-closed family of graphs; here, k = max  uv r ( uv ) is the maximum connectivity requirement. This improves upon the O ( k log  n )-approximation known for node-weighted EC-SNDP and Elem-SNDP in general graphs [31]. We also obtain an O (1) approximation for node-weighted VC-SNDP when the connectivity requirements are in {0, 1, 2}; for higher connectivity our result for Elem-SNDP can be used in a black-box fashion to obtain a logarithmic factor improvement over currently known general graph results. Our results are inspired by, and generalize, the work of Demaine, Hajiaghayi, and Klein [13], who obtained constant factor approximations for node-weighted Steiner tree and Steiner forest problems in planar graphs and proper minor-closed families of graphs via a primal-dual algorithm. 

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