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1. Maximal k-Edge-Connected Subgraphs in Almost-Linear Time for Small k

   https://doi.org/10.4230/LIPIcs.ESA.2023.92  

   Saranurak, Thatchaphol; Yuan, Wuwei (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

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

   We give the first almost-linear time algorithm for computing the maximal k-edge-connected subgraphs of an undirected unweighted graph for any constant k. More specifically, given an n-vertex m-edge graph G = (V,E) and a number k = log^o(1) n, we can deterministically compute in O(m+n^{1+o(1)}) time the unique vertex partition {V_1,… ,V_z} such that, for every i, V_i induces a k-edge-connected subgraph while every superset V'_i ⊃ V_{i} does not. Previous algorithms with linear time work only when k ≤ 2 [Tarjan SICOMP'72], otherwise they all require Ω(m+n√n) time even when k = 3 [Chechik et al. SODA'17; Forster et al. SODA'20]. Our algorithm also extends to the decremental graph setting; we can deterministically maintain the maximal k-edge-connected subgraphs of a graph undergoing edge deletions in m^{1+o(1)} total update time. Our key idea is a reduction to the dynamic algorithm supporting pairwise k-edge-connectivity queries [Jin and Sun FOCS'20]. 

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2. 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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3. Improved Approximation Algorithms for Capacitated Network Design and Flexible Graph Connectivity

   https://doi.org/10.4230/lipics.icalp.2025.20  

   Bansal, Ishan; Cheriyan, Joe; Khanna, Sanjeev; Simmons, Miles (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Censor-Hillel, Keren; Grandoni, Fabrizio; Ouaknine, Joel; Puppis, Gabriele (Ed.)

   Abstract: We present improved approximation algorithms for some problems in the related areas of Capacitated Network Design and Flexible Graph Connectivity. In the Cap-k-ECSS problem, we are given a graph G = (V,E) whose edges have non-negative costs and positive integer capacities, and the goal is to find a minimum-cost edge-set F such that every non-trivial cut of the graph G' = (V,F) has capacity at least k. Let n = |V| and let u_{min} (respectively, u_{max}) denote the minimum (respectively, maximum) capacity of an edge; assume that u_{max} ≤ k. We present an O(log({k}/u_{min}))-approximation algorithm for the Cap-k-ECSS problem, asymptotically improving upon the previous best approximation ratio of min(O(log{n}), k, 2u_{max}, 6 ⋅ {⌈ k/u_{min} ⌉}) whenever log(k/u_{min}) = o(log{n}) and u_{max} is sufficiently large. In the (p,q)-Flexible Graph Connectivity problem, denoted (p,q)-FGC, the input is a graph G = (V, E) where E is partitioned into safe and unsafe edges, and the goal is to find a minimum-cost edge-set F such that the subgraph G' = (V, F) remains p-edge connected upon removal of any q unsafe edges from F. We present an 8-approximation algorithm for the (1,q)-FGC problem that improves upon the previous best approximation ratio of (q+1). Both of our results are obtained by using natural LP relaxations strengthened with the knapsack-cover inequalities, and then, during the rounding process, utilizing a recent O(1)-approximation algorithm for the Cover Small Cuts problem. In the latter problem, the goal is to find a minimum-cost set of links such that each non-trivial cut of capacity less than a specified value is covered by a link. We also show that the problem of covering small cuts inherently arises in another variant of (p,q)-FGC. Specifically, we give Cook reductions that preserve approximation ratios within O(1) factors between the (2,q)-FGC problem and the 2-Cover Small Cuts problem; in the latter problem, each small cut needs to be covered by two links. 

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4. Online Balanced Allocation of Dynamic Components

   https://doi.org/10.4230/lipics.itcs.2025.81  

   Rajaraman, Rajmohan; Wasim, Omer (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We introduce Online Balanced Allocation of Dynamic Components (OBADC), a problem motivated by the practical challenge of dynamic resource allocation for large-scale distributed applications. In OBADC, we need to allocate a dynamic set of at most k𝓁 vertices (representing processes) in 𝓁 > 0 clusters. We consider an over-provisioned setup in which each cluster can hold at most k(1+ε) vertices, for an arbitrary constant ε > 0. The communication requirements among the vertices are modeled by the notion of a dynamically changing component, which is a subset of vertices that need to be co-located in the same cluster. At each time t, a request r_t of one of the following types arrives: 1) insertion of a vertex v forming a singleton component v at unit cost. 2) merge of (u,v) requiring that the components containing u and v be merged and co-located thereafter. 3) deletion of an existing vertex v at zero cost. Before serving any request, an algorithm can migrate vertices from one cluster to another, at a unit migration cost per vertex. We seek an online algorithm to minimize the total migration cost incurred for an arbitrary request sequence σ = (r_t)_{t > 0}, while simultaneously minimizing the number of clusters utilized. We analyze competitiveness with respect to an optimal clairvoyant offline algorithm with identical (over-provisioned) capacity constraints. We give an O(log k)-competitive algorithm for OBADC, and a matching lower-bound. The number of clusters utilized by our algorithm is always within a (2+ε) factor of the minimum. Furthermore, in a resource augmented setting where the optimal offline algorithm is constrained to capacity k per cluster, our algorithm obtains O(log k) competitiveness and utilizes a number of clusters within (1+ε) factor of the minimum. We also consider OBADC in the context of machine-learned predictions, where for each newly inserted vertex v at time t: i) with probability η > 0, the set of vertices (that exist at time t) in the component of v is revealed and, ii) with probability 1-η, no information is revealed. For OBADC with predictions, we give a O(1)-consistent and O(min(log 1/(η), log k))-robust algorithm. 

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5. From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

   https://doi.org/10.4230/LIPIcs.ESA.2024.42  

   Chekuri, Chandra; Jain, Rhea; Kulkarni, Shubhang; Zheng, Da Wei; Zhu, Weihao (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Chan, Timothy; Fischer, Johannes; Iacono, John; Herman, Grzegorz (Ed.)

   In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph G = (V,E), a root vertex r and a set S ⊆ V of k terminals. The goal is to find a min-cost subgraph that connects r to each of the terminals. DST admits an O(log² k/log log k)-approximation in quasi-polynomial time [Grandoni et al., 2022; Rohan Ghuge and Viswanath Nagarajan, 2022], and an O(k^{ε})-approximation for any fixed ε > 0 in polynomial-time [Alexander Zelikovsky, 1997; Moses Charikar et al., 1999]. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [Zachary Friggstad and Ramin Mousavi, 2023] obtained a simple and elegant polynomial-time O(log k)-approximation for DST in planar digraphs via Thorup’s shortest path separator theorem [Thorup, 2004]. We build on their work and obtain several new results on DST and related problems. - We develop a tree embedding technique for rooted problems in planar digraphs via an interpretation of the recursion in [Zachary Friggstad and Ramin Mousavi, 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree [Naveen Garg et al., 2000], Covering Steiner Tree [Goran Konjevod et al., 2002] and the Polymatroid Steiner Tree [Gruia Călinescu and Alexander Zelikovsky, 2005] problems in planar digraphs. All these problems are hard to approximate to within a factor of Ω(log² n/log log n) even in trees [Eran Halperin and Robert Krauthgamer, 2003; Grandoni et al., 2022]. - We prove that the natural cut-based LP relaxation for DST has an integrality gap of O(log² k) in planar digraphs. This is in contrast to general graphs where the integrality gap of this LP is known to be Ω(√k) [Leonid Zosin and Samir Khuller, 2002] and Ω(n^{δ}) for some fixed δ > 0 [Shi Li and Bundit Laekhanukit, 2022]. - We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the O(R + log k) approximation of [Zachary Friggstad and Ramin Mousavi, 2023] when R = ω(log² k). 

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