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1. On Approximability of 𝓁₂² Min-Sum Clustering

   https://doi.org/10.4230/lipics.socg.2025.62  

   S, Karthik C; Lee, Euiwoong; Rabani, Yuval; Schwiegelshohn, Chris; Zhou, Samson (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Aichholzer, Oswin; Wang, Haitao (Ed.)

   The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0. 

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2. Parameterized Approximation Scheme for Feedback Vertex Set

   https://doi.org/10.4230/LIPIcs.MFCS.2023.56  

   Jana, Satyabrata; Lokshtanov, Daniel; Mandal, Soumen; Rai, Ashutosh; Saurabh, Saket (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Leroux, Jérôme; Lombardy, Sylvain; Peleg, David (Ed.)

   Feedback Vertex Set (FVS) is one of the most studied vertex deletion problems in the field of graph algorithms. In the decision version of the problem, given a graph G and an integer k, the question is whether there exists a set S of at most k vertices in G such that G-S is acyclic. It is one of the first few problems which were shown to be NP-complete, and has been extensively studied from the viewpoint of approximation and parameterized algorithms. The best-known polynomial time approximation algorithm for FVS is a 2-factor approximation, while the best known deterministic and randomized FPT algorithms run in time 𝒪^*(3.460^k) and 𝒪^*(2.7^k) respectively. In this paper, we contribute to the newly established area of parameterized approximation, by studying FVS in this paradigm. In particular, we combine the approaches of parameterized and approximation algorithms for the study of FVS, and achieve an approximation guarantee with a factor better than 2 in randomized FPT running time, that improves over the best known parameterized algorithm for FVS. We give three simple randomized (1+ε) approximation algorithms for FVS, running in times 𝒪^*(2^{εk}⋅ 2.7^{(1-ε)k}), 𝒪^*(({(4/(1+ε))^{(1+ε)}}⋅{(ε/3)^ε})^k), and 𝒪^*(4^{(1-ε)k}) respectively for every ε ∈ (0,1). Combining these three algorithms, we obtain a factor (1+ε) approximation algorithm for FVS, which has better running time than the best-known (randomized) FPT algorithm for every ε ∈ (0, 1). This is the first attempt to look at a parameterized approximation of FVS to the best of our knowledge. Our algorithms are very simple, and they rely on some well-known reduction rules used for arriving at FPT algorithms for FVS. 

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3. 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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4. Approximating Red-Blue Set Cover and Minimum Monotone Satisfying Assignment

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.11  

   Chlamtáč, Eden; Makarychev, Yury; Vakilian, Ali (September 2023, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023))

   Megow, Nicole; Smith, Adam (Ed.)

   We provide new approximation algorithms for the Red-Blue Set Cover and Circuit Minimum Monotone Satisfying Assignment (MMSA) problems. Our algorithm for Red-Blue Set Cover achieves Õ(m^{1/3})-approximation improving on the Õ(m^{1/2})-approximation due to Elkin and Peleg (where m is the number of sets). Our approximation algorithm for MMSA_t (for circuits of depth t) gives an Õ(N^{1-δ}) approximation for δ = 1/3 2^{3-⌈t/2⌉}, where N is the number of gates and variables. No non-trivial approximation algorithms for MMSA_t with t ≥ 4 were previously known. We complement these results with lower bounds for these problems: For Red-Blue Set Cover, we provide a nearly approximation preserving reduction from Min k-Union that gives an Ω(m^{1/4 - ε}) hardness under the Dense-vs-Random conjecture, while for MMSA we sketch a proof that an SDP relaxation strengthened by Sherali-Adams has an integrality gap of N^{1-ε} where ε → 0 as the circuit depth t → ∞. 

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5. On Cyclic Solutions to the Min-Max Latency Multi-Robot Patrolling Problem

   https://doi.org/10.4230/LIPIcs.SoCG.2022.2  

   Afshani, Peyman; de Berg, Mark; Buchin, Kevin; Gao, Jie; L; Nayyeri, Amir; Raichel, Benjamin; Sarkar, Rik; Wang, Haotian; Yang, Hao-Tsung (January 2022, International Symposium on Computational Geometry (SOCG))

   Goaoc, Xavier; Kerber, Michael (Ed.)

   We consider the following surveillance problem: Given a set P of n sites in a metric space and a set R of k robots with the same maximum speed, compute a patrol schedule of minimum latency for the robots. Here a patrol schedule specifies for each robot an infinite sequence of sites to visit (in the given order) and the latency L of a schedule is the maximum latency of any site, where the latency of a site s is the supremum of the lengths of the time intervals between consecutive visits to s. When k = 1 the problem is equivalent to the travelling salesman problem (TSP) and thus it is NP-hard. For k ≥ 2 (which is the version we are interested in) the problem becomes even more challenging; for example, it is not even clear if the decision version of the problem is decidable, in particular in the Euclidean case. We have two main results. We consider cyclic solutions in which the set of sites must be partitioned into 𝓁 groups, for some 𝓁 ≤ k, and each group is assigned a subset of the robots that move along the travelling salesman tour of the group at equal distance from each other. Our first main result is that approximating the optimal latency of the class of cyclic solutions can be reduced to approximating the optimal travelling salesman tour on some input, with only a 1+ε factor loss in the approximation factor and an O((k/ε) ^k) factor loss in the runtime, for any ε > 0. Our second main result shows that an optimal cyclic solution is a 2(1-1/k)-approximation of the overall optimal solution. Note that for k = 2 this implies that an optimal cyclic solution is optimal overall. We conjecture that this is true for k ≥ 3 as well. The results have a number of consequences. For the Euclidean version of the problem, for instance, combining our results with known results on Euclidean TSP, yields a PTAS for approximating an optimal cyclic solution, and it yields a (2(1-1/k)+ε)-approximation of the optimal unrestricted (not necessarily cyclic) solution. If the conjecture mentioned above is true, then our algorithm is actually a PTAS for the general problem in the Euclidean setting. Similar results can be obtained by combining our results with other known TSP algorithms in non-Euclidean metrics. 

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