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1. Improved Hitting Set for Orbit of ROABPs

   https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.30  

   Bhargava, Vishwas ; Ghosh, Sumanta ( September 2021 , Leibniz international proceedings in informatics)

   Wootters, Mary ; Sanita, Laura (Ed.)

   The orbit of an n-variate polynomial f(x) over a field 𝔽 is the set {f(Ax+b) ∣ A ∈ GL(n, 𝔽) and b ∈ 𝔽ⁿ}, and the orbit of a polynomial class is the union of orbits of all the polynomials in it. In this paper, we give improved constructions of hitting-sets for the orbit of read-once oblivious algebraic branching programs (ROABPs) and a related model. Over fields with characteristic zero or greater than d, we construct a hitting set of size (ndw)^{O(w²log n⋅ min{w², dlog w})} for the orbit of ROABPs in unknown variable order where d is the individual degree and w is the width of ROABPs. We also give a hitting set of size (ndw)^{O(min{w²,dlog w})} for the orbit of polynomials computed by w-width ROABPs in any variable order. Our hitting sets improve upon the results of Saha and Thankey [Chandan Saha and Bhargav Thankey, 2021] who gave an (ndw)^{O(dlog w)} size hitting set for the orbit of commutative ROABPs (a subclass of any-order ROABPs) and (nw)^{O(w⁶log n)} size hitting set for the orbit of multilinear ROABPs. Designing better hitting sets in large individual degree regime, for instance d > n, was asked as an open problem by [Chandan Saha and Bhargav Thankey, 2021] and this work solves it in small width setting. We prove some new rank concentration results by establishing low-cone concentration for the polynomials over vector spaces, and they strengthen some previously known low-support based rank concentrations shown in [Michael A. Forbes et al., 2013]. These new low-cone concentration results are crucial in our hitting set construction, and may be of independent interest. To the best of our knowledge, this is the first time when low-cone rank concentration has been used for designing hitting sets. 

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2. On the fine-grained complexity of small-size geometric set cover and discrete k-center for small k

   https://doi.org/10.4230/LIPIcs.ICALP.2023.34  

   Chan, Timothy M. ; He, Qizheng ; Yu, Yuancheng ( July 2023 , Proc. 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Etessami, Kousha ; Feige, Uriel ; Puppis, Gabriele (Ed.)

   We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time. - We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}). - We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2. - We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2. - We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound. 

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3. Geometric Hitting Set for Line-Constrained Disks

   Liu, Gang ; Wang, Haitao ( August 2023 , Proceedings of the 18th Algorithms and Data Structures Symposium (WADS 2023))

   Given a set P of n weighted points and a set S of m disks in the plane, the hitting set problem is to compute a subset 𝑃′ of points of P such that each disk contains at least one point of 𝑃′ and the total weight of all points of 𝑃′ is minimized. The problem is known to be NP-hard. In this paper, we consider a line-constrained version of the problem in which all disks are centered on a line ℓ. We present an 𝑂((𝑚+𝑛)log(𝑚+𝑛)+𝜅log𝑚) time algorithm for the problem, where 𝜅 is the number of pairs of disks that intersect. For the unit-disk case where all disks have the same radius, the running time can be reduced to 𝑂((𝑛+𝑚)log(𝑚+𝑛)). In addition, we solve the problem in 𝑂((𝑚+𝑛)log(𝑚+𝑛)) time in the 𝐿∞ and 𝐿1 metrics, in which a disk is a square and a diamond, respectively. 

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4. Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem

   Angelidakis, H ; Awasthi, P. ; Blum, A. ; Chatziafratis, V. ; Dan, C. ( September 2019 , 27th Annual European Symposium on Algorithms (ESA 2019))

   We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is γ-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most γ ≥ 1. The goal then is to efficiently recover this “pronounced” optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving \tilde{O}(∆/sqrt(log ∆))-stable instances on graphs of maximum degree ∆, (k − 1)-stable instances on k-colorable graphs and (1 + ε)-stable instances on planar graphs (for any fixed ε > 0), using both combinatorial techniques as well as LPs and the Sherali-Adams hierarchy. For general graphs, we give an algorithm for (εn)-stable instances, for any fixed ε > 0, and lower bounds based on the planted clique conjecture. As a by-product of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a γ-certified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of γ-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of γ ≥ 1 (hence, such algorithms not only solve γ-stable instances optimally, but also have guarantees even on unstable instances). Here, we obtain ∆-certified algorithms for Independent Set on graphs of maximum degree ∆, and (1 + ε)-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((∆+1)/3 + ε)-certified algorithm for Independent Set on graphs of maximum degree ∆ where all weights are equal to 1. 

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5. Leibniz International Proceedings in Informatics (LIPIcs):32nd International Symposium on Algorithms and Computation (ISAAC 2021)

   https://doi.org/10.4230/LIPIcs.ISAAC.2021.5  

   Gibson-Lopez, Matt ; Yang, Zhongxiu ( November 2021 , Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ahn, Hee-Kap ; Sadakane, Kunihiko (Ed.)

   Visibility problems are fundamental to computational geometry, and many versions of geometric set cover where coverage is based on visibility have been considered. In most settings, points can see "infinitely far" so long as visibility is not "blocked" by some obstacle. In many applications, this may be an unreasonable assumption. In this paper, we consider a new model of visibility where no point can see any other point beyond a sight radius ρ. In particular, we consider this visibility model in the context of terrains. We show that the VC-dimension of limited visibility terrains is exactly 7. We give lower bound construction that shatters a set of 7 points, and we prove that shattering 8 points is not possible. 

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