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1. Adaptive Out-Orientations with Applications

   Chekuri, Chandra ; Christiansen, Aleksander Bjørn ; Holm, Jacob ; van der Hoog, Ivor ; Quanrud, Kent ; Rotenberg, Eva ; Schwiegelshohn, Chris. ( January 2024 , Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out-degree is proportional to the arboricity $\alpha$ of the graph, in, either, an amortised update time of $O(\log^2 n \log \alpha)$, or a worst-case update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worst-case, for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $n$ and $\alpha$. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$, of the dynamic graph. Our algorithms have update times of $O(\varepsilon^{-6}\log^3 n \log \rho)$ worst-case, and $O(\varepsilon^{-4}\log^2 n \log \rho)$ amortised, respectively. We may output a subgraph $H$ of the input graph where its density is a $(1+\varepsilon)$ approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the $O(\varepsilon^{-6}\log ^4 n)$ algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an $O(\varepsilon^{-6}\log^3 n \log \alpha)$ worst-case update time algorithm for maintaining a $(1~+~\varepsilon)\textnormal{OPT} + 2$ approximation of the optimal out-orientation of a graph with adaptive arboricity $\alpha$, improving the $O(\varepsilon^{-6}\alpha^2 \log^3 n)$ algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, $\Delta+1$ colouring, and matrix vector multiplication. All update times are worst-case $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time $O(\alpha^2 + \log^2 n)$, and by Neiman and Solomon from STOC 2013 runs in time $O(\sqrt{m})$. We give improved running times whenever the arboricity $\alpha \in \omega( \log n\sqrt{\log\log n})$. 

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2. Constructing many faces in arrangements of lines and segments

   https://doi.org/10.20382/jocg.v14i1a11  

   Wang, Haitao ( December 2023 , Journal of computational geometry)

   We present new algorithms for computing many faces in arrangements of lines and segments. Given a set $S$ of $n$ lines (resp., segments) and a set $P$ of $m$ points in the plane, the problem is to compute the faces of the arrangements of $S$ that contain at least one point of $P$. For the line case, we give a deterministic algorithm of $O(m^{2/3}n^{2/3}\log^{2/3} (n/\sqrt{m})+(m+n)\log n)$ time. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $\log^{2.22}n$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $\log^{1/3}n$ in certain cases (e.g., when $m=\Theta(n)$). For the segment case, we present a deterministic algorithm of $O(n^{2/3}m^{2/3}\log n+\tau(n\alpha^2(n)+n\log m+m)\log n)$ time, where $\tau=\min\{\log m,\log (n/\sqrt{m})\}$ and $\alpha(n)$ is the inverse Ackermann function. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $\log^{2.11}n$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $\log n$ in certain cases (e.g., when $m=\Theta(n)$). We also give a randomized algorithm of $O(m^{2/3}K^{1/3}\log n+\tau(n\alpha(n)+n\log m+m)\log n\log K)$ expected time, where $K$ is the number of intersections of all segments of $S$. In addition, we consider the query version of the problem, that is, preprocess $S$ to compute the face of the arrangement of $S$ that contains any given query point. We present new results that improve the previous work for both the line and the segment cases. In particulary, for the line case, we build a data structure of $O(n\log n)$ space in $O(n\log n)$ randomized time, so that the face containing the query point can be obtained in $O(\sqrt{n\log n})$ time with high probability (more specifically, the query returns a binary search tree representing the face so that standard binary-search-based queries on the face can be handled in $O(\log n)$ time each and the face itself can be output explicitly in time linear in its size). 

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3. A Weighted Approach to the Maximum Cardinality Bipartite Matching Problem with Applications in Geometric Settings

   Lahn, Nathaniel ; Raghvendra, Sharath ( June 2019 , Leibniz international proceedings in informatics)

   We present a weighted approach to compute a maximum cardinality matching in an arbitrary bipartite graph. Our main result is a new algorithm that takes as input a weighted bipartite graph G(A cup B,E) with edge weights of 0 or 1. Let w <= n be an upper bound on the weight of any matching in G. Consider the subgraph induced by all the edges of G with a weight 0. Suppose every connected component in this subgraph has O(r) vertices and O(mr/n) edges. We present an algorithm to compute a maximum cardinality matching in G in O~(m(sqrt{w} + sqrt{r} + wr/n)) time. When all the edge weights are 1 (symmetrically when all weights are 0), our algorithm will be identical to the well-known Hopcroft-Karp (HK) algorithm, which runs in O(m sqrt{n}) time. However, if we can carefully assign weights of 0 and 1 on its edges such that both w and r are sub-linear in n and wr=O(n^{gamma}) for gamma < 3/2, then we can compute maximum cardinality matching in G in o(m sqrt{n}) time. Using our algorithm, we obtain a new O~(n^{4/3}/epsilon^4) time algorithm to compute an epsilon-approximate bottleneck matching of A,B subsetR^2 and an 1/(epsilon^{O(d)}}n^{1+(d-1)/(2d-1)}) poly log n time algorithm for computing epsilon-approximate bottleneck matching in d-dimensions. All previous algorithms take Omega(n^{3/2}) time. Given any graph G(A cup B,E) that has an easily computable balanced vertex separator for every subgraph G'(V',E') of size |V'|^{delta}, for delta in [1/2,1), we can apply our algorithm to compute a maximum matching in O~(mn^{delta/1+delta}) time improving upon the O(m sqrt{n}) time taken by the HK-Algorithm. 

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4. A Spanner for the Day After

   https://doi.org/10.1007/s00454-020-00228-6  

   Buchin, Kevin ; Har-Peled, Sariel ; Oláh, Dániel ( December 2020 , Discrete & Computational Geometry)

   null (Ed.)

   Abstract We show how to construct a $$(1+\varepsilon )$$ ( 1 + ε ) -spanner over a set $${P}$$ P of n points in $${\mathbb {R}}^d$$ R d that is resilient to a catastrophic failure of nodes. Specifically, for prescribed parameters $${\vartheta },\varepsilon \in (0,1)$$ ϑ , ε ∈ ( 0 , 1 ) , the computed spanner $${G}$$ G has $$\begin{aligned} {{\mathcal {O}}}\bigl (\varepsilon ^{-O(d)} {\vartheta }^{-6} n(\log \log n)^6 \log n \bigr ) \end{aligned}$$ O ( ε - O ( d ) ϑ - 6 n ( log log n ) 6 log n ) edges. Furthermore, for any k , and any deleted set $${{B}}\subseteq {P}$$ B ⊆ P of k points, the residual graph $${G}\setminus {{B}}$$ G \ B is a $$(1+\varepsilon )$$ ( 1 + ε ) -spanner for all the points of $${P}$$ P except for $$(1+{\vartheta })k$$ ( 1 + ϑ ) k of them. No previous constructions, beyond the trivial clique with $${{\mathcal {O}}}(n^2)$$ O ( n 2 ) edges, were known with this resilience property (i.e., only a tiny additional fraction of vertices, $$\vartheta |B|$$ ϑ | B | , lose their distance preserving connectivity). Our construction works by first solving the exact problem in one dimension, and then showing a surprisingly simple and elegant construction in higher dimensions, that uses the one-dimensional construction in a black-box fashion. 

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5. A Distanced Matching Game, Decremental {APSP} in Expanders, and Faster Deterministic Algorithms for Graph Cut Problems

   https://doi.org/10.1137/1.9781611977554.ch82  

   Chuzhoy, Julia ( January 2023 , Proceedings of the 2023 {ACM-SIAM} Symposium on Discrete Algorithms,{SODA} 2023)

   Expander graphs play a central role in graph theory and algorithms. With a number of powerful algorithmic tools developed around them, such as the Cut-Matching game, expander pruning, expander decomposition, and algorithms for decremental All-Pairs Shortest Paths (APSP) in expanders, to name just a few, the use of expanders in the design of graph algorithms has become ubiquitous. Specific applications of interest to us are fast deterministic algorithms for cut problems in static graphs, and algorithms for dynamic distance-based graph problems, such as APSP. Unfortunately, the use of expanders in these settings incurs a number of drawbacks. For example, the best currently known algorithm for decremental APSP in constant-degree expanders can only achieve a (log n) O(1/ 2 ) -approximation with n 1+O( ) total update time for any . All currently known algorithms for the Cut Player in the Cut-Matching game are either randomized, or provide rather weak guarantees: expansion 1/(log n) 1/ with running time n 1+O( ) . This, in turn, leads to somewhat weak algorithmic guarantees for several central cut problems: the best current almost linear time deterministic algorithms for Sparsest Cut, Lowest Conductance Cut, and Balanced Cut can only achieve approximation factor (log n) ω(1). Lastly, when relying on expanders in distancebased problems, such as dynamic APSP, via current methods, it seems inevitable that one has to settle for approximation factors that are at least Ω(log n). In contrast, we do not have any negative results that rule out a factor-5 approximation with near-linear total update time. In this paper we propose the use of well-connected graphs, and introduce a new algorithmic toolkit for such graphs that, in a sense, mirrors the above mentioned algorithmic tools for expanders. One of these new tools is the Distanced Matching game, an analogue of the Cut-Matching game for well-connected graphs. We demonstrate the power of these new tools by obtaining better results for several of the problems mentioned above. First, we design an algorithm for decremental APSP in expanders with significantly better guarantees: in a constant-degree expander, the algorithm achieves (log n) 1+o(1)-approximation, with total update time n 1+o(1). We also obtain a deterministic algorithm for the Cut Player in the Cut-Matching game that achieves expansion 1 (log n) 5+o(1) in time n 1+o(1), deterministic almost linear-time algorithms for Sparsest Cut, Lowest-Conductance Cut, and Minimum Balanced Cut with approximation factors O(poly log n), as well as improved deterministic algorithm for Expander Decomposition. We believe that the use of well-connected graphs instead of expanders in various dynamic distance-based problems (such as APSP in general graphs) has the potential of providing much stronger guarantees, since we are no longer necessarily restricted to superlogarithmic approximation factors. 

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