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1. Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

   https://doi.org/10.1109/FOCS46700.2020.00109  

   Chen, Li ; Goranci, Gramoz ; Henzinger, Monika ; 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 present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $\tilde{O}(n^{2/3})$ amortized time against an oblivious adversary, and $\tilde{O}(m^{3/4})$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $n^{o(1)}$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $\tilde{O}(n^{2/3+o(1)})$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $(1+\eps)$ factor in $\tilde{O}(n^{2/3+o(1)} \epsilon^{-O(1)})$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy. 

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

   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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3. 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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4. A Subpolynomial Approximation Algorithm for Graph Crossing Number in Low-Degree Graphs

   https://doi.org/10.1145/3519935.3519984  

   Chuzhoy, Julia ; Tan, Zihan ( June 2022 , STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing)

   We consider the classical Minimum Crossing Number problem: given an n-vertex graph G, compute a drawing of G in the plane, while minimizing the number of crossings between the images of its edges. This is a fundamental and extensively studied problem, whose approximability status is widely open. In all currently known approximation algorithms, the approximation factor depends polynomially on Δ – the maximum vertex degree in G. The best current approximation algorithm achieves an O(n1/2−· (Δ·logn))-approximation, for a small fixed constant є, while the best negative result is APX-hardness, leaving a large gap in our understanding of this basic problem. In this paper we design a randomized O(2O((logn)7/8loglogn)·(Δ))-approximation algorithm for Minimum Crossing Number. This is the first approximation algorithm for the problem that achieves a subpolynomial in n approximation factor (albeit only in graphs whose maximum vertex degree is subpolynomial in n). In order to achieve this approximation factor, we design a new algorithm for a closely related problem called Crossing Number with Rotation System, in which, for every vertex v∈ V(G), the circular ordering, in which the images of the edges incident to v must enter the image of v in the drawing is fixed as part of input. Combining this result with the recent reduction of [Chuzhoy, Mahabadi, Tan ’20] immediately yields the improved approximation algorithm for Minimum Crossing Number. 

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5. Efficient algorithms for dynamic bidirected Dyck-reachability

   https://doi.org/10.1145/3498724  

   Li, Yuanbo ; Satya, Kris ; Zhang, Qirun ( January 2022 , Proceedings of the ACM on Programming Languages)

   Dyck-reachability is a fundamental formulation for program analysis, which has been widely used to capture properly-matched-parenthesis program properties such as function calls/returns and field writes/reads. Bidirected Dyck-reachability is a relaxation of Dyck-reachability on bidirected graphs where each edge u → ( i v labeled by an open parenthesis “( i ” is accompanied with an inverse edge v → ) i u labeled by the corresponding close parenthesis “) i ”, and vice versa. In practice, many client analyses such as alias analysis adopt the bidirected Dyck-reachability formulation. Bidirected Dyck-reachability admits an optimal reachability algorithm. Specifically, given a graph with n nodes and m edges, the optimal bidirected Dyck-reachability algorithm computes all-pairs reachability information in O ( m ) time. This paper focuses on the dynamic version of bidirected Dyck-reachability. In particular, we consider the problem of maintaining all-pairs Dyck-reachability information in bidirected graphs under a sequence of edge insertions and deletions. Dynamic bidirected Dyck-reachability can formulate many program analysis problems in the presence of code changes. Unfortunately, solving dynamic graph reachability problems is challenging. For example, even for maintaining transitive closure, the fastest deterministic dynamic algorithm requires O ( n 2 ) update time to achieve O (1) query time. All-pairs Dyck-reachability is a generalization of transitive closure. Despite extensive research on incremental computation, there is no algorithmic development on dynamic graph algorithms for program analysis with worst-case guarantees. Our work fills the gap and proposes the first dynamic algorithm for Dyck reachability on bidirected graphs. Our dynamic algorithms can handle each graph update ( i.e. , edge insertion and deletion) in O ( n ·α( n )) time and support any all-pairs reachability query in O (1) time, where α( n ) is the inverse Ackermann function. We have implemented and evaluated our dynamic algorithm on an alias analysis and a context-sensitive data-dependence analysis for Java. We compare our dynamic algorithms against a straightforward approach based on the O ( m )-time optimal bidirected Dyck-reachability algorithm and a recent incremental Datalog solver. Experimental results show that our algorithm achieves orders of magnitude speedup over both approaches. 

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