arXiv subjects

Julia Chuzhoy

Publications and source records attributed to Julia Chuzhoy.

At least 19 recordsLinked to original sources

arXiv2026

A Faster Deterministic Algorithm for Fully Dynamic Maximal Matching

In the fully dynamic maximal matching problem, the goal is to maintain a maximal matching in a graph undergoing an online sequence of edge insertions and deletions. The problem has been studied extensively in the oblivious-adversary setting, where randomized algorithms with polylogarithmic worst-case and constant amortized update time have been known for some time. A major challenge in this area has been designing an algorithm with non-trivial update time against an adaptive adversary. In a recent breakthrough, Bernstein, Bhattacharya, Kiss, and Saranurak (STOC 2025; hereafter, BBKS25) obtained the first algorithms with sublinear update time for this setting: namely, a randomized algorithm with $\tilde{O}(n^{3/4})$ amortized update time, and a deterministic algorithm with $\tilde{O}(n^{8/9})$ amortized update time. Our main result is a deterministic algorithm for fully dynamic maximal matching with amortized update time $n^{1/2+o(1)}$. A powerful tool in dynamic matching is the use of matching sparsifiers: sparse subgraphs that preserve enough information to recover matchings with desired properties. Sparsifiers, such as the EDCS data structure, have been successfully used for approximate maximum matching. For maximal matching, however, this paradigm is not as natural, since maximality must hold with respect to the entire graph. Nevertheless, BBKS25 showed that EDCS can be repurposed as a verification-and-repair mechanism for fully dynamic maximal matching against adaptive adversaries. We introduce a new deterministic framework, referred to as the subgraph system, which, in contrast to EDCS, is purpose-built for verification and maintenance of maximality. It is also designed to allow efficient recursive refinements leading to stronger and stronger parameters, that yield our deterministic algorithm with $n^{1/2+o(1)}$ amortized update time.

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arXiv2026

Fully Dynamic Algorithms for Graph Spanners via Low-Diameter Router Decomposition

A $t$-spanner of an undirected $n$-vertex graph $G$ is a sparse subgraph $H$ of $G$ that preserves all pairwise distances between its vertices to within multiplicative factor $t$, also called the \emph{stretch}. We investigate the problem of maintaining spanners in the fully dynamic setting with an adaptive adversary. Despite a long line of research, this problem is still poorly understood: no algorithm achieving a sublogarithmic stretch, a sublinear in $n$ update time, and a strongly subquadratic in $n$ spanner size is currently known. One of our main results is a deterministic algorithm, that, for any $512 \leq k \leq (\log n)^{1/49}$ and $1/k\leq δ\leq 1/400$, maintains a spanner $H$ of a fully dynamic graph with stretch $poly(k)\cdot 2^{O(1/δ^6)}$ and size $|E(H)|\leq O(n^{1+O(1/k)})$, with worst-case update time $n^{O(δ)}$ and recourse $n^{O(1/k)}$. Our algorithm relies on a new technical tool that we develop, called low-diameter router decomposition. We design a deterministic algorithm that maintains a decomposition of a fully dynamic graph into edge-disjoint clusters with bounded vertex overlap, where each cluster $C$ is a bounded-diameter router, meaning that any reasonable multicommodity demand over the vertices of $C$ can be routed along short paths and with low congestion. A similar graph decomposition notion was introduced by [Haeupler et al., STOC 2022] and strengthened by [Haeupler et al., FOCS 2024]. However, in contrast to these and other prior works, the decomposition that our algorithm maintains is proper, ensuring that the routing paths between the pairs of vertices of each cluster $C$ are contained inside $C$, rather than in the entire graph $G$. We show additional applications of our router decomposition, including dynamic algorithms for fault-tolerant spanners and low-congestion spanners.

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arXiv2025

Faster Algorithms for Global Minimum Vertex-Cut in Directed Graphs

We study the directed global minimum vertex-cut problem: given a directed vertex-weighted graph $G$, compute a vertex-cut $(L,S,R)$ in $G$ of minimum value, which is defined to be the total weight of all vertices in $S$. The problem, together with its edge-based variant, is one of the most basic in graph theory and algorithms, and has been studied extensively. The fastest currently known algorithm for directed global minimum vertex-cut (Henzinger, Rao and Gabow, FOCS 1996 and J. Algorithms 2000) has running time $\tilde{O}(mn)$, where $m$ and $n$ denote the number of edges and vertices in the input graph, respectively. A long line of work over the past decades led to faster algorithms for other main versions of the problem, including the undirected edge-based setting (Karger, STOC 1996 and J. ACM 2000), directed edge-based setting (Cen et al., FOCS 2021), and undirected vertex-based setting (Chuzhoy and Trabelsi, STOC 2025). However, for the vertex-based version in directed graphs, the 29 year-old $\tilde{O}(mn)$-time algorithm of Henzinger, Rao and Gabow remains the state of the art to this day, in all edge-density regimes. In this paper we break the $Θ(mn)$ running time barrier for the first time, by providing a randomized algorithm for directed global minimum vertex-cut, with running time $O\left(mn^{0.976}\cdot\operatorname{polylog} W\right)$ where $W$ is the ratio of largest to smallest vertex weight. Additionally, we provide a randomized $O\left(\min\left\{m^{1+o(1)}\cdot k,n^{2+o(1)}\right\}\right)$-time algorithm for the unweighted version of directed global minimum vertex-cut, where $k$ is the value of the optimal solution. The best previous algorithm for the problem achieved running time $\tilde O\left(\min\left\{k^2 \cdot m, mn^{11/12+o(1)}, n^{2+o(1)}\right\}\right)$ (Forster et al., SODA 2020, Li et al., STOC 2021).

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arXiv2025

Breaking the O(mn)-Time Barrier for Vertex-Weighted Global Minimum Cut

We consider the Global Minimum Vertex-Cut problem: given an undirected vertex-weighted graph $G$, compute a minimum-weight subset of its vertices whose removal disconnects $G$. The problem is closely related to Global Minimum Edge-Cut, where the weights are on the graph edges instead of vertices, and the goal is to compute a minimum-weight subset of edges whose removal disconnects the graph. Global Minimum Cut is one of the most basic and extensively studied problems in combinatorial optimization and graph theory. While an almost-linear time algorithm was known for the edge version of the problem for awhile (Karger, STOC 1996 and J. ACM 2000), the fastest previous algorithm for the vertex version (Henzinger, Rao and Gabow, FOCS 1996 and J. Algorithms 2000) achieves a running time of $\tilde{O}(mn)$, where $m$ and $n$ denote the number of edges and vertices in the input graph, respectively. For the special case of unit vertex weights, this bound was broken only recently (Li {et al.}, STOC 2021); their result, combined with the recent breakthrough almost-linear time algorithm for Maximum $s$-$t$ Flow (Chen {et al.}, FOCS 2022, van den Brand {et al.}, FOCS 2023), yields an almost-linear time algorithm for Global Minimum Vertex-Cut with unit vertex weights. In this paper we break the $28$ years old bound of Henzinger {et al.} for the general weighted Global Minimum Vertex-Cut, by providing a randomized algorithm for the problem with running time $O(\min\{mn^{0.99+o(1)},m^{1.5+o(1)}\})$.

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arXiv2024

Maximum Bipartite Matching in $n^{2+o(1)}$ Time via a Combinatorial Algorithm

Maximum bipartite matching (MBM) is a fundamental problem in combinatorial optimization with a long and rich history. A classic result of Hopcroft and Karp (1973) provides an $O(m \sqrt{n})$-time algorithm for the problem, where $n$ and $m$ are the number of vertices and edges in the input graph, respectively. For dense graphs, an approach based on fast matrix multiplication achieves a running time of $O(n^{2.371})$. For several decades, these results represented state-of-the-art algorithms, until, in 2013, Madry introduced a powerful new approach for solving MBM using continuous optimization techniques. This line of research led to several spectacular results, culminating in a breakthrough $m^{1+o(1)}$-time algorithm for min-cost flow, that implies an $m^{1+o(1)}$-time algorithm for MBM as well. These striking advances naturally raise the question of whether combinatorial algorithms can match the performance of the algorithms that are based on continuous techniques for MBM. A recent work of the authors (2024) made progress on this question by giving a combinatorial $\tilde{O}(m^{1/3}n^{5/3})$-time algorithm for MBM, thus outperforming both the Hopcroft-Karp algorithm and matrix multiplication based approaches, on sufficiently dense graphs. Still, a large gap remains between the running time of their algorithm and the almost linear-time achievable by algorithms based on continuous techniques. In this work, we take another step towards narrowing this gap, and present a randomized $n^{2+o(1)}$-time combinatorial algorithm for MBM. Thus in dense graphs, our algorithm essentially matches the performance of algorithms that are based on continuous methods. We also obtain a randomized $n^{2+o(1)}$-time combinatorial algorithm for maximum vertex-capacitated $s$-$t$ flow in directed graphs when all vertex capacities are identical, using a standard reduction from this problem to MBM.

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arXiv2023

A Faster Combinatorial Algorithm for Maximum Bipartite Matching

The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp (1973) shows that maximum bipartite matching can be solved in $O(m \sqrt{n})$ time on a graph with $n$ vertices and $m$ edges. For the case of very dense graphs, a fast matrix multiplication based approach gives a running time of $O(n^{2.371})$. These results represented the fastest known algorithms for the problem until 2013, when Madry introduced a new approach based on continuous techniques achieving much faster runtime in sparse graphs. This line of research has culminated in a spectacular recent breakthrough due to Chen et al. (2022) that gives an $m^{1+o(1)}$ time algorithm for maximum bipartite matching (and more generally, for min cost flows). This raises a natural question: are continuous techniques essential to obtaining fast algorithms for the bipartite matching problem? Our work makes progress on this question by presenting a new, purely combinatorial algorithm for bipartite matching, that runs in $\tilde{O}(m^{1/3}n^{5/3})$ time, and hence outperforms both Hopcroft-Karp and the fast matrix multiplication based algorithms on moderately dense graphs. Using a standard reduction, we also obtain an $\tilde{O}(m^{1/3}n^{5/3})$ time deterministic algorithm for maximum vertex-capacitated $s$-$t$ flow in directed graphs when all vertex capacities are identical.

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arXiv2023

A New Deterministic Algorithm for Fully Dynamic All-Pairs Shortest Paths

We study the fully dynamic All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. Given an $n$-vertex graph $G$ with non-negative edge lengths, that undergoes an online sequence of edge insertions and deletions, the goal is to support approximate distance queries and shortest-path queries. We provide a deterministic algorithm for this problem, that, for a given precision parameter $ε$, achieves approximation factor $(\log\log n)^{2^{O(1/ε^3)}}$, and has amortized update time $O(n^ε\log L)$ per operation, where $L$ is the ratio of longest to shortest edge length. Query time for distance-query is $O(2^{O(1/ε)}\cdot \log n\cdot \log\log L)$, and query time for shortest-path query is $O(|E(P)|+2^{O(1/ε)}\cdot \log n\cdot \log\log L)$, where $P$ is the path that the algorithm returns. To the best of our knowledge, even allowing any $o(n)$-approximation factor, no adaptive-update algorithms with better than $Θ(m)$ amortized update time and better than $Θ(n)$ query time were known prior to this work. We also note that our guarantees are stronger than the best current guarantees for APSP in decremental graphs in the adaptive-adversary setting.

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arXiv2022

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

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 $(\log n)^{O(1/ε^2)}$-approximation with $n^{1+O(ε)}$ total update time for any $ε$. All currently known algorithms for the Cut Player in the Cut-Matching game are either randomized, or provide rather weak guarantees. This, in turn, leads to somewhat weak algorithmic guarantees for several central cut problems: for example, the best current almost linear time deterministic algorithm for Sparsest Cut can only achieve approximation factor $(\log n)^{ω(1)}$. Lastly, when relying on expanders in distance-based 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 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.

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arXiv2022

A New Conjecture on Hardness of Low-Degree 2-CSP's with Implications to Hardness of Densest $k$-Subgraph and Other Problems

We propose a new conjecture on hardness of low-degree $2$-CSP's, and show that new hardness of approximation results for Densest $k$-Subgraph and several other problems, including a graph partitioning problem, and a variation of the Graph Crossing Number problem, follow from this conjecture. The conjecture can be viewed as occupying a middle ground between the $d$-to-$1$ conjecture, and hardness results for $2$-CSP's that can be obtained via standard techniques, such as Parallel Repetition combined with standard $2$-prover protocols for the 3SAT problem. We hope that this work will motivate further exploration of hardness of $2$-CSP's in the regimes arising from the conjecture. We believe that a positive resolution of the conjecture will provide a good starting point for further hardness of approximation proofs. Another contribution of our work is proving that the problems that we consider are roughly equivalent from the approximation perspective. Some of these problems arose in previous work, from which it appeared that they may be related to each other. We formalize this relationship in this work.

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arXiv2022

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

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 $O(n^{1/2-\varepsilon}\cdot \text{poly}(Δ\cdot\log n))$-approximation, for a small fixed constant $ε$, 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 $O\left(2^{O((\log n)^{7/8}\log\log n)}\cdot\text{poly}(Δ)\right )$-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 $v\in V(G)$, 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 the input. Combining this result with the recent reduction of [Chuzhoy, Mahabadi, Tan '20] immediately yields the improved approximation algorithm for Minimum Crossing Number. We introduce several new technical tools, that we hope will be helpful in obtaining better algorithms for the problem in the future.

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arXiv2021

Decremental All-Pairs Shortest Paths in Deterministic Near-Linear Time

We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an $n$-vertex $m$-edge graph $G$ with non-negative edge lengths, that undergoes a sequence of edge deletions. The goal is to support approximate shortest-path queries: given a pair $x,y$ of vertices of $G$, return a path $P$ connecting $x$ to $y$, whose length is within factor $α$ of the length of the shortest $x$-$y$ path, in time $\tilde O(|E(P)|)$, where $α$ is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees for the oblivious-adversary setting. Unfortunately, adaptive-adversary setting is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS '13, SICOMP '16] achieves a $(1+ε)$-approximation with total update time $\tilde O(mn/ε)$; the best current total update time of $n^{2.5+O(ε)}$ is achieved by the deterministic algorithm of [Chuzhoy, Saranurak, SODA'21], with $2^{O(1/ε)}$-multiplicative and $2^{O(\log^{3/4}n/ε)}$-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time $O(n^{3}\log L/ε)$, achieving a $(1+ε)$-approximation. Here, L is the ratio of longest to shortest edge lengths. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any $Ω(1/\log\log m)\leq ε< 1$, achieves approximation factor $(\log m)^{2^{O(1/ε)}}$, with total update time $O\left (m^{1+O(ε)}\cdot (\log m)^{O(1/ε^2)}\cdot \log L\right )$.

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arXiv2021

Towards Better Approximation of Graph Crossing Number

Graph Crossing Number is a fundamental problem with various applications. In this problem, the goal is to draw an input graph $G$ in the plane so as to minimize the number of crossings between the images of its edges. Despite extensive work, non-trivial approximation algorithms are only known for bounded-degree graphs. Even for this special case, the best current algorithm achieves a $\tilde O(\sqrt n)$-approximation, while the best current negative result is APX-hardness. All current approximation algorithms for the problem build on the same paradigm: compute a set $E'$ of edges (called a \emph{planarizing set}) such that $G\setminus E'$ is planar; compute a planar drawing of $G\setminus E'$; then add the drawings of the edges of $E'$ to the resulting drawing. Unfortunately, there are examples of graphs, in which any implementation of this method must incur $Ω(\text{OPT}^2)$ crossings, where $\text{OPT}$ is the value of the optimal solution. This barrier seems to doom the only known approach to designing approximation algorithms for the problem, and to prevent it from yielding a better than $O(\sqrt n)$-approximation. In this paper we propose a new paradigm that allows us to overcome this barrier. We show an algorithm that, given a bounded-degree graph $G$ and a planarizing set $E'$ of its edges, computes another set $E''$ with $E'\subseteq E''$, such that $|E''|$ is relatively small, and there exists a near-optimal drawing of $G$ in which only edges of $E''$ participate in crossings. This allows us to reduce the Crossing Number problem to \emph{Crossing Number with Rotation System} -- a variant in which the ordering of the edges incident to every vertex is fixed as part of input. We show a randomized algorithm for this new problem, that allows us to obtain an $O(n^{1/2-ε})$-approximation for Crossing Number on bounded-degree graphs, for some constant $ε>0$.

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arXiv2020

Deterministic Algorithms for Decremental Shortest Paths via Layered Core Decomposition

In the decremental single-source shortest paths (SSSP) problem, the input is an undirected graph $G=(V,E)$ with $n$ vertices and $m$ edges undergoing edge deletions, together with a fixed source vertex $s\in V$. The goal is to maintain a data structure that supports shortest-path queries: given a vertex $v\in V$, quickly return an (approximate) shortest path from $s$ to $v$. The decremental all-pairs shortest paths (APSP) problem is defined similarly, but now the shortest-path queries are allowed between any pair of vertices of $V$. Both problems have been studied extensively since the 80's, and algorithms with near-optimal total update time and query time have been discovered for them. Unfortunately, all these algorithms are randomized and, more importantly, they need to assume an oblivious adversary. Our first result is a deterministic algorithm for the decremental SSSP problem on weighted graphs with $O(n^{2+o(1)})$ total update time, that supports $(1+ε)$-approximate shortest-path queries, with query time $O(|P|\cdot n^{o(1)})$, where $P$ is the returned path. This is the first $(1+ε)$-approximation algorithm against an adaptive adversary that supports shortest-path queries in time below $O(n)$, that breaks the $O(mn)$ total update time bound of the classical algorithm of Even and Shiloah from 1981. Our second result is a deterministic algorithm for the decremental APSP problem on unweighted graphs that achieves total update time $O(n^{2.5+δ})$, for any constant $δ>0$, supports approximate distance queries in $O(\log\log n)$ time; the algorithm achieves an $O(1)$-multiplicative and $n^{o(1)}$-additive approximation on the path length. All previous algorithms for APSP either assume an oblivious adversary or have an $Ω(n^{3})$ total update time when $m=Ω(n^{2})$.

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arXiv2020

On Packing Low-Diameter Spanning Trees

Edge connectivity of a graph is one of the most fundamental graph-theoretic concepts. The celebrated tree packing theorem of Tutte and Nash-Williams from 1961 states that every $k$-edge connected graph $G$ contains a collection $\cal{T}$ of $\lfloor k/2 \rfloor$ edge-disjoint spanning trees, that we refer to as a tree packing; the diameter of the tree packing $\cal{T}$ is the largest diameter of any tree in $\cal{T}$. A desirable property of a tree packing, that is both sufficient and necessary for leveraging the high connectivity of a graph in distributed communication, is that its diameter is low. Yet, despite extensive research in this area, it is still unclear how to compute a tree packing, whose diameter is sublinear in $|V(G)|$, in a low-diameter graph $G$, or alternatively how to show that such a packing does not exist. In this paper we provide first non-trivial upper and lower bounds on the diameter of tree packing. First, we show that, for every $k$-edge connected $n$-vertex graph $G$ of diameter $D$, there is a tree packing $\cal{T}$ of size $Ω(k)$, diameter $O((101k\log n)^D)$, that causes edge-congestion at most $2$. Second, we show that for every $k$-edge connected $n$-vertex graph $G$ of diameter $D$, the diameter of $G[p]$ is $O(k^{D(D+1)/2})$ with high probability, where $G[p]$ is obtained by sampling each edge of $G$ independently with probability $p=Θ(\log n/k)$. This provides a packing of $Ω(k/\log n)$ edge-disjoint trees of diameter at most $O(k^{(D(D+1)/2)})$ each. We then prove that these two results are nearly tight. Lastly, we show that if every pair of vertices in a graph has $k$ edge-disjoint paths of length at most $D$ connecting them, then there is a tree packing of size $k$, diameter $O(D\log n)$, causing edge-congestion $O(\log n)$. We also provide several applications of low-diameter tree packing in distributed computation.

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arXiv2020

A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

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 $(\log m)^{1/ε}$-approximation in time $m^{1+O(1/\sqrtε)}$ for any constant $ε$, 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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arXiv2019

Pinning Down the Strong Wilber 1 Bound for Binary Search Trees

The dynamic optimality conjecture, postulating the existence of an $O(1)$-competitive online algorithm for binary search trees (BSTs), is among the most fundamental open problems in dynamic data structures. Despite extensive work and some notable progress, including, for example, the Tango Trees (Demaine et al., FOCS 2004), that give the best currently known $O(\log \log n)$-competitive algorithm, the conjecture remains widely open. One of the main hurdles towards settling the conjecture is that we currently do not have approximation algorithms achieving better than an $O(\log \log n)$-approximation, even in the offline setting. All known non-trivial algorithms for BST's so far rely on comparing the algorithm's cost with the so-called Wilber's first bound (WB-1). Therefore, establishing the worst-case relationship between this bound and the optimal solution cost appears crucial for further progress, and it is an interesting open question in its own right. Our contribution is two-fold. First, we show that the gap between the WB-1 bound and the optimal solution value can be as large as $Ω(\log \log n/ \log \log \log n)$; in fact, the gap holds even for several stronger variants of the bound. Second, we provide a simple algorithm, that, given an integer $D>0$, obtains an $O(D)$-approximation in time $\exp\left(O\left (n^{1/2^{Ω(D)}}\log n\right )\right )$. In particular, this gives a constant-factor approximation sub-exponential time algorithm. Moreover, we obtain a simpler and cleaner efficient $O(\log \log n)$-approximation algorithm that can be used in an online setting. Finally, we suggest a new bound, that we call {\em Guillotine Bound}, that is stronger than WB, while maintaining its algorithm-friendly nature, that we hope will lead to better algorithms. All our results use the geometric interpretation of the problem, leading to cleaner and simpler analysis.

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arXiv2019

A New Algorithm for Decremental Single-Source Shortest Paths with Applications to Vertex-Capacitated Flow and Cut Problems

We study the vertex-decremental Single-Source Shortest Paths (SSSP) problem: given an undirected graph $G=(V,E)$ with lengths $\ell(e)\geq 1$ on its edges and a source vertex $s$, we need to support (approximate) shortest-path queries in $G$, as $G$ undergoes vertex deletions. In a shortest-path query, given a vertex $v$, we need to return a path connecting $s$ to $v$, whose length is at most $(1+ε)$ times the length of the shortest such path, where $ε$ is a given accuracy parameter. The problem has many applications, for example to flow and cut problems in vertex-capacitated graphs. Our main result is a randomized algorithm for vertex-decremental SSSP with total expected update time $O(n^{2+o(1)}\log L)$, that responds to each shortest-path query in $O(n\log L)$ time in expectation, returning a $(1+ε)$-approximate shortest path. The algorithm works against an adaptive adversary. The main technical ingredient of our algorithm is an $\tilde O(|E(G)|+ n^{1+o(1)})$-time algorithm to compute a \emph{core decomposition} of a given dense graph $G$, which allows us to compute short paths between pairs of query vertices in $G$ efficiently. We believe that this core decomposition algorithm may be of independent interest. We use our result for vertex-decremental SSSP to obtain $(1+ε)$-approximation algorithms for maximum $s$-$t$ flow and minimum $s$-$t$ cut in vertex-capacitated graphs, in expected time $n^{2+o(1)}$, and an $O(\log^4n)$-approximation algorithm for the vertex version of the sparsest cut problem with expected running time $n^{2+o(1)}$. These results improve upon the previous best known results for these problems in the regime where $m= ω(n^{1.5 + o(1)})$.

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arXiv2019

Large Minors in Expanders

In this paper we study expander graphs and their minors. Specifically, we attempt to answer the following question: what is the largest function $f(n,α,d)$, such that every $n$-vertex $α$-expander with maximum vertex degree at most $d$ contains {\bf every} graph $H$ with at most $f(n,α,d)$ edges and vertices as a minor? Our main result is that there is some universal constant $c$, such that $f(n,α,d)\geq \frac{n}{c\log n}\cdot \left(\fracα{d}\right )^c$. This bound achieves a tight dependence on $n$: it is well known that there are bounded-degree $n$-vertex expanders, that do not contain any grid with $Ω(n/\log n)$ vertices and edges as a minor. The best previous result showed that $f(n,α,d) \geq Ω(n/\log^κn)$, where $κ$ depends on both $α$ and $d$. Additionally, we provide a randomized algorithm, that, given an $n$-vertex $α$-expander with maximum vertex degree at most $d$, and another graph $H$ containing at most $\frac{n}{c\log n}\cdot \left(\fracα{d}\right )^c$ vertices and edges, with high probability finds a model of $H$ in $G$, in time poly$(n)\cdot (d/α)^{O\left( \log(d/α) \right)}$. We note that similar but stronger results were independently obtained by Krivelevich and Nenadov: they show that $f(n,α,d)=Ω\left(\frac{nα^2}{d^2\log n} \right)$, and provide an efficient algorithm, that, given an $n$-vertex $α$-expander of maximum vertex degree at most $d$, and a graph $H$ with $O\left( \frac{nα^2}{d^2\log n} \right)$ vertices and edges, finds a model of $H$ in $G$. Finally, we observe that expanders are the `most minor-rich' family of graphs in the following sense: for every $n$-vertex and $m$-edge graph $G$, there exists a graph $H$ with $O \left( \frac{n+m}{\log n} \right)$ vertices and edges, such that $H$ is not a minor of $G$.

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