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Christian Konrad

Publications and source records attributed to Christian Konrad.

At least 19 recordsLinked to original sources
arXiv2026

On the Assadi Liu Tarjan Auction Algorithm for Bipartite Matching: Simplification, Alternative Analysis, and Hard Instance

Assadi, Liu, and Tarjan [SOSA'21] gave an auction algorithm that outputs a $(1-\epsilon)$-approximation to Maximum Matching in bipartite graphs. Their algorithm computes a sequence of $O(\frac{1}{\epsilon^2})$ maximal matchings in subgraphs of the input graph and can be implemented in the multi-pass streaming setting with $O(\frac{1}{\epsilon^2})$ passes in a straightforward manner, which constitutes the state-of-the-art pass/approximation trade-off result in the multi-pass streaming setting. Their analysis uses tools from combinatorial auctions and, at its heart, relies on a clever potential function argument. Their proof, however, provides only limited insight into the inner workings of the algorithm. In this paper, we revisit the ALT-algorithm and present the following contributions. Simplification: The ALT-algorithm is built upon a freezing mechanism where vertices on one side of the bipartition that have already been rematched $\Theta(\frac{1}{\epsilon})$ times over the course of the algorithm remain matched to their current partner forever. We show that this mechanism is in fact unnecessary, i.e., no special treatment of such vertices is needed. Alternative Analysis: We give an alternative analysis of the algorithm that is based on augmenting paths. Our analysis allows for a reinterpretation as one that follows the traditional approach of searching for and eliminating augmenting paths. Our analysis also copes with the removal of the freezing mechanism in a natural way, whereas the analysis of Assadi et al. strictly depends on its use. Hard Instance: We provide the first hard instance on which the algorithm requires $\Omega(\frac{1}{\epsilon^2})$ iterations/maximal matching computations. The instance is a simple path graph, where we exhibit a cyclic behaviour that prevents fast progress.

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arXiv2026

Fault-Tolerant Distance Oracles Below the $n \cdot f$ Barrier

Fault-tolerant spanners are fundamental objects that preserve distances in graphs even under edge failures. A long line of work culminating in Bodwin, Dinitz, Robelle (SODA 2022) gives $(2k-1)$-stretch, $f$-fault-tolerant spanners with $O(k^2 f^{\frac{1}{2}-\frac{1}{2k}} n^{1+\frac{1}{k}} + k f n)$ edges for any odd $k$. For any $k = \tilde{O}(1)$, this bound is essentially optimal for deterministic spanners in part due to a known folklore lower bound that \emph{any} $f$-fault-tolerant spanner requires $\Omega(nf)$ edges in the worst case. For $f \geq n$, this $\Omega(nf)$ barrier means that any $f$-fault tolerant spanners are trivial in size. Crucially however, this folklore lower bound exploits that the spanner \emph{is itself a subgraph}. It does not rule out distance-reporting data structures that may not be subgraphs. This leads to our central question: can one beat the $n \cdot f$ barrier with fault-tolerant distance oracles? We give a strong affirmative answer to this question. As our first contribution, we construct $f$-fault-tolerant distance oracles with stretch $O(\log(n)\log\log(n))$ that require only $\widetilde{O}(n\sqrt{f})$ bits of space; substantially below the spanner barrier of $n \cdot f$. Beyond this, in the regime $n \leq f \leq n^{3/2}$ we show that by using our new \emph{high-degree, low-diameter} decomposition in combination with tools from sparse recovery, we can even obtain stretch $7$ distance oracles in space $\widetilde{O}(n^{3/2}f^{1/3})$ bits. We also show that our techniques are sufficiently general to yield randomized sketches for fault-tolerant ``oblivious'' spanners and fault-tolerant deterministic distance oracles in bounded-deletion streams, with space below the $nf$ barrier in both settings.

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arXiv2026

Unit Interval Selection in Random Order Streams

We consider the \textsf{Unit Interval Selection} problem in the one-pass random order streaming model. Here, an algorithm is presented a sequence of $n$ unit-length intervals on the line that arrive in uniform random order, and the objective is to output a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, in this space constraint, a $(2/3)$-approximation can be achieved, and this is also best possible, i.e. any improvement requires space $\Omega(n)$ [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, with the expectation taken over the stream order. Specifically, we give a one-pass streaming algorithm with expected approximation factor $0.7401$ using space $O(|OPT|)$, where $OPT$ denotes an optimal solution. We also show that algorithms with expected approximation factor above $8/9$ require space $\Omega(n)$, and algorithms that compute a better than $2/3$-approximation with probability above $2/3$ also require $\Omega(n)$ space. On a technical note, we design an algorithm for the restricted domain $[0,\Delta)$, for some constant $\Delta$, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain $[0,\Delta)$, we run $O(\Delta)$ recursive instances of our algorithm, with each instance targeting the situation where a specific interval from $OPT$ arrives first. We establish the interesting property that our algorithm performs worst when the input stream is precisely a set of independent intervals. We then analyse the algorithm on these instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds by [Chakrabarti et al., Theory Comput. 2016].

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arXiv2025

Semi-Robust Communication Complexity of Maximum Matching

We study the one-way two-party communication complexity of Maximum Matching in the semi-robust setting where the edges of a maximum matching are randomly partitioned between Alice and Bob, but all remaining edges of the input graph are adversarially partitioned between the two parties. We show that the simple protocol where Alice solely communicates a lexicographically-first maximum matching of their edges to Bob is surprisingly powerful: We prove that it yields a $3/4$-approximation in expectation and that our analysis is tight. The semi-robust setting is at least as hard as the fully robust setting. In this setting, all edges of the input graph are randomly partitioned between Alice and Bob, and the state-of-the-art result is a fairly involved $5/6$-approximation protocol that is based on the computation of edge-degree constrained subgraphs [Azarmehr, Behnezhad, ICALP'23]. Our protocol also immediately yields a $3/4$-approximation in the fully robust setting. One may wonder whether an improved analysis of our protocol in the fully robust setting is possible: While we cannot rule this out, we give an instance where our protocol only achieves a $0.832 < 5/6 = 0.83$-approximation. Hence, while our simple protocol performs surprisingly well, it cannot be used to improve over the state-of-the-art in the fully robust setting.

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arXiv2025

Constructing Long Paths in Graph Streams

In the graph stream model of computation, an algorithm processes the edges of an input graph in one or more sequential passes while using a memory sublinear in the input size. This model poses significant challenges for constructing long paths. Many known algorithms tasked with extending an existing path as a subroutine require an entire pass to add a single additional edge. This raises a fundamental question: Are multiple passes inherently necessary to construct paths of non-trivial lengths, or can a single pass suffice? To address this question, we study the Longest Path problem in the one-pass streaming model. In this problem, given a desired approximation factor $\alpha$, the objective is to compute a path of length at least $\lp(G) / \alpha$, where $\lp(G)$ is the length of a longest path in the input graph. We give algorithms as well as space lower bounds for both undirected and directed graphs. Our results include: We show that for undirected graphs, in both the insertion-only and the insertion-deletion models, there are semi-streaming algorithms, that compute a path of length at least $d /3$ with high probability, where $d$ is the average degree of the graph. These algorithms can also yield an $\alpha$-approximation to Longest Path using space $\tilde{O}(n^2 / \alpha)$. Next, we show that such a result cannot be achieved for directed graphs, even in the insertion-only model. We show that computing a $(n^{1 - o(1)})$-approximation to Longest Path in directed graphs in the insertion-only model requires space $\Omega(n^2)$. We further show two additional lower bounds. First, we show that semi-streaming space is insufficient for small constant factor approximations to Longest Path for undirected graphs in the insertion-only model. Last, in undirected graphs in the insertion-deletion model, we show that computing an $\alpha$-approximation requires space $\Omega(n^2 / \alpha^3)$.

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arXiv2025

Streaming Maximal Matching with Bounded Deletions

We initiate the study of the Maximal Matching problem in bounded-deletion graph streams. In this setting, a graph $G$ is revealed as an arbitrary sequence of edge insertions and deletions, where the number of insertions is unrestricted but the number of deletions is guaranteed to be at most $K$, for some given parameter $K$. The single-pass streaming space complexity of this problem is known to be $\Theta(n^2)$ when $K$ is unrestricted, where $n$ is the number of vertices of the input graph. In this work, we present new randomized and deterministic algorithms and matching lower bound results that together give a tight understanding (up to poly-log factors) of how the space complexity of Maximal Matching evolves as a function of the parameter $K$: The randomized space complexity of this problem is $\tilde{\Theta}(n \cdot \sqrt{K})$, while the deterministic space complexity is $\tilde{\Theta}(n \cdot K)$. We further show that if we relax the maximal matching requirement to an $\alpha$-approximation to Maximum Matching, for any constant $\alpha > 2$, then the space complexity for both, deterministic and randomized algorithms, strikingly changes to $\tilde{\Theta}(n + K)$.

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arXiv2024

Settling the Pass Complexity of Approximate Matchings in Dynamic Graph Streams

A semi-streaming algorithm in dynamic graph streams processes any $n$-vertex graph by making one or multiple passes over a stream of insertions and deletions to edges of the graph and using $O(n \cdot \mbox{polylog}(n))$ space. Semi-streaming algorithms for dynamic streams were first obtained in the seminal work of Ahn, Guha, and McGregor in 2012, alongside the introduction of the graph sketching technique, which remains the de facto way of designing algorithms in this model and a highly popular technique for designing graph algorithms in general. We settle the pass complexity of approximating maximum matchings in dynamic streams via semi-streaming algorithms by improving the state-of-the-art in both upper and lower bounds. We present a randomized sketching based semi-streaming algorithm for $O(1)$-approximation of maximum matching in dynamic streams using $O(\log\log{n})$ passes. The approximation ratio of this algorithm can be improved to $(1+ε)$ for any fixed $ε> 0$ even on weighted graphs using standard techniques. This exponentially improves upon several $O(\log{n})$ pass algorithms developed for this problem since the introduction of the dynamic graph streaming model. In addition, we prove that any semi-streaming algorithm (not only sketching based) for $O(1)$-approximation of maximum matching in dynamic streams requires $Ω(\log\log{n})$ passes. This presents the first multi-pass lower bound for this problem, which is already also optimal, settling a longstanding open question in this area.

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arXiv2024

Interval Selection in Sliding Windows

We initiate the study of the Interval Selection problem in the (streaming) sliding window model of computation. In this problem, an algorithm receives a potentially infinite stream of intervals on the line, and the objective is to maintain at every moment an approximation to a largest possible subset of disjoint intervals among the $L$ most recent intervals, for some integer $L$. We give the following results: - In the unit-length intervals case, we give a $2$-approximation sliding window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, and we show that any sliding window algorithm that computes a $(2-\varepsilon)$-approximation requires space $\Omega(L)$, for any $\varepsilon > 0$. - In the arbitrary-length case, we give a $(\frac{11}{3}+\varepsilon)$-approximation sliding window algorithm with space $\tilde{\mathrm{O}}(|OPT|)$, for any constant $\varepsilon > 0$, which constitutes our main result. We also show that space $\Omega(L)$ is needed for algorithms that compute a $(2.5-\varepsilon)$-approximation, for any $\varepsilon > 0$. Our main technical contribution is an improvement over the smooth histogram technique, which consists of running independent copies of a traditional streaming algorithm with different start times. By employing the one-pass $2$-approximation streaming algorithm by Cabello and P\'{e}rez-Lantero [Theor. Comput. Sci. '17] for Interval Selection on arbitrary-length intervals as the underlying algorithm, the smooth histogram technique immediately yields a $(4+\varepsilon)$-approximation in this setting. Our improvement is obtained by forwarding the structure of the intervals identified in a run to the subsequent run, which constrains the shape of an optimal solution and allows us to target optimal intervals differently.

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arXiv2024

Graph Reconstruction via MIS Queries

In the Graph Reconstruction (GR) problem, a player initially only knows the vertex set $V$ of an input graph $G=(V, E)$ and is required to learn its set of edges $E$. To this end, the player submits queries to an oracle and must deduce $E$ from the oracle's answers. In this paper, we initiate the study of GR via Maximal Independent Set (MIS) queries, a more powerful variant of Independent Set (IS) queries. Given a query $U \subseteq V$, the oracle responds with any, potentially adversarially chosen, maximal independent set $I \subseteq U$ in the induced subgraph $G[U]$. We show that, for GR, MIS queries are strictly more powerful than IS queries when parametrized by the maximum degree $\Delta$ of the input graph. We give tight (up to poly-logarithmic factors) upper and lower bounds for this problem: 1. We observe that the simple strategy of taking uniform independent random samples of $V$ and submitting those to the oracle yields a non-adaptive randomized algorithm that executes $O(\Delta^2 \cdot \log n)$ queries and succeeds with high probability. Furthermore, combining the strategy of taking uniform random samples of $V$ with the probabilistic method, we show the existence of a deterministic non-adaptive algorithm that executes $O(\Delta^3 \cdot \log(\frac{n}{\Delta}))$ queries. 2. Regarding lower bounds, we prove that the additional $\Delta$ factor when going from randomized non-adaptive algorithms to deterministic non-adaptive algorithms is necessary. We show that every non-adaptive deterministic algorithm requires $\Omega(\Delta^3 / \log^2 \Delta)$ queries. For arbitrary randomized adaptive algorithms, we show that $\Omega(\Delta^2)$ queries are necessary in graphs of maximum degree $\Delta$, and that $\Omega(\log n)$ queries are necessary, even when the input graph is an $n$-vertex cycle.

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arXiv2023

$\mathcal{O}(\log\log{n})$ Passes is Optimal for Semi-Streaming Maximal Independent Set

In the semi-streaming model for processing massive graphs, an algorithm makes multiple passes over the edges of a given $n$-vertex graph and is tasked with computing the solution to a problem using $O(n \cdot \text{polylog}(n))$ space. Semi-streaming algorithms for Maximal Independent Set (MIS) that run in $O(\log\log{n})$ passes have been known for almost a decade, however, the best lower bounds can only rule out single-pass algorithms. We close this large gap by proving that the current algorithms are optimal: Any semi-streaming algorithm for finding an MIS with constant probability of success requires $Ω(\log\log{n})$ passes. This settles the complexity of this fundamental problem in the semi-streaming model, and constitutes one of the first optimal multi-pass lower bounds in this model. We establish our result by proving an optimal round vs communication tradeoff for the (multi-party) communication complexity of MIS. The key ingredient of this result is a new technique, called hierarchical embedding, for performing round elimination: we show how to pack many but small hard $(r-1)$-round instances of the problem into a single $r$-round instance, in a way that enforces any $r$-round protocol to effectively solve all these $(r-1)$-round instances also. These embeddings are obtained via a novel application of results from extremal graph theory -- in particular dense graphs with many disjoint unique shortest paths -- together with a newly designed graph product, and are analyzed via information-theoretic tools such as direct-sum and message compression arguments.

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arXiv2023

Improved Weighted Matching in the Sliding Window Model

We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set $V$ of size $n$. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the $L$ most recent edges, for some integer $L$, using as little space as possible. Prior to our work, the state-of-the-art results were a $(3.5+\varepsilon)$-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a $(3+\varepsilon)$-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space $\tilde{O}(n)$. We give the following results: 1. We give a $(2+\varepsilon)$-approximation algorithm for MWM with space $\tilde{O}(\sqrt{nL})$. Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space $\tilde{O}(n \sqrt{n})$. 2. In the $\tilde{O}(n)$ space regime, we give a $(3+\varepsilon)$-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.'s MWM algorithm, both our algorithms execute multiple instances of the $(2+\varepsilon)$-approximation $\tilde{O}(n)$-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our $(2+\varepsilon)$-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements.

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arXiv2022

Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

We present $O(\log\log n)$-round algorithms in the Massively Parallel Computation (MPC) model, with $\tilde{O}(n)$ memory per machine, that compute a maximal independent set, a $1+ε$ approximation of maximum matching, and a $2+ε$ approximation of minimum vertex cover, for any $n$-vertex graph and any constant $ε>0$. These improve the state of the art as follows: - Our MIS algorithm leads to a simple $O(\log\log Δ)$-round MIS algorithm in the Congested Clique model of distributed computing, which improves on the $\tilde{O}(\sqrt{\log Δ})$-round algorithm of Ghaffari [PODC'17]. - Our $O(\log\log n)$-round $(1+ε)$-approximate maximum matching algorithm simplifies or improves on the following prior work: $O(\log^2\log n)$-round $(1+ε)$-approximation algorithm of Czumaj et al. [STOC'18] and $O(\log\log n)$-round $(1+ε)$-approximation algorithm of Assadi et al. [SODA'19]. - Our $O(\log\log n)$-round $(2+ε)$-approximate minimum vertex cover algorithm improves on an $O(\log\log n)$-round $O(1)$-approximation of Assadi et al. [arXiv'17].

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arXiv2021

On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

We study two-pass streaming algorithms for Maximum Bipartite Matching (MBM). All known two-pass streaming algorithms for MBM operate in a similar fashion: They compute a maximal matching in the first pass and find 3-augmenting paths in the second in order to augment the matching found in the first pass. Our aim is to explore the limitations of this approach and to determine whether current techniques can be used to further improve the state-of-the-art algorithms. We give the following results: We show that every two-pass streaming algorithm that solely computes a maximal matching in the first pass and outputs a $(2/3+ε)$-approximation requires $n^{1+Ω(\frac{1}{\log \log n})}$ space, for every $ε> 0$, where $n$ is the number of vertices of the input graph. This result is obtained by extending the Ruzsa-Szemerédi graph construction of [GKK, SODA'12] so as to ensure that the resulting graph has a close to perfect matching, the key property needed in our construction. This result may be of independent interest. Furthermore, we combine the two main techniques, i.e., subsampling followed by the Greedy matching algorithm [Konrad, MFCS'18] which gives a $2-\sqrt{2} \approx 0.5857$-approximation, and the computation of \emph{degree-bounded semi-matchings} [EHM, ICDMW'16][KT, APPROX'17] which gives a $\frac{1}{2} + \frac{1}{12} \approx 0.5833$-approximation, and obtain a meta-algorithm that yields Konrad's and Esfandiari et al.'s algorithms as special cases. This unifies two strands of research. By optimizing parameters, we discover that Konrad's algorithm is optimal for the implied class of algorithms and, perhaps surprisingly, that there is a second optimal algorithm. We show that the analysis of our meta-algorithm is best possible. Our results imply that further improvements, if possible, require new techniques.

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arXiv2021

The Complexity of Symmetry Breaking in Massive Graphs

The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related $β$-ruling set problem, in two computational models suited for large-scale graph processing, namely the $k$-machine model and the graph streaming model. We present a number of results. For MIS in the $k$-machine model, we improve the $\tilde{O}(m/k^2 + Δ/k)$-round upper bound of Klauck et al. (SODA 2015) by presenting an $\tilde{O}(m/k^2)$-round algorithm. We also present an $\tildeΩ(n/k^2)$ round lower bound for MIS, the first lower bound for a symmetry breaking problem in the $k$-machine model. For $β$-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the $k$-machine model and also in the graph streaming model. More specifically, we obtain a $k$-machine algorithm that runs in $\tilde{O}(βnΔ^{1/β}/k^2)$ rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use $O(β\cdot n^{1+1/2^{β-1}})$ space. The latter result establishes a clear separation between MIS, which is known to require $Ω(n^2)$ space (Cormode et al., ICALP 2019), and $β$-ruling sets, even for $β= 2$. Finally, we present an even faster 2-ruling set algorithm in the $k$-machine model, one that runs in $\tilde{O}(n/k^{2-ε} + k^{1-ε})$ rounds for any $ε$, $0 \le ε\le 1$.

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arXiv2021

Robust Lower Bounds for Graph Problems in the Blackboard Model of Communication

We give lower bounds on the communication complexity of graph problems in the multi-party blackboard model. In this model, the edges of an $n$-vertex input graph are partitioned among $k$ parties, who communicate solely by writing messages on a shared blackboard that is visible to every party. We show that any non-trivial graph problem on $n$-vertex graphs has blackboard communication complexity $Ω(n)$ bits, even if the edges of the input graph are randomly assigned to the $k$ parties. We say that a graph problem is non-trivial if the output cannot be computed in a model where every party holds at most one edge and no communication is allowed. Our lower bound thus holds for essentially all key graph problems relevant to distributed computing, including Maximal Independent Set (MIS), Maximal Matching, ($Δ+1$)-coloring, and Dominating Set. In many cases, e.g., MIS, Maximal Matching, and $(Δ+1)$-coloring, our lower bounds are optimal, up to poly-logarithmic factors.

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arXiv2021

Frequent Elements with Witnesses in Data Streams

Detecting frequent elements is among the oldest and most-studied problems in the area of data streams. Given a stream of $m$ data items in $\{1, 2, \dots, n\}$, the objective is to output items that appear at least $d$ times, for some threshold parameter $d$, and provably optimal algorithms are known today. However, in many applications, knowing only the frequent elements themselves is not enough: For example, an Internet router may not only need to know the most frequent destination IP addresses of forwarded packages, but also the timestamps of when these packages appeared or any other meta-data that "arrived" with the packages, e.g., their source IP addresses. In this paper, we introduce the witness version of the frequent elements problem: Given a desired approximation guarantee $α\ge 1$ and a desired frequency $d \le Δ$, where $Δ$ is the frequency of the most frequent item, the objective is to report an item together with at least $d / α$ timestamps of when the item appeared in the stream (or any other meta-data that arrived with the items). We give provably optimal algorithms for both the insertion-only and insertion-deletion stream settings: In insertion-only streams, we show that space $\tilde{O}(n + d \cdot n^{\frac{1}α})$ is necessary and sufficient for every integral $1 \le α\le \log n$. In insertion-deletion streams, we show that space $\tilde{O}(\frac{n \cdot d}{α^2})$ is necessary and sufficient, for every $α\le \sqrt{n}$.

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arXiv2020

Constructing Large Matchings via Query Access to a Maximal Matching Oracle

Multi-pass streaming algorithm for Maximum Matching have been studied since more than 15 years and various algorithmic results are known today, including $2$-pass streaming algorithms that break the $1/2$-approximation barrier, and $(1-ε)$-approximation streaming algorithms that run in $O(\text{poly} \frac{1}ε)$ passes in bipartite graphs and in $O( (\frac{1}ε)^{\frac{1}ε})$ or $O(\text{poly} (\frac{1}ε) \cdot \log n)$ passes in general graphs, where $n$ is the number of vertices of the input graph. However, proving impossibility results for such algorithms has so far been elusive, and, for example, even the existence of $2$-pass small space streaming algorithms with approximation factor $0.999$ has not yet been ruled out. The key building block of all multi-pass streaming algorithms for Maximum Matching is the Greedy matching algorithm. Our aim is to understand the limitations of this approach: How many passes are required if the algorithm solely relies on the invocation of the Greedy algorithm? In this paper, we initiate the study of lower bounds for restricted families of multi-pass streaming algorithms for Maximum Matching. We focus on the simple yet powerful class of algorithms that in each pass run Greedy on a vertex-induced subgraph of the input graph. In bipartite graphs, we show that $3$ passes are necessary and sufficient to improve on the trivial approximation factor of $1/2$: We give a lower bound of $0.6$ on the approximation ratio of such algorithms, which is optimal. We further show that $Ω( \frac{1}ε)$ passes are required for computing a $(1-ε)$-approximation, even in bipartite graphs. Last, the considered class of algorithms is not well-suited to general graphs: We show that $Ω(n)$ passes are required in order to improve on the trivial approximation factor of $1/2$.

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arXiv2020

Optimal Lower Bounds for Matching and Vertex Cover in Dynamic Graph Streams

In this paper, we give simple optimal lower bounds on the one-way two-party communication complexity of approximate Maximum Matching and Minimum Vertex Cover with deletions. In our model, Alice holds a set of edges and sends a single message to Bob. Bob holds a set of edge deletions, which form a subset of Alice's edges, and needs to report a large matching or a small vertex cover in the graph spanned by the edges that are not deleted. Our results imply optimal space lower bounds for insertion-deletion streaming algorithms for Maximum Matching and Minimum Vertex Cover. Previously, Assadi et al. [SODA 2016] gave an optimal space lower bound for insertion-deletion streaming algorithms for Maximum Matching via the simultaneous model of communication. Our lower bound is simpler and stronger in several aspects: The lower bound of Assadi et al. only holds for algorithms that (1) are able to process streams that contain a triple exponential number of deletions in $n$, the number of vertices of the input graph; (2) are able to process multi-graphs; and (3) never output edges that do not exist in the input graph when the randomized algorithm errs. In contrast, our lower bound even holds for algorithms that (1) rely on short ($O(n^2)$-length) input streams; (2) are only able to process simple graphs; and (3) may output non-existing edges when the algorithm errs.

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