arXiv subjects

Soheil Behnezhad

Publications and source records attributed to Soheil Behnezhad.

At least 19 recordsLinked to original sources

arXiv2026

Streaming Algorithms for Monotonicity Testing

Consider a poset - or equivalently an $n$-vertex DAG $G=(V, E)$ - and a boolean function $f: V \rightarrow \{0, 1\}$ on its vertex set. We say $f$ is monotone if $f(u) \leq f(v)$ for all $(u, v) \in E$. While there is extensive literature on the query complexity of testing monotonicity, we focus instead on the space complexity and initiate the study of this problem in the streaming setting. Namely, the edges of $G$ arrive in an arbitrary order, and the goal is to estimate distance to monotonicity of a given function $f$ using $\widetilde{O}(n)$ space. Note that while this space allows receiving and storing $f$, it is much smaller than the input graph $G$ which could have up to $\Omega(n^2)$ edges. Our main result is an algorithm that $(1+\epsilon)$-approximates distance to monotonicity in $\sqrt{n}^{1+o(1)}$ passes. We also prove that this is the best pass-complexity one can hope for, for any $O(1)$-approximation, short of improving the state-of-the-art streaming algorithm for $st$-reachability, which is a very well-studied problem. On the technical side, our algorithm approximates the size of maximum matching in (a subgraph of) the transitive closure of $G$. While the maximum matching problem has received significant attention in the streaming setting, the fact that we are computing it in the transitive closure requires very different ideas. In fact, a main contribution of our work is to connect sublinear time algorithms for estimating the maximum matching size to the streaming setting for the first time. While existing off-the-shelf sublinear time algorithms only result in an $n\sqrt{n}^{1+o(1)}$ pass algorithm in our setting, we show how to significantly improve upon them by allowing stronger queries (such as vertex and subset queries) that can be implemented just as efficiently as more standard adjacency matrix and list queries for our problem.

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arXiv2026

Single-Pass Streaming CSPs via Two-Tier Sampling

We study the maximum constraint satisfaction problem, Max-CSP, in the streaming setting. Given $n$ variables, the constraints arrive sequentially in an arbitrary order, with each constraint involving only a small subset of the variables. The objective is to approximate the maximum fraction of constraints that can be satisfied by an optimal assignment in a single pass. The problem admits a trivial near-optimal solution with $O(n)$ space, so the major open problem in the literature has been the best approximation achievable when limiting the space to $o(n)$. The answer to the question above depends heavily on the CSP instance at hand. The integrality gap $α$ of an LP relaxation, known as the BasicLP, plays a central role. In particular, a major conjecture of the area is that in the single-pass streaming setting, for any fixed $\varepsilon > 0$, (i) an $(α-\varepsilon)$-approximation can be achieved with $o(n)$ space, and (ii) any $(α+\varepsilon)$-approximation requires $Ω(n)$ space. In this work, we fully resolve the first side of the conjecture by proving that an $(α- \varepsilon)$-approximation of Max-CSP can indeed be achieved using $n^{1-Ω_\varepsilon(1)}$ space and in a single pass. Given that Max-DiCut is a special case of Max-CSP, our algorithm fully recovers the recent result of [ABFS26, STOC'26] via a completely different algorithm and proof. On a technical level, our algorithm simulates a suitable local algorithm on a reduced graph using a technique that we call *two-tier sampling*: the algorithm combines both edge sampling and vertex sampling to handle high- and low-degree vertices at the same time.

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arXiv2026

Half-Approximating Maximum Dicut in the Streaming Setting

We study streaming algorithms for the maximum directed cut problem. The edges of an $n$-vertex directed graph arrive one by one in an arbitrary order, and the goal is to estimate the value of the maximum directed cut using a single pass and small space. With $O(n)$ space, a $(1-\varepsilon)$-approximation can be trivially obtained for any fixed $\varepsilon > 0$ using additive cut sparsifiers. The question that has attracted significant attention in the literature is the best approximation achievable by algorithms that use truly sublinear (i.e., $n^{1-Ω(1)}$) space. A lower bound of Kapralov and Krachun (STOC'19) implies .5-approximation is the best one can hope for. The current best algorithm for general graphs obtains a .485-approximation due to the work of Saxena, Singer, Sudan, and Velusamy (FOCS'23). The same authors later obtained a $(1/2-\varepsilon)$-approximation, assuming that the graph is constant-degree (SODA'25). In this paper, we show that for any $\varepsilon > 0$, a $(1/2-\varepsilon)$-approximation of maximum dicut value can be obtained with $n^{1-Ω_\varepsilon(1)}$ space in *general graphs*. This shows that the lower bound of Kapralov and Krachun is generally tight, settling the approximation complexity of this fundamental problem. The key to our result is a careful analysis of how correlation propagates among high- and low-degree vertices, when simulating a suitable local algorithm.

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arXiv2026

Caterpillar of Thoughts: The Optimal Test-Time Algorithm for Large Language Models

Large language models (LLMs) can often produce substantially better outputs when allowed to use additional test-time computation, such as sampling, chain of thought, backtracking, or revising partial solutions. Despite the growing empirical success of such techniques, there is limited theoretical understanding of how inference time computation should be structured, or what constitutes an optimal use of a fixed computation budget. We model test-time computation as an algorithm interacting with a Markov chain: at any point, the algorithm may resume generation from any previously observed state. That is, unlike standard Markov chains where the states are drawn passively, we allow the algorithm to backtrack to any previously observed state of the Markov chain at any time. Many of the existing test-time algorithms, such as Chain-of-Thought (CoT) (Wei et al., 2023), Tree-of-Thoughts (ToT) (Yao et al., 2023), or Best-of-$k$ (Brown et al., 2024) could be seen as specific algorithms in this model. We prove that while backtracking can reduce the number of generations exponentially, a very limited form of backtracking is theoretically sufficient. Namely, we show that the optimal algorithm always generates a caterpillar tree. That is, if we remove the leaves of the state tree generated by the optimal algorithm, we obtain a path. Motivated by our characterization of the optimal algorithm, we present Caterpillar of Thoughts (CaT), a new test-time computation algorithm, reducing the number of token/state generations. Our empirical evaluation shows that CaT, compared to ToT, achieves a better success rate while also reducing the number of token generations.

cs.LG↗

arXiv2026

Markov Chains with Rewinding

Motivated by techniques developed in recent progress on lower bounds for sublinear time algorithms (Behnezhad, Roghani and Rubinstein, STOC 2023, FOCS 2023, and STOC 2024) we introduce and study a new class of randomized algorithmic processes that we call Markov Chains with Rewinding. In this setting, an algorithm interacts with a (partially observable) Markovian random evolution by strategically rewinding the Markov chain to previous states. Depending on the application, this may lead the evolution to desired states faster, or allow the agent to efficiently learn or test properties of the underlying Markov chain that may be infeasible or inefficient with passive observation. We study the task of identifying the initial state in a given partially observable Markov chain. Analysis of this question in specific Markov chains is the central ingredient in the above cited works and we aim to systematize the analysis in our work. Our first result is that any pair of states distinguishable with any rewinding strategy can also be distinguished with a non-adaptive rewinding strategy (one whose rewinding choices are determined before observing any outcomes of the chain). Therefore, while rewinding strategies can be shown to be strictly more powerful than passive strategies (those that do not rewind back to previous states), adaptivity does not give additional power to a rewinding strategy in the absence of efficiency considerations. The difference becomes apparent however when we introduce a natural efficiency measure, namely the query complexity (i.e., the number of observations they need to identify distinguishable states). Our second main contribution is to quantify this efficiency gap. We present a non-adaptive rewinding strategy whose query complexity is within a polynomial of that of the optimal (adaptive) strategy, and show that such a polynomial loss is necessary in general.

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arXiv2025

Tight Pair Query Lower Bounds for Matching and Earth Mover's Distance

How many adjacency matrix queries (also known as pair queries) are required to estimate the size of a maximum matching in an $n$-vertex graph $G$? We study this fundamental question in this paper. On the upper bound side, an algorithm of Bhattacharya, Kiss, and Saranurak [FOCS'23] gives an estimate that is within $εn$ of the right bound with $n^{2-Ω_ε(1)}$ queries, which is subquadratic in $n$ (and thus sublinear in the matrix size) for any fixed $ε> 0$. On the lower bound side, while there has been a lot of progress in the adjacency list model, no non-trivial lower bound has been established for algorithms with adjacency matrix query access. In particular, the only known lower bound is a folklore bound of $Ω(n)$, leaving a huge gap. In this paper, we present the first superlinear in $n$ lower bound for this problem. In fact, we close the gap mentioned above entirely by showing that the algorithm of [BKS'23] is optimal. Formally, we prove that for any fixed $δ> 0$, there is a fixed $ε> 0$ such that an estimate that is within $εn$ of the true bound requires $Ω(n^{2-δ})$ adjacency matrix queries. Our lower bound also has strong implications for estimating the earth mover's distance between distributions. For this problem, Beretta and Rubinstein [STOC'24] gave an $n^{2-Ω_ε(1)}$ time algorithm that obtains an additive $ε$-approximation and works for any distance function. Whether this can be improved generally, or even for metric spaces, had remained open. Our lower bound rules out the possibility of any improvements over this bound, even under the strong assumption that the underlying distances are in a (1, 2)-metric.

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arXiv2025

Vizing's Theorem in Deterministic Almost-Linear Time

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors. Vizing's original proof is easily translated into a deterministic $O(mn)$ time algorithm. This deterministic time bound was subsequently improved to $\tilde O(m \sqrt n)$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. A series of recent papers improved the time bound of $\tilde O(m\sqrt{n})$ using randomization, culminating in the randomized near-linear time $(Δ+1)$-coloring algorithm by [Assadi, Behnezhad, Bhattacharya, Costa, Solomon, and Zhang, 2025]. At the heart of all of these recent improvements, there is some form of a sublinear time algorithm. Unfortunately, sublinear time algorithms as a whole almost always require randomization. This raises a natural question: can the deterministic time complexity of the problem be reduced below the $\tilde O(m\sqrt{n})$ barrier? In this paper, we answer this question in the affirmative. We present a deterministic almost-linear time $(Δ+1)$-coloring algorithm, namely, an algorithm running in $m \cdot 2^{O(\sqrt{\log Δ})} \cdot \log n = m^{1+o(1)}$ time. Our main technical contribution is to entirely forego sublinear time algorithms. We do so by presenting a new deterministic color-type sparsification approach that runs in almost-linear (instead of sublinear) time, but can be used to color a much larger set of edges.

cs.DS↗

arXiv2025

Vizing's Theorem in Near-Linear Time

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a $(Δ+1)$-edge coloring in near-linear time -- in fact, only $O(m\logΔ)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem.

cs.DS↗

arXiv2025

Massively Parallel Minimum Spanning Tree in General Metric Spaces

We study the minimum spanning tree (MST) problem in the massively parallel computation (MPC) model. Our focus is particularly on the *strictly sublinear* regime of MPC where the space per machine is $O(n^δ)$. Here $n$ is the number of vertices and constant $δ\in (0, 1)$ can be made arbitrarily small. The MST problem admits a simple and folklore $O(\log n)$-round algorithm in the MPC model. When the weights can be arbitrary, this matches a conditional lower bound of $Ω(\log n)$ which follows from a well-known 1vs2-Cycle conjecture. As such, much of the literature focuses on breaking the logarithmic barrier in more structured variants of the problem, such as when the vertices correspond to points in low- [ANOY14, STOC'14] or high-dimensional Euclidean spaces [JMNZ, SODA'24]. In this work, we focus more generally on metric spaces. Namely, all pairwise weights are provided and guaranteed to satisfy the triangle inequality, but are otherwise unconstrained. We show that for any $\varepsilon > 0$, a $(1+\varepsilon)$-approximate MST can be found in $O(\log \frac{1}{\varepsilon} + \log \log n)$ rounds, which is the first $o(\log n)$-round algorithm for finding any constant approximation in this setting. Other than being applicable to more general weight functions, our algorithm also slightly improves the $O(\log \log n \cdot \log \log \log n)$ round-complexity of [JMNZ24, SODA'24] and significantly improves its approximation from a large constant to $1+\varepsilon$. On the lower bound side, we prove that under the 1vs2-Cycle conjecture, $Ω(\log \frac{1}{\varepsilon})$ rounds are needed for finding a $(1+\varepsilon)$-approximate MST in general metrics. It is worth noting that while many existing lower bounds in the MPC model under the 1vs2-Cycle conjecture only hold against "component stable" algorithms, our lower bound applies to *all* algorithms.

cs.DS↗

arXiv2025

Correlation Clustering Beyond the Pivot Algorithm

We study the classic correlation clustering in the dynamic setting. Given $n$ objects and a complete labeling of the object-pairs as either similar or dissimilar, the goal is to partition the objects into arbitrarily many clusters while minimizing disagreements with the labels. In the dynamic setting, an update consists of a flip of a label of an edge. In a breakthrough result, [BDHSS, FOCS'19] showed how to maintain a 3-approximation with polylogarithmic update time by providing a dynamic implementation of the Pivot algorithm of [ACN, STOC'05]. Since then, it has been a major open problem to determine whether the 3-approximation barrier can be broken in the fully dynamic setting. In this paper, we resolve this problem. Our algorithm, Modified Pivot, locally improves the output of Pivot by moving some vertices to other existing clusters or new singleton clusters. We present an analysis showing that this modification does indeed improve the approximation to below 3. We also show that its output can be maintained in polylogarithmic time per update.

cs.DS↗

arXiv2025

Stochastic Matching via In-n-Out Local Computation Algorithms

Consider the following stochastic matching problem. Given a graph $G=(V, E)$, an unknown subgraph $G_p = (V, E_p)$ is realized where $E_p$ includes every edge of $E$ independently with some probability $p \in (0, 1]$. The goal is to query a sparse subgraph $H$ of $G$, such that the realized edges in $H$ include an approximate maximum matching of $G_p$. This problem has been studied extensively over the last decade due to its numerous applications in kidney exchange, online dating, and online labor markets. For any fixed $ε> 0$, [BDH STOC'20] showed that any graph $G$ has a subgraph $H$ with $\text{quasipoly}(1/p) = (1/p)^{\text{poly}(\log(1/p))}$ maximum degree, achieving a $(1-ε)$-approximation. A major open question is the best approximation achievable with $\text{poly}(1/p)$-degree subgraphs. A long line of work has progressively improved the approximation in the $\text{poly}(1/p)$-degree regime from .5 [BDH+ EC'15] to .501 [AKL EC'17], .656 [BHFR SODA'19], .666 [AB SOSA'19], .731 [BBD SODA'22] (bipartite graphs), and most recently to .68 [DS '24]. In this work, we show that a $\text{poly}(1/p)$-degree subgraph can obtain a $(1-ε)$-approximation for any desirably small fixed $ε> 0$, achieving the best of both worlds. Beyond its quantitative improvement, a key conceptual contribution of our work is to connect local computation algorithms (LCAs) to the stochastic matching problem for the first time. While prior work on LCAs mainly focuses on their out-queries (the number of vertices probed to produce the output of a given vertex), our analysis also bounds the in-queries (the number of vertices that probe a given vertex). We prove that the outputs of LCAs with bounded in- and out-queries (in-n-out LCAs for short) have limited correlation, a property that our analysis crucially relies on and might find applications beyond stochastic matchings.

cs.DS↗

arXiv2025

Lower Bounds for Non-adaptive Local Computation Algorithms

We study *non-adaptive* Local Computation Algorithms (LCA). A reduction of Parnas and Ron (TCS'07) turns any distributed algorithm into a non-adaptive LCA. Plugging known distributed algorithms, this leads to non-adaptive LCAs for constant approximations of maximum matching (MM) and minimum vertex cover (MVC) with complexity $Δ^{O(\log Δ/ \log \log Δ)}$, where $Δ$ is the maximum degree of the graph. Allowing adaptivity, this bound can be significantly improved to $\text{poly}(Δ)$, but is such a gap necessary or are there better non-adaptive LCAs? Adaptivity as a resource has been studied extensively across various areas. Beyond this, we further motivate the study of non-adaptive LCAs by showing that even a modest improvement over the Parnas-Ron bound for the MVC problem would have major implications in the Massively Parallel Computation (MPC) setting; It would lead to faster truly sublinear space MPC algorithms for approximate MM, a major open problem of the area. Our main result is a lower bound that rules out this avenue for progress. We prove that $Δ^{Ω(\log Δ/ \log \log Δ)}$ queries are needed for any non-adaptive LCA computing a constant approximation of MM or MVC. This is the first separation between non-adaptive and adaptive LCAs, and already matches (up to constants in the exponent) the algorithm obtained by the black-box reduction of Parnas and Ron. Our proof blends techniques from two separate lines of work: sublinear time lower bounds and distributed lower bounds. Particularly, we adopt techniques such as couplings over acyclic subgraphs from the recent sublinear time lower bounds of Behnezhad, Roghani, and Rubinstein (STOC'23, FOCS'23, STOC'24). We apply these techniques to a very different instance, (a modified version of) the construction of Kuhn, Moscibroda and Wattenhoffer (JACM'16) from distributed computing.

cs.DS↗

arXiv2025

Fully Dynamic (Δ+1) Coloring Against Adaptive Adversaries

Over the years, there has been extensive work on fully dynamic algorithms for classic graph problems that admit greedy solutions. Examples include $(Δ+1)$ vertex coloring, maximal independent set, and maximal matching. For all three problems, there are randomized algorithms that maintain a valid solution after each edge insertion or deletion to the $n$-vertex graph by spending $\polylog n$ time, provided that the adversary is oblivious. However, none of these algorithms work against adaptive adversaries whose updates may depend on the output of the algorithm. In fact, even breaking the trivial bound of $O(n)$ against adaptive adversaries remains open for all three problems. For instance, in the case of $(Δ+1)$ vertex coloring, the main challenge is that an adaptive adversary can keep inserting edges between vertices of the same color, necessitating a recoloring of one of the endpoints. The trivial algorithm would simply scan all neighbors of one endpoint to find a new available color (which always exists) in $O(n)$ time. In this paper, we break this linear barrier for the $(Δ+1)$ vertex coloring problem. Our algorithm is randomized, and maintains a valid $(Δ+1)$ vertex coloring after each edge update by spending $\widetilde{O}(n^{8/9})$ time with high probability.

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arXiv2024

Fully Dynamic Matching and Ordered Ruzsa-Szemerédi Graphs

We study the fully dynamic maximum matching problem. In this problem, the goal is to efficiently maintain an approximate maximum matching of a graph that is subject to edge insertions and deletions. Our focus is on algorithms that maintain the edges of a $(1-ε)$-approximate maximum matching for an arbitrarily small constant $ε> 0$. Until recently, the fastest known algorithm for this problem required $Θ(n)$ time per update where $n$ is the number of vertices. This bound was slightly improved to $n/(\log^* n)^{Ω(1)}$ by Assadi, Behnezhad, Khanna, and Li [STOC'23] and very recently to $n/2^{Ω(\sqrt{\log n})}$ by Liu [FOCS'24]. Whether this can be improved to $n^{1-Ω(1)}$ remains a major open problem. In this paper, we introduce {\em Ordered Ruzsa-Szemerédi (ORS)} graphs (a generalization of Ruzsa-Szemerédi graphs) and show that the complexity of dynamic matching is closely tied to them. For $δ> 0$, define $ORS(δn)$ to be the maximum number of matchings $M_1, \ldots, M_t$, each of size $δn$, that one can pack in an $n$-vertex graph such that each matching $M_i$ is an {\em induced matching} in subgraph $M_1 \cup \ldots \cup M_{i}$. We show that there is a randomized algorithm that maintains a $(1-ε)$-approximate maximum matching of a fully dynamic graph in $$ \widetilde{O}\left( \sqrt{n^{1+ε} \cdot ORS(Θ_ε(n))} \right) $$ amortized update-time. While the value of $ORS(Θ(n))$ remains unknown and is only upper bounded by $n^{1-o(1)}$, the densest construction known from more than two decades ago only achieves $ORS(Θ(n)) \geq n^{1/Θ(\log \log n)} = n^{o(1)}$ [Fischer et al. STOC'02]. If this is close to the right bound, then our algorithm achieves an update-time of $\sqrt{n^{1+O(ε)}}$, resolving the aforementioned longstanding open problem in dynamic algorithms in a strong sense.

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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.

cs.DS↗

arXiv2024

Approximating Maximum Matching Requires Almost Quadratic Time

We study algorithms for estimating the size of maximum matching. This problem has been subject to extensive research. For $n$-vertex graphs, Bhattacharya, Kiss, and Saranurak [FOCS'23] (BKS) showed that an estimate that is within $\varepsilon n$ of the optimal solution can be achieved in $n^{2-Ω_\varepsilon(1)}$ time, where $n$ is the number of vertices. While this is subquadratic in $n$ for any fixed $\varepsilon > 0$, it gets closer and closer to the trivial $Θ(n^2)$ time algorithm that reads the entire input as $\varepsilon$ is made smaller and smaller. In this work, we close this gap and show that the algorithm of BKS is close to optimal. In particular, we prove that for any fixed $δ> 0$, there is another fixed $\varepsilon = \varepsilon(δ) > 0$ such that estimating the size of maximum matching within an additive error of $\varepsilon n$ requires $Ω(n^{2-δ})$ time in the adjacency list model.

cs.DS↗

arXiv2024

Bipartite Matching in Massive Graphs: A Tight Analysis of EDCS

Maximum matching is one of the most fundamental combinatorial optimization problems with applications in various contexts such as balanced clustering, data mining, resource allocation, and online advertisement. In many of these applications, the input graph is massive. The sheer size of these inputs makes it impossible to store the whole graph in the memory of a single machine and process it there. Graph sparsification has been an extremely powerful tool to alleviate this problem. In this paper, we study a highly successful and versatile sparsifier for the matching problem: the *edge-degree constrained subgraph (EDCS)* introduced first by Bernstein and Stein [ICALP'15]. The EDCS has a parameter $β\geq 2$ which controls the density of the sparsifier. It has been shown through various proofs in the literature that by picking a subgraph with $O(nβ)$ edges, the EDCS includes a matching of size at least $2/3-O(1/β)$ times the maximum matching size. As such, by increasing $β$ the approximation ratio of EDCS gets closer and closer to $2/3$. In this paper, we propose a new approach for analyzing the approximation ratio of EDCS. Our analysis is *tight* for any value of $β$. Namely, we pinpoint the precise approximation ratio of EDCS for any sparsity parameter $β$. Our analysis reveals that one does not necessarily need to increase $β$ to improve approximation, as suggested by previous analysis. In particular, the best choice turns out to be $β= 6$, which achieves an approximation ratio of $.677$! This is arguably surprising as it is even better than $2/3 \sim .666$, the bound that was widely believed to be the limit for EDCS.

cs.DS↗

arXiv2024

Sublinear Algorithms for TSP via Path Covers

We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related {\em maximum path cover} problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed $ε> 0$, there is an algorithm that $(1/2 - ε)$-approximates the maximum path cover size of an $n$-vertex graph in $\widetilde{O}(n)$ time. This improves upon a $(3/8-ε)$-approximate $\widetilde{O}(n \sqrt{n})$-time algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give an $\widetilde{O}(n)$ time algorithm that estimates the cost of $(1,2)$-TSP within a factor of $(1.5+ε)$ which is an improvement over a folklore $(1.75 + ε)$-approximate $\widetilde{O}(n)$-time algorithm, as well as a $(1.625+ε)$-approximate $\widetilde{O}(n\sqrt{n})$-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an $\widetilde{O}(n)$ algorithm that estimates the cost of graphic TSP within a factor of $1.83$ which is an improvement over a $1.92$-approximate $\widetilde{O}(n)$ time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. We show that the approximation can be further improved to $1.66$ using $n^{2-Ω(1)}$ time. All of our $\widetilde{O}(n)$ time algorithms are information-theoretically time-optimal up to poly log n factors. Additionally, we show that our approximation guarantees for path cover and $(1,2)$-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

cs.DS↗