In this paper, we study Ranking, a well-known randomized greedy matching algorithm, for general graphs. The algorithm was originally introduced by Karp, Vazirani, and Vazirani [STOC 1990] for the online bipartite matching problem with one-sided vertex arrivals, where it achieves a tight approximation ratio of \(1 -1/e\). It was later extended to bipartite graphs with random vertex arrivals by Mahdian and Yan [STOC 2011] and to general graphs by Goel and Tripathi [FOCS 2012]. The Ranking algorithm for general graphs is as follows: a permutation \(\sigma\) over the vertices is chosen uniformly at random. The vertices are then processed sequentially according to this order, with each vertex being matched to the first available neighbor (if any) according to the same permutation \(\sigma\).

In this work we consider the Metric Steiner Forest problem in the sublinear time model. Given a set \(V\) of \(n\) points in a metric space where distances are provided by means of query access to an \(n \times n\) distance matrix, along with a set of \(k\) terminal pairs (\((s_1, t_1), \ldots, (s_k, t_k), \in V \times V\), the goal is to find a minimum-weight subset of edges that connects each terminal pair. Although sublinear time algorithms have been studied for estimating the weight of a minimum spanning tree in both general and metric settings, as well as for the metric Steiner Tree problem, no sublinear time algorithm was known for the metric Steiner Forest problem.

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 $\varepsilon n$ of the right bound with $n^{2-\Omega_{\varepsilon}(1)}$ queries, which is subquadratic in n (and thus sublinear in the matrix size) for any fixed $\varepsilon\gt0$. 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 $\Omega(n)$, 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 $\delta\gt0$, there is a fixed $\varepsilon\gt0$ such that an estimate that is within $\varepsilon n$ of the true bound requires $\Omega\left(n^{2-\delta}\right)$ 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 $n^{2-\Omega_{\varepsilon}(1)}$ time algorithm that obtains an additive $\varepsilon$-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.

We study the problem of estimating the size of a maximum matching in sublinear time. The problem has been studied extensively in the literature and various algorithms and lower bounds are known for it. Our result is a $0.5109$-approximation algorithm with a running time of $\tilde{O}(n\sqrt{n})$. All previous algorithms either provide only a marginal improvement (e.g., $2^{-280}$) over the $0.5$-approximation that arises from estimating a \emph{maximal} matching, or have a running time that is nearly $n^2$. Our approach is also arguably much simpler than other algorithms beating $0.5$-approximation.

In several two-sided markets, including labor and dating, agents typically have limited information about their preferences prior to mutual interactions. This issue can result in matching frictions, as arising in the labor market for medical residencies, where high application rates are followed by a large number of interviews. Yet, the extensive literature on two-sided matching primarily focuses on models where agents know their preferences, leaving the interactions necessary for preference discovery largely overlooked. This paper studies this problem using an algorithmic approach, extending Gale-Shapley’s deferred acceptance to this context. Two algorithms are proposed. The first is an adaptive algorithm that expands upon Gale-Shapley’s deferred acceptance by incorporating interviews between applicants and positions. Similar to deferred acceptance, one side sequentially proposes to the other. However, the order of proposals is carefully chosen to ensure an interim stable matching is found. Furthermore, with high probability, the number of interviews conducted by each applicant or position is limited to O(log² n). In many seasonal markets, interactions occur more simultaneously, consisting of an initial interview phase followed by a clearing stage. We present a non-adaptive algorithm for generating a single stage set of in tiered random markets. The algorithm finds an interim stable matching in such markets while assigning no more than O(log³ n) interviews to each applicant or position.

We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of $n$ points $V$ in a metric space given to us by means of query access to an $n\times n$ matrix $w$, and a set of terminals $T\subseteq V$, the goal is to find the minimum-weight subset of the edges that connects all the terminal vertices. Recently, Chen, Khanna and Tan [SODA'23] gave an algorithm that uses $\widetilde{O}(n^{13/7})$ queries and outputs a $(2-η)$-estimate of the metric Steiner tree weight, where $η>0$ is a universal constant. A key component in their algorithm is a sublinear algorithm for a particular set cover problem where, given a set system $(U, F)$, the goal is to provide a multiplicative-additive estimate for $|U|-\textsf{SC}(U, F)$. Here $U$ is the set of elements, $F$ is the collection of sets, and $\textsf{SC}(U, F)$ denotes the optimal set cover size of $(U, F)$. In particular, their algorithm returns a $(1/4, \varepsilon\cdot|U|)$-multiplicative-additive estimate for this set cover problem using $\widetilde{O}(|F|^{7/4})$ membership oracle queries (querying whether a set $S$ contains an $e$), where $\varepsilon$ is a fixed constant. In this work, we improve the query complexity of $(2-η)$-estimating the metric Steiner tree weight to $\widetilde{O}(n^{5/3})$ by showing a $(1/2, \varepsilon \cdot |U|)$-estimate for the above set cover problem using $\widetilde{O}(|F|^{5/3})$ membership queries. To design our set cover algorithm, we estimate the size of a random greedy maximal matching for an auxiliary multigraph that the algorithm constructs implicitly, without access to its adjacency list or matrix.

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdős, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar ( G , H ) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar ( G , P k ) , where P k is a path of length k . First, we observe that when k is close to n (the number of vertices in G ), the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar ( G , P k ) for every integer k ≥ 3 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k .

Given a so called “Sperner coloring” of a triangulation of the$D$-dimensional simplex, Sperner's lemma guarantees the existence of a rainbow simplex, i.e. a simplex colored by all$D+1$colors. However, finding a rainbow simplex was the first problem to be proven PPAD-complete in Papadimitriou's classical paper introducing the class PPAD [1]. In this paper, we prove that the problem does not become easier if we relax “all -${D}+1$colors” to allow some fraction of missing colors: in fact, for any constant$D$, finding even a simplex with just three colors remains PPAD-complete! Our result has an interesting application for the envy-free cake cutting from fair division. It is known that if agents value pieces of cake using general continuous functions satisfying a simple boundary condition (“a non-empty piece is better than an empty piece of cake”), there exists an envy-free allocation with connected pieces. We show that for any constant number of agents it is PPAD-complete to find an allocation -even using any constant number of possibly disconnected pieces- that makes just three agents envy-free. Our results extend to super-constant dimension, number of agents, and number of pieces, as long as they are asymptotically bounded by any$\log^{1-\Omega(1)}(\varepsilon)$, where$\varepsilon$is the precision parameter (side length for Sperner and approximate envy-free for cake cutting).

We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related 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 Õ(n) time. This improves upon a (3/8-ε)-approximate Õ(n √n)-time algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give an Õ(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 Õ(n)-time algorithm, as well as a (1.625+ε)-approximate Õ(n√n)-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an Õ(n) algorithm that estimates the cost of graphic TSP within a factor of 1.83 which is an improvement over a 1.92-approximate Õ(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 Õ(n) time algorithms are information-theoretically time-optimal up to polylog 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.

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 & Stein 2015 The EDCS has a parameter $\beta \geq 2$ which controls the density of the sparsifier. It has been shown through various proofs in the literature that by picking a subgraph with $O(n\beta)$ edges, the EDCS includes a matching of size at least $2/3-O(1/\beta)$ times the maximum matching size. As such, by increasing $\beta$ 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 $\beta$. Namely, we pinpoint the precise approximation ratio of EDCS for any sparsity parameter $\beta$. Our analysis reveals that one does not necessarily need to increase $\beta$ to improve approximation, as suggested by previous analysis. In particular, the best choice turns out to be $\beta = 6$, 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.

We study maximum matchings in fully dynamic graphs, which are graphs that undergo both edge insertions and deletions. Our focus is on algorithms that estimate the size of maximum matching after each update while spending a small 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 є n of the optimal solution can be achieved in n2−Ωє(1) time, where n is the number of vertices. While this is subquadratic in n for any fixed є > 0, it gets closer and closer to the trivial Θ(n2) time algorithm that reads the entire input as є 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 є = є(δ) > 0 such that estimating the size of maximum matching within an additive error of є n requires Ω(n2−δ) time in the adjacency list model.

We study local computation algorithms (LCA) for maximum matching. An LCA does not return its output entirely, but reveals parts of it upon query. For matchings, each query is a vertex v; the LCA should return whether v is matched—and if so to which neighbor—while spending a small time per query. In this paper, we prove that any LCA that computes a matching that is at most an additive of $\epsilon n$ smaller than the maximum matching in n-vertex graphs of maximum degree $\Delta$ must take at least $\Delta^{\Omega(1 / \varepsilon)}$ time. This comes close to the existing upper bounds that take $(\Delta / \epsilon)^{O\left(1 / \epsilon^{2}\right)} \operatorname{polylog}(n)$ time. In terms of sublinear time algorithms, our techniques imply that any algorithm that estimates the size of maximum matching up to an additive error of $\epsilon n$ must take $\Delta^{\Omega(1 / \epsilon)}$ time. This negatively resolves a decade old open problem of the area (see Open Problem 39 of sublinear.info) on whether such estimates can be achieved in $\operatorname{poly}(\Delta / \epsilon)$ time.

We study sublinear time algorithms for estimating the size of maximum matching in graphs. Our main result is a (½ + Ω(1))-approximation algorithm which can be implemented in O(n1+ε) time, where n is the number of vertices and the constant ε > 0 can be made arbitrarily small. The best known lower bound for the problem is Ω(n), which holds for any constant approximation. Existing algorithms either obtain the greedy bound of ½-approximation [Behnezhad FOCS'21], or require some assumption on the maximum degree to run in o(n2)-time [Yoshida, Yamamoto, and Ito STOC'09]. We improve over these by designing a less “adaptive” augmentation algorithm for maximum matching that might be of independent interest.

Sublinear time algorithms for approximating maximum matching size have long been studied. Much of the progress over the last two decades on this problem has been on the algorithmic side. For instance, an algorithm of [Behnezhad; FOCS’21] obtains a 1/2-approximation in O(n) time for n-vertex graphs. A more recent algorithm by [Behnezhad, Roghani, Rubinstein, and Saberi; SODA’23] obtains a slightly-better-than-1/2 approximation in O(n1+є) time (for arbitrarily small constant ε>0). On the lower bound side, [Parnas and Ron; TCS’07] showed 15 years ago that obtaining any constant approximation of maximum matching size requires Ω(n) time. Proving any super-linear in n lower bound, even for (1−є)-approximations, has remained elusive since then.

The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore algorithm, which maintains a maximal (and hence $2$-approximate) matching in $O(n)$ worst-case update time in $n$-node graphs. We present the first deterministic algorithm which outperforms the folklore algorithm in terms of {\em both} approximation ratio and worst-case update time. Specifically, we give a $(2-Ω(1))$-approximate algorithm with $O(m^{3/8})=O(n^{3/4})$ worst-case update time in $n$-node, $m$-edge graphs. For sufficiently small constant $ε>0$, no deterministic $(2+ε)$-approximate algorithm with worst-case update time $O(n^{0.99})$ was known. Our second result is the first deterministic $(2+ε)$-approximate weighted matching algorithm with $O_ε(1)\cdot O(\sqrt[4]{m}) = O_ε(1)\cdot O(\sqrt{n})$ worst-case update time. Our main technical contributions are threefold: first, we characterize the tight cases for \emph{kernels}, which are the well-studied matching sparsifiers underlying much of the $(2+ε)$-approximate dynamic matching literature. This characterization, together with multiple ideas -- old and new -- underlies our result for breaking the approximation barrier of $2$. Our second technical contribution is the first example of a dynamic matching algorithm whose running time is improved due to improving the \emph{recourse} of other dynamic matching algorithms. Finally, we show how to use dynamic bipartite matching algorithms as black-box subroutines for dynamic matching in general graphs without incurring the natural $\frac{3}{2}$ factor in the approximation ratio which such approaches naturally incur.

No abstract available.

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{-1/2 - ε} is hard already in 3-partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.