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A $(1 \pm \epsilon)$-sparsifier of a hypergraph $G(V,E)$ is a (weighted) subgraph that preserves the value of every cut to within a $(1 \pm \epsilon)$-factor. It is known that every hypergraph with $n$ vertices admits a $(1 \pm \epsilon)$-sparsifier with $\tilde{O}(n/\epsilon^2)$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a \emph{linear sketch}) over the hyperedges of $G$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $\widetilde{O}(n r \log(m) / \epsilon^2)$ bits which with high probability contains sufficient information to recover a $(1 \pm \epsilon)$ cut-sparsifier with $\tilde{O}(n/\epsilon^2)$ hyperedges for any hypergraph with at most $m$ edges each of which has arity bounded by $r$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $\widetilde{O}(n r^2 \log^4(m) / \epsilon^2)$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $\epsilon \in (0,1)$, one needs $\Omega(nr \log(m/n) / \log(n))$ bits to construct a $(1 \pm \epsilon)$-sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $r$ and $\log(m)$.

In a very recent breakthrough, Behnezhad and Ghafari [FOCS'24] developed a novel fully dynamic randomized algorithm for maintaining a $(1-\epsilon)$-approximation of maximum matching with amortized update time potentially much better than the trivial $O(n)$ update time. The runtime of the BG algorithm is parameterized via the following graph theoretical concept: * For any $n$, define $ORS(n)$ -- standing for Ordered RS Graph -- to be the largest number of edge-disjoint matchings $M_1,\ldots,M_t$ of size $\Theta(n)$ in an $n$-vertex graph such that for every $i \in [t]$, $M_i$ is an induced matching in the subgraph $M_{i} \cup M_{i+1} \cup \ldots \cup M_t$. Then, for any fixed $\epsilon > 0$, the BG algorithm runs in \[ O\left( \sqrt{n^{1+O(\epsilon)} \cdot ORS(n)} \right) \] amortized update time with high probability, even against an adaptive adversary. $ORS(n)$ is a close variant of a more well-known quantity regarding RS graphs (which require every matching to be induced regardless of the ordering). It is currently only known that $n^{o(1)} \leqslant ORS(n) \leqslant n^{1-o(1)}$, and closing this gap appears to be a notoriously challenging problem. In this work, we further strengthen the result of Behnezhad and Ghafari and push it to limit to obtain a randomized algorithm with amortized update time of \[ n^{o(1)} \cdot ORS(n) \] with high probability, even against an adaptive adversary. In the limit, i.e., if current lower bounds for $ORS(n) = n^{o(1)}$ are almost optimal, our algorithm achieves an $n^{o(1)}$ update time for $(1-\epsilon)$-approximation of maximum matching, almost fully resolving this fundamental question. In its current stage also, this fully reduces the algorithmic problem of designing dynamic matching algorithms to a purely combinatorial problem of upper bounding $ORS(n)$ with no algorithmic considerations., Comment: 24 pages, 2 figures. In SODA 2025

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.

CSP sparsification, introduced by Kogan and Krauthgamer (ITCS 2015), considers the following question: when can an instance of a constraint satisfaction problem be sparsified (by retaining a weighted subset of the constraints) while still roughly capturing the weight of constraints satisfied by {\em every} assignment. CSP sparsification generalizes and abstracts other commonly studied problems including graph cut-sparsification, hypergraph cut-sparsification and hypergraph XOR-sparsification. A central question here is to understand what properties of a constraint predicate $P:\Sigma^r \to \{0,1\}$ (where variables are assigned values in $\Sigma$) allow for nearly linear-size sparsifiers (in the number of variables). In this work (1) we significantly extend the class of CSPs for which nearly linear-size, and other non-trivial, sparsifications exist and give classifications in some broad settings and (2) give a polynomial-time algorithm to extract this sparsification. Our results captured in item (1) completely classify all symmetric Boolean predicates $P$ (i.e., on the Boolean domain $\Sigma = \{0,1\}$) that allow nearly-linear-size sparsifications. Symmetric Boolean CSPs already capture all the special classes of sparisifcation listed above including hypergraph cut-sparsification and variants. Our study of symmetric CSPs reveals an inherent, previously undetected, number-theoretic phenomenon that determines near-linear size sparsifiability. We also completely classify the set of Boolean predicates $P$ that allow non-trivial ($o(n^r)$-size) sparsifications, thus answering an open question from the work of Kogan and Krauthgamer.

We present a parallel algorithm for the $(1-\epsilon)$-approximate maximum flow problem in capacitated, undirected graphs with $n$ vertices and $m$ edges, achieving $O(\epsilon^{-3}\text{polylog} n)$ depth and $O(m \epsilon^{-3} \text{polylog} n)$ work in the PRAM model. Although near-linear time sequential algorithms for this problem have been known for almost a decade, no parallel algorithms that simultaneously achieved polylogarithmic depth and near-linear work were known. At the heart of our result is a polylogarithmic depth, near-linear work recursive algorithm for computing congestion approximators. Our algorithm involves a recursive step to obtain a low-quality congestion approximator followed by a "boosting" step to improve its quality which prevents a multiplicative blow-up in error. Similar to Peng [SODA'16], our boosting step builds upon the hierarchical decomposition scheme of R\"acke, Shah, and T\"aubig [SODA'14]. A direct implementation of this approach, however, leads only to an algorithm with $n^{o(1)}$ depth and $m^{1+o(1)}$ work. To get around this, we introduce a new hierarchical decomposition scheme, in which we only need to solve maximum flows on subgraphs obtained by contracting vertices, as opposed to vertex-induced subgraphs used in R\"acke, Shah, and T\"aubig [SODA'14]. In particular, we are able to directly extract congestion approximators for the subgraphs from a congestion approximator for the entire graph, thereby avoiding additional recursion on those subgraphs. Along the way, we also develop a parallel flow-decomposition algorithm that is crucial to achieving polylogarithmic depth and may be of independent interest.

Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.

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.

We introduce a notion of code sparsification that generalizes the notion of cut sparsification in graphs. For a (linear) code $\mathcal{C} \subseteq \mathbb{F}_q^n$ of dimension $k$ a $(1 \pm \epsilon)$-sparsification of size $s$ is given by a weighted set $S \subseteq [n]$ with $|S| \leq s$ such that for every codeword $c \in \mathcal{C}$ the projection $c|_S$ of $c$ to the set $S$ has (weighted) hamming weight which is a $(1 \pm \epsilon)$ approximation of the hamming weight of $c$. We show that for every code there exists a $(1 \pm \epsilon)$-sparsification of size $s = \widetilde{O}(k \log (q) / \epsilon^2)$. This immediately implies known results on graph and hypergraph cut sparsification up to polylogarithmic factors (with a simple unified proof). One application of our result is near-linear size sparsifiers for constraint satisfaction problems (CSPs) over $\mathbb{F}_p$-valued variables whose unsatisfying assignments can be expressed as the zeros of a linear equation modulo a prime $p$. Building on this, we obtain a complete characterization of ternary Boolean CSPs that admit near-linear size sparsification. Finally, by connections between the eigenvalues of the Laplacians of Cayley graphs over $\mathbb{F}_2^k$ to the weights of codewords, we also give the first proof of the existence of spectral Cayley graph sparsifiers over $\mathbb{F}_2^k$ by Cayley graphs, i.e., where we sparsify the set of generators to nearly-optimal size.

We study the query complexity of the metric Steiner Tree problem, where we are given an $n \times n$ metric on a set $V$ of vertices along with a set $T \subseteq V$ of $k$ terminals, and the goal is to find a tree of minimum cost that contains all terminals in $T$. The query complexity for the related minimum spanning tree (MST) problem is well-understood: for any fixed $\varepsilon > 0$, one can estimate the MST cost to within a $(1+\varepsilon)$-factor using only $\tilde{O}(n)$ queries, and this is known to be tight. This implies that a $(2 + \varepsilon)$-approximate estimate of Steiner Tree cost can be obtained with $\tilde{O}(k)$ queries by simply applying the MST cost estimation algorithm on the metric induced by the terminals. Our first result shows that any (randomized) algorithm that estimates the Steiner Tree cost to within a $(5/3 - \varepsilon)$-factor requires $\Omega(n^2)$ queries, even if $k$ is a constant. This lower bound is in sharp contrast to an upper bound of $O(nk)$ queries for computing a $(5/3)$-approximate Steiner Tree, which follows from previous work by Du and Zelikovsky. Our second main result, and the main technical contribution of this work, is a sublinear query algorithm for estimating the Steiner Tree cost to within a strictly better-than-$2$ factor, with query complexity $\tilde{O}(n^{12/7} + n^{6/7}\cdot k)=\tilde{O}(n^{13/7})=o(n^2)$. We complement this result by showing an $\tilde{\Omega}(n + k^{6/5})$ query lower bound for any algorithm that estimates Steiner Tree cost to a strictly better than $2$ factor. Thus $\tilde{\Omega}(n^{6/5})$ queries are needed to just beat $2$-approximation when $k = \Omega(n)$; a sharp contrast to MST cost estimation where a $(1+o(1))$-approximate estimate of cost is achievable with only $\tilde{O}(n)$ queries.

A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a cut sparsifier of an unweighted undirected graph by taking a near-linear number of linear measurements on the graph. Subsequent works also studied computing other graph sparsifiers using linear sketching, and obtained near-linear upper bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21]. All these linear sketching algorithms, however, only work on unweighted graphs. In this paper, we initiate the study of weighted graph sparsification by linear sketching by investigating a natural class of linear sketches that we call incidence sketches, in which each measurement is a linear combination of the weights of edges incident on a single vertex. Our results are: 1. Weighted cut sparsification: We give an algorithm that computes a $(1 + \epsilon)$-cut sparsifier using $\tilde{O}(n \epsilon^{-3})$ linear measurements, which is nearly optimal. 2. Weighted spectral sparsification: We give an algorithm that computes a $(1 + \epsilon)$-spectral sparsifier using $\tilde{O}(n^{6/5} \epsilon^{-4})$ linear measurements. Complementing our algorithm, we then prove a superlinear lower bound of $\Omega(n^{21/20-o(1)})$ measurements for computing some $O(1)$-spectral sparsifier using incidence sketches. 3. Weighted spanner computation: We focus on graphs whose largest/smallest edge weights differ by an $O(1)$ factor, and prove that, for incidence sketches, the upper bounds obtained by~[Filtser-Kapralov-Nouri, SODA'21] are optimal up to an $n^{o(1)}$ factor.

We present a new approach for finding matchings in dense graphs by building on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain non-trivial albeit slight improvements over longstanding bounds for matchings in streaming and dynamic graphs. In particular, we establish the following results for $n$-vertex graphs: * A deterministic single-pass streaming algorithm that finds a $(1-o(1))$-approximate matching in $o(n^2)$ bits of space. This constitutes the first single-pass algorithm for this problem in sublinear space that improves over the $\frac{1}{2}$-approximation of the greedy algorithm. * A randomized fully dynamic algorithm that with high probability maintains a $(1-o(1))$-approximate matching in $o(n)$ worst-case update time per each edge insertion or deletion. The algorithm works even against an adaptive adversary. This is the first $o(n)$ update-time dynamic algorithm with approximation guarantee arbitrarily close to one. Given the use of regularity lemma, the improvement obtained by our algorithms over trivial bounds is only by some $(\log^*{n})^{\Theta(1)}$ factor. Nevertheless, in each case, they show that the ``right'' answer to the problem is not what is dictated by the previous bounds. Finally, in the streaming model, we also present a randomized $(1-o(1))$-approximation algorithm whose space can be upper bounded by the density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have been used extensively to prove streaming lower bounds, ours is the first to use them as an upper bound tool for designing improved streaming algorithms.

Hierarchical clustering over graphs is a fundamental task in data mining and machine learning with applications in domains such as phylogenetics, social network analysis, and information retrieval. Specifically, we consider the recently popularized objective function for hierarchical clustering due to Dasgupta. Previous algorithms for (approximately) minimizing this objective function require linear time/space complexity. In many applications the underlying graph can be massive in size making it computationally challenging to process the graph even using a linear time/space algorithm. As a result, there is a strong interest in designing algorithms that can perform global computation using only sublinear resources. The focus of this work is to study hierarchical clustering for massive graphs under three well-studied models of sublinear computation which focus on space, time, and communication, respectively, as the primary resources to optimize: (1) (dynamic) streaming model where edges are presented as a stream, (2) query model where the graph is queried using neighbor and degree queries, (3) MPC model where the graph edges are partitioned over several machines connected via a communication channel. We design sublinear algorithms for hierarchical clustering in all three models above. At the heart of our algorithmic results is a view of the objective in terms of cuts in the graph, which allows us to use a relaxed notion of cut sparsifiers to do hierarchical clustering while introducing only a small distortion in the objective function. Our main algorithmic contributions are then to show how cut sparsifiers of the desired form can be efficiently constructed in the query model and the MPC model. We complement our algorithmic results by establishing nearly matching lower bounds that rule out the possibility of designing better algorithms in each of these models.

The stochastic $K$-armed bandit problem has been studied extensively due to its applications in various domains ranging from online advertising to clinical trials. In practice however, the number of arms can be very large resulting in large memory requirements for simultaneously processing them. In this paper we consider a streaming setting where the arms are presented in a stream and the algorithm uses limited memory to process these arms. Here, the goal is not only to minimize regret, but also to do so in minimal memory. Previous algorithms for this problem operate in one of the two settings: they either use $\Omega(\log \log T)$ passes over the stream (Rathod, 2021; Chaudhuri and Kalyanakrishnan, 2020; Liau et al., 2018), or just a single pass (Maiti et al., 2021). In this paper we study the trade-off between memory and regret when $B$ passes over the stream are allowed, for any $B \geq 1$, and establish tight regret upper and lower bounds for any $B$-pass algorithm. Our results uncover a surprising *sharp transition phenomenon*: $O(1)$ memory is sufficient to achieve $\widetilde\Theta\Big(T^{\frac{1}{2} + \frac{1}{2^{B+2}-2}}\Big)$ regret in $B$ passes, and increasing the memory to any quantity that is $o(K)$ has almost no impact on further reducing this regret, unless we use $\Omega(K)$ memory. Our main technical contribution is our lower bound which requires the use of information-theoretic techniques as well as ideas from round elimination to show that the *residual problem* remains challenging over subsequent passes.

We consider the design of sublinear space and query complexity algorithms for estimating the cost of a minimum spanning tree (MST) and the cost of a minimum traveling salesman (TSP) tour in a metric on $n$ points. We first consider the $o(n)$-space regime and show that, when the input is a stream of all $\binom{n}{2}$ entries of the metric, for any $\alpha \ge 2$, both MST and TSP cost can be $\alpha$-approximated using $\tilde{O}(n/\alpha)$ space, and that $\Omega(n/\alpha^2)$ space is necessary for this task. Moreover, we show that even if the streaming algorithm is allowed $p$ passes over a metric stream, it still requires $\tilde{\Omega}(\sqrt{n/\alpha p^2})$ space. We next consider the semi-streaming regime, where computing even the exact MST cost is easy and the main challenge is to estimate TSP cost to within a factor that is strictly better than $2$. We show that, if the input is a stream of all edges of the weighted graph that induces the underlying metric, for any $\varepsilon > 0$, any one-pass $(2-\varepsilon)$-approximation of TSP cost requires $\Omega(\varepsilon^2 n^2)$ space; on the other hand, there is an $\tilde{O}(n)$ space two-pass algorithm that approximates the TSP cost to within a factor of 1.96. Finally, we consider the query complexity of estimating metric TSP cost to within a factor that is strictly better than $2$, when the algorithm is given access to a matrix that specifies pairwise distances between all points. For MST estimation in this model, it is known that a $(1+\varepsilon)$-approximation is achievable with $\tilde{O}(n/\varepsilon^{O(1)})$ queries. We design an algorithm that performs $\tilde{O}(n^{1.5})$ distance queries and achieves a strictly better than $2$-approximation when either the metric is known to contain a spanning tree supported on weight-$1$ edges or the algorithm is given access to a minimum spanning tree of the graph.

We study the maximum matching problem in fully dynamic graphs: a graph is undergoing both edge insertions and deletions, and the goal is to efficiently maintain a large matching after each edge update. This problem has received considerable attention in recent years. The known algorithms naturally exhibit a trade-off between the quality of the matching maintained (i.e., the approximation ratio) and the time needed per update. While several interesting results have been obtained, the optimal behavior of this trade-off remains largely unclear. Our main contribution is a new approach to designing fully dynamic approximate matching algorithms that in a unified manner not only (essentially) recovers all previously known trade-offs that were achieved via very different techniques, but reveals some new ones as well. As our main tool to achieve this, we introduce a generalization of the edge-degree constrained subgraph (EDCS) of Bernstein and Stein (2015) that we call the hierarchical EDCS (HEDCS).

Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making $\mathrm{poly}(N)$ queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where $N$ denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least $N$ rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with $\mathrm{poly}(N)$ queries using only poly-logarithmic rounds of adaptivity. An important step towards understanding the adaptivity needed for efficient parallel SFM was taken recently in the work of Balkanski and Singer who showed that any SFM algorithm making $\mathrm{poly}(N)$ queries necessarily requires $\Omega(\log N/\log \log N)$ rounds. This left open the possibility of efficient SFM algorithms in poly-logarithmic rounds. For matroid intersection, even the possibility of a constant round, $\mathrm{poly}(N)$ query algorithm was not hitherto ruled out. In this work, we prove that any, possibly randomized, algorithm for submodular function minimization or matroid intersection making $\mathrm{poly}(N)$ queries requires $\tilde{\Omega}\left(N^{1/3}\right)$ rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make at most $2^{N^{1-\delta}}$ queries, for any constant $\delta> 0$. Therefore, even though SFM and matroid intersection are efficiently solvable, they are not highly parallelizable in the oracle model., Comment: An extended abstract will appear in the Proceedings of IEEE FOCS 2021

The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Bencz\'ur and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate cut sparsifier} of $G$. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require $\Omega(n + m)$ time where $n$ denotes the number of vertices and $m$ denotes the number of hyperedges in the hypergraph. Since $m$ can be exponentially large in $n$, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in $n$, {\em independent of the number of edges}. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph., Comment: ICALP 2021

Cuts in graphs are a fundamental object of study, and play a central role in the study of graph algorithms. The problem of sparsifying a graph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Bencz\'ur and Karger (1996) showed that given any $n$-vertex undirected weighted graph $G$ and a parameter $\varepsilon \in (0,1)$, there is a near-linear time algorithm that outputs a weighted subgraph $G'$ of $G$ of size $\tilde{O}(n/\varepsilon^2)$ such that the weight of every cut in $G$ is preserved to within a $(1 \pm \varepsilon)$-factor in $G'$. The graph $G'$ is referred to as a {\em $(1 \pm \varepsilon)$-approximate cut sparsifier} of $G$. A natural question is if such cut-preserving sparsifiers also exist for hypergraphs. Kogan and Krauthgamer (2015) initiated a study of this question and showed that given any weighted hypergraph $H$ where the cardinality of each hyperedge is bounded by $r$, there is a polynomial-time algorithm to find a $(1 \pm \varepsilon)$-approximate cut sparsifier of $H$ of size $\tilde{O}(\frac{nr}{\varepsilon^2})$. Since $r$ can be as large as $n$, in general, this gives a hypergraph cut sparsifier of size $\tilde{O}(n^2/\varepsilon^2)$, which is a factor $n$ larger than the Bencz\'ur-Karger bound for graphs. It has been an open question whether or not Bencz\'ur-Karger bound is achievable on hypergraphs. In this work, we resolve this question in the affirmative by giving a new polynomial-time algorithm for creating hypergraph sparsifiers of size $\tilde{O}(n/\varepsilon^2)$., Comment: FOCS 2020

We study the problem of finding a spanning forest in an undirected, $n$-vertex multi-graph under two basic query models. One is the Linear query model which are linear measurements on the incidence vector induced by the edges; the other is the weaker OR query model which only reveals whether a given subset of plausible edges is empty or not. At the heart of our study lies a fundamental problem which we call the {\em single element recovery} problem: given a non-negative real vector $x$ in $N$ dimension, return a single element $x_j > 0$ from the support. Queries can be made in rounds, and our goals is to understand the trade-offs between the query complexity and the rounds of adaptivity needed to solve these problems, for both deterministic and randomized algorithms. These questions have connections and ramifications to multiple areas such as sketching, streaming, graph reconstruction, and compressed sensing. Our main results are: * For the single element recovery problem, it is easy to obtain a deterministic, $r$-round algorithm which makes $(N^{1/r}-1)$-queries per-round. We prove that this is tight: any $r$-round deterministic algorithm must make $\geq (N^{1/r} - 1)$ linear queries in some round. In contrast, a $1$-round $O(\log^2 N)$-query randomized algorithm which succeeds 99% of the time is known to exist. * We design a deterministic $O(r)$-round, $\tilde{O}(n^{1+1/r})$-OR query algorithm for graph connectivity. We complement this with an $\tilde{\Omega}(n^{1 + 1/r})$-lower bound for any $r$-round deterministic algorithm in the OR-model. * We design a randomized, $2$-round algorithm for the graph connectivity problem which makes $\tilde{O}(n)$-OR queries. In contrast, we prove that any $1$-round algorithm (possibly randomized) requires $\tilde{\Omega}(n^2)$-OR queries.

We give the first polynomial-time approximation scheme (PTAS) for the stochastic load balancing problem when the job sizes follow Poisson distributions. This improves upon the 2-approximation algorithm due to Goel and Indyk (FOCS'99). Moreover, our approximation scheme is an efficient PTAS that has a running time double exponential in $1/\epsilon$ but nearly-linear in $n$, where $n$ is the number of jobs and $\epsilon$ is the target error. Previously, a PTAS (not efficient) was only known for jobs that obey exponential distributions (Goel and Indyk, FOCS'99). Our algorithm relies on several probabilistic ingredients including some (seemingly) new results on scaling and the so-called "focusing effect" of maximum of Poisson random variables which might be of independent interest.

We consider the problem of designing sublinear time algorithms for estimating the cost of a minimum metric traveling salesman (TSP) tour. Specifically, given access to a $n \times n$ distance matrix $D$ that specifies pairwise distances between $n$ points, the goal is to estimate the TSP cost by performing only sublinear (in the size of $D$) queries. For the closely related problem of estimating the weight of a metric minimum spanning tree (MST), it is known that for any $\varepsilon > 0$, there exists an $\tilde{O}(n/\varepsilon^{O(1)})$ time algorithm that returns a $(1 + \varepsilon)$-approximate estimate of the MST cost. This result immediately implies an $\tilde{O}(n/\varepsilon^{O(1)})$ time algorithm to estimate the TSP cost to within a $(2 + \varepsilon)$ factor for any $\varepsilon > 0$. However, no $o(n^2)$ time algorithms are known to approximate metric TSP to a factor that is strictly better than $2$. On the other hand, there were also no known barriers that rule out the existence of $(1 + \varepsilon)$-approximate estimation algorithms for metric TSP with $\tilde{O}(n)$ time for any fixed $\varepsilon > 0$. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost. We also show that the problem of estimating metric TSP cost is closely connected to the problem of estimating the size of a maximum matching in a graph., Comment: ICALP 2020

Suppose a graph $G$ is stochastically created by uniformly sampling vertices along a line segment and connecting each pair of vertices with a probability that is a known decreasing function of their distance. We ask if it is possible to reconstruct the actual positions of the vertices in $G$ by only observing the generated unlabeled graph. We study this question for two natural edge probability functions -- one where the probability of an edge decays exponentially with the distance and another where this probability decays only linearly. We initiate our study with the weaker goal of recovering only the order in which vertices appear on the line segment. For a segment of length $n$ and a precision parameter $\delta$, we show that for both exponential and linear decay edge probability functions, there is an efficient algorithm that correctly recovers (up to reflection symmetry) the order of all vertices that are at least $\delta$ apart, using only $\tilde{O}(\frac{n}{\delta ^ 2})$ samples (vertices). Building on this result, we then show that $O(\frac{n^2 \log n}{\delta ^2})$ vertices (samples) are sufficient to additionally recover the location of each vertex on the line to within a precision of $\delta$. We complement this result with an $\Omega (\frac{n^{1.5}}{\delta})$ lower bound on samples needed for reconstructing positions (even by a computationally unbounded algorithm), showing that the task of recovering positions is information-theoretically harder than recovering the order. We give experimental results showing that our algorithm recovers the positions of almost all points with high accuracy., Comment: Web Conference 2020

We study a network formation game where agents receive benefits by forming connections to other agents but also incur both direct and indirect costs from the formed connections. Specifically, once the agents have purchased their connections, an attack starts at a randomly chosen vertex in the network and spreads according to the independent cascade model with a fixed probability, destroying any infected agents. The utility or welfare of an agent in our game is defined to be the expected size of the agent's connected component post-attack minus her expenditure in forming connections. Our goal is to understand the properties of the equilibrium networks formed in this game. Our first result concerns the edge density of equilibrium networks. A network connection increases both the likelihood of remaining connected to other agents after an attack as well the likelihood of getting infected by a cascading spread of infection. We show that the latter concern primarily prevails and any equilibrium network in our game contains only $O(n\log n)$ edges where $n$ denotes the number of agents. On the other hand, there are equilibrium networks that contain $\Omega(n)$ edges showing that our edge density bound is tight up to a logarithmic factor. Our second result shows that the presence of attack and its spread through a cascade does not significantly lower social welfare as long as the network is not too dense. We show that any non-trivial equilibrium network with $O(n)$ edges has $\Theta(n^2)$ social welfare, asymptotically similar to the social welfare guarantee in the game without any attacks., Comment: The short version of this paper appears in the proceedings of IJCAI-19

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 $(1+\epsilon)$ times the length of the shortest such path, where $\epsilon$ 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 $(1+\epsilon)$-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 $(1+\epsilon)$-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 $m= \omega(n^{1.5 + o(1)})$.

We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum $s$-$t$ cut in an $n$-vertex undirected graph requires $n^{2-o(1)}$ space unless it makes $n^{\Omega(1)}$ passes over the stream. To prove our lower bounds, we introduce and analyze a new four-player communication problem that we refer to as the hidden-pointer chasing problem. This is a problem in spirit of the standard pointer chasing problem with the key difference that the pointers in this problem are hidden to players and finding each one of them requires solving another communication problem, namely the set intersection problem. Our lower bounds for graph problems are then obtained by reductions from the hidden-pointer chasing problem. Our hidden-pointer chasing problem appears flexible enough to find other applications and is therefore interesting in its own right. To showcase this, we further present an interesting application of this problem beyond streaming algorithms. Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make $n^{2-o(1)}$ value queries to the function unless it has a polynomial degree of adaptivity.

In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the closely related subgraph enumeration problem that has received significant attention in the database community as the database join problem. The AGM bound shows that the maximum number of occurrences of any arbitrary subgraph $H$ in a graph $G$ with $m$ edges is $O(m^{\rho(H)})$, where $\rho(H)$ is the fractional edge-cover of $H$, and enumeration algorithms with matching runtime are known for any $H$. We bridge this gap between subgraph counting and subgraph enumeration by designing a sublinear-time algorithm that can estimate the number of any arbitrary subgraph $H$ in $G$, denoted by $\#H$, to within a $(1\pm \epsilon)$-approximation w.h.p. in $O(\frac{m^{\rho(H)}}{\#H}) \cdot poly(\log{n},1/\epsilon)$ time. Our algorithm is allowed the standard set of queries for general graphs, namely degree queries, pair queries and neighbor queries, plus an additional edge-sample query that returns an edge chosen uniformly at random. The performance of our algorithm matches those of Eden et.al. [FOCS 2015, STOC 2018] for counting triangles and cliques and extend them to all choices of subgraph $H$ under the additional assumption of edge-sample queries. We further show that our algorithm works for the more general database join size estimation problem and prove a matching lower bound for this problem.

In the submodular cover problem, we are given a non-negative monotone submodular function $f$ over a ground set $E$ of items, and the goal is to choose a smallest subset $S \subseteq E$ such that $f(S) = Q$ where $Q = f(E)$. In the stochastic version of the problem, we are given $m$ stochastic items which are different random variables that independently realize to some item in $E$, and the goal is to find a smallest set of stochastic items whose realization $R$ satisfies $f(R) = Q$. The problem captures as a special case the stochastic set cover problem and more generally, stochastic covering integer programs. We define an $r$-round adaptive algorithm to be an algorithm that chooses a permutation of all available items in each round $k \in [r]$, and a threshold $\tau_k$, and realizes items in the order specified by the permutation until the function value is at least $\tau_k$. The permutation for each round $k$ is chosen adaptively based on the realization in the previous rounds, but the ordering inside each round remains fixed regardless of the realizations seen inside the round. Our main result is that for any integer $r$, there exists a poly-time $r$-round adaptive algorithm for stochastic submodular cover whose expected cost is $\tilde{O}(Q^{{1}/{r}})$ times the expected cost of a fully adaptive algorithm. Prior to our work, such a result was not known even for the case of $r=1$ and when $f$ is the coverage function. On the other hand, we show that for any $r$, there exist instances of the stochastic submodular cover problem where no $r$-round adaptive algorithm can achieve better than $\Omega(Q^{{1}/{r}})$ approximation to the expected cost of a fully adaptive algorithm. Our lower bound result holds even for coverage function and for algorithms with unbounded computational power.

We consider the problem of testing graph cluster structure: given access to a graph $G=(V, E)$, can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph? This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish between the graph having good expansion (i.e.\ being a good single cluster) and the graph having a sparse cut (i.e.\ being a union of at least two clusters). A recent work of Czumaj, Peng, and Sohler (STOC'15) gave an ingenious sublinear time algorithm for testing $k$-clusterability in time $\tilde{O}(n^{1/2} \text{poly}(k))$: their algorithm implicitly embeds a random sample of vertices of the graph into Euclidean space, and then clusters the samples based on estimates of Euclidean distances between the points. This yields a very efficient testing algorithm, but only works if the cluster structure is very strong: it is necessary to assume that the gap between conductances of accepted and rejected graphs is at least logarithmic in the size of the graph $G$. In this paper we show how one can leverage more refined geometric information, namely angles as opposed to distances, to obtain a sublinear time tester that works even when the gap is a sufficiently large constant. Our tester is based on the singular value decomposition of a natural matrix derived from random walk transition probabilities from a small sample of seed nodes. We complement our algorithm with a matching lower bound on the query complexity of testing clusterability. Our lower bound is based on a novel property testing problem, which we analyze using Fourier analytic tools. As a byproduct of our techniques, we also achieve new lower bounds for the problem of approximating MAX-CUT value in sublinear time., Comment: Appears in FOCS 2018

Any graph with maximum degree $\Delta$ admits a proper vertex coloring with $\Delta + 1$ colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm? We answer this fundamental question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. In particular, we design: * A single-pass semi-streaming algorithm in dynamic streams using $\tilde{O}(n)$ space. The only known semi-streaming algorithm prior to our work was a folklore O(log n)-pass algorithm obtained by simulating classical distributed algorithms in the streaming model. * A sublinear-time algorithm in the standard query model that allows neighbor queries and pair queries using $\tilde{O}(n\sqrt{n})$ time. We further show that any algorithm that outputs a valid coloring with sufficiently large constant probability requires $\Omega(n\sqrt{n})$ time. No non-trivial sublinear time algorithms were known prior to our work. * A parallel algorithm in the massively parallel computation (MPC) model using $\tilde{O}(n)$ memory per machine and $O(1)$ MPC rounds. Our number of rounds significantly improves upon the recent $O(\log\log{\Delta}\cdot\log^*{(n)})$-round algorithm of Parter [ICALP 2018]. At the core of our results is a remarkably simple meta-algorithm for the $(\Delta+1)$ coloring problem: Sample $O(\log{n})$ colors for each vertex from the $\Delta+1$ colors; find a proper coloring of the graph using only the sampled colors. We prove that the sampled set of colors with high probability contains a proper coloring of the input graph. The sublinear algorithms are then obtained by designing efficient algorithms for finding a proper coloring of the graph from the sampled colors in the corresponding models.

We study the maximum $k$-set coverage problem in the following distributed setting. A collection of sets $S_1,\ldots,S_m$ over a universe $[n]$ is partitioned across $p$ machines and the goal is to find $k$ sets whose union covers the most number of elements. The computation proceeds in synchronous rounds. In each round, all machines simultaneously send a message to a central coordinator who then communicates back to all machines a summary to guide the computation for the next round. At the end, the coordinator outputs the answer. The main measures of efficiency in this setting are the approximation ratio of the returned solution, the communication cost of each machine, and the number of rounds of computation. Our main result is an asymptotically tight bound on the tradeoff between these measures for the distributed maximum coverage problem. We first show that any $r$-round protocol for this problem either incurs a communication cost of $ k \cdot m^{\Omega(1/r)}$ or only achieves an approximation factor of $k^{\Omega(1/r)}$. This implies that any protocol that simultaneously achieves good approximation ratio ($O(1)$ approximation) and good communication cost ($\widetilde{O}(n)$ communication per machine), essentially requires logarithmic (in $k$) number of rounds. We complement our lower bound result by showing that there exist an $r$-round protocol that achieves an $\frac{e}{e-1}$-approximation (essentially best possible) with a communication cost of $k \cdot m^{O(1/r)}$ as well as an $r$-round protocol that achieves a $k^{O(1/r)}$-approximation with only $\widetilde{O}(n)$ communication per each machine (essentially best possible). We further use our results in this distributed setting to obtain new bounds for the maximum coverage problem in two other main models of computation for massive datasets, namely, the dynamic streaming model and the MapReduce model.

Given a non-negative $n \times m$ real matrix $A$, the {\em matrix scaling} problem is to determine if it is possible to scale the rows and columns so that each row and each column sums to a specified target value for it. This problem arises in many algorithmic applications, perhaps most notably as a preconditioning step in solving a linear system of equations. One of the most natural and by now classical approach to matrix scaling is the Sinkhorn-Knopp algorithm (also known as the RAS method) where one alternately scales either all rows or all columns to meet the target values. In addition to being extremely simple and natural, another appeal of this procedure is that it easily lends itself to parallelization. A central question is to understand the rate of convergence of the Sinkhorn-Knopp algorithm. In this paper, we present an elementary convergence analysis for the Sinkhorn-Knopp algorithm that improves upon the previous best bound. In a nutshell, our approach is to show a simple bound on the number of iterations needed so that the KL-divergence between the current row-sums and the target row-sums drops below a specified threshold $\delta$, and then connect the KL-divergence with $\ell_1$ and $\ell_2$ distances. For $\ell_1$, we can use Pinsker's inequality. For $\ell_2$, we develop a strengthening of Pinsker's inequality, called (KL vs $\ell_1/\ell_2$) in the paper, which lower bounds the KL-divergence by a combination of $\ell_1$ and $\ell_2$ distance. This inequality may be of independent interest. The idea of studying Sinkhorn-Knopp convergence via KL-divergence is not new and has indeed been previously explored. Our contribution is an elementary, self-contained presentation of this approach and an interesting new inequality that yields a significantly stronger convergence guarantee for the extensively studied $\ell_2$-error.

A common approach for designing scalable algorithms for massive data sets is to distribute the computation across, say $k$, machines and process the data using limited communication between them. A particularly appealing framework here is the simultaneous communication model whereby each machine constructs a small representative summary of its own data and one obtains an approximate/exact solution from the union of the representative summaries. If the representative summaries needed for a problem are small, then this results in a communication-efficient and round-optimal protocol. While many fundamental graph problems admit efficient solutions in this model, two prominent problems are notably absent from the list of successes, namely, the maximum matching problem and the minimum vertex cover problem. Indeed, it was shown recently that for both these problems, even achieving a polylog$(n)$ approximation requires essentially sending the entire input graph from each machine. The main insight of our work is that the intractability of matching and vertex cover in the simultaneous communication model is inherently connected to an adversarial partitioning of the underlying graph across machines. We show that when the underlying graph is randomly partitioned across machines, both these problems admit randomized composable coresets of size $\widetilde{O}(n)$ that yield an $\widetilde{O}(1)$-approximate solution. This results in an $\widetilde{O}(1)$-approximation simultaneous protocol for these problems with $\widetilde{O}(nk)$ total communication when the input is randomly partitioned across $k$ machines. We further prove the optimality of our results. Finally, by a standard application of composable coresets, our results also imply MapReduce algorithms with the same approximation guarantee in one or two rounds of communication

In the stochastic matching problem, we are given a general (not necessarily bipartite) graph $G(V,E)$, where each edge in $E$ is realized with some constant probability $p > 0$ and the goal is to compute a bounded-degree (bounded by a function depending only on $p$) subgraph $H$ of $G$ such that the expected maximum matching size in $H$ is close to the expected maximum matching size in $G$. The algorithms in this setting are considered non-adaptive as they have to choose the subgraph $H$ without knowing any information about the set of realized edges in $G$. Originally motivated by an application to kidney exchange, the stochastic matching problem and its variants have received significant attention in recent years. The state-of-the-art non-adaptive algorithms for stochastic matching achieve an approximation ratio of $\frac{1}{2}-\epsilon$ for any $\epsilon > 0$, naturally raising the question that if $1/2$ is the limit of what can be achieved with a non-adaptive algorithm. In this work, we resolve this question by presenting the first algorithm for stochastic matching with an approximation guarantee that is strictly better than $1/2$: the algorithm computes a subgraph $H$ of $G$ with the maximum degree $O(\frac{\log{(1/ p)}}{p})$ such that the ratio of expected size of a maximum matching in realizations of $H$ and $G$ is at least $1/2+\delta_0$ for some absolute constant $\delta_0 > 0$. The degree bound on $H$ achieved by our algorithm is essentially the best possible (up to an $O(\log{(1/p)})$ factor) for any constant factor approximation algorithm, since an $\Omega(\frac{1}{p})$ degree in $H$ is necessary for a vertex to acquire at least one incident edge in a realization.

We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams and dynamic streams and present new upper and lower bound results for both models. On the upper bound front, we show that an $\alpha$-approximate estimate of the matching size can be computed in dynamic streams using $\widetilde{O}({n^2/\alpha^4})$ space, and in insertion-only streams using $\widetilde{O}(n/\alpha^2)$-space. On the lower bound front, we prove that any $\alpha$-approximation algorithm for estimating matching size in dynamic graph streams requires $\Omega(\sqrt{n}/\alpha^{2.5})$ bits of space, even if the underlying graph is both sparse and has arboricity bounded by $O(\alpha)$. We further improve our lower bound to $\Omega(n/\alpha^2)$ in the case of dense graphs. Furthermore, we prove that a $(1+\epsilon)$-approximation to matching size in insertion-only streams requires RS$(n) \cdot n^{1-O(\epsilon)}$ space; here, RS${n}$ denotes the maximum number of edge-disjoint induced matchings of size $\Theta(n)$ in an $n$-vertex graph. It is a major open problem to determine the value of RS$(n)$, and current results leave open the possibility that RS$(n)$ may be as large as $n/\log n$. We also show how to avoid the dependency on the parameter RS$(n)$ in proving lower bound for dynamic streams and present a near-optimal lower bound of $n^{2-O(\epsilon)}$ for $(1+\epsilon)$-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal $n^{2-O(\epsilon)}$ bit lower bound for $(1+\epsilon)$-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an $\alpha$-approximate set cover (for any $\alpha= o(\sqrt{n})$) using a single-pass streaming algorithm, we show that $\Theta(mn/\alpha)$ space is both sufficient and necessary (up to an $O(\log{n})$ factor); here $m$ denotes number of the sets and $n$ denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk et al. (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of $\alpha$ saving in the space is achievable in this case and that this is the best possible. In other words, we show that $\Theta(mn/\alpha^2)$ space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of $\alpha$. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.

In this paper, a 3D conjugated heat transfer model for Nano-Encapsulated Phase Change Materials (NEPCMs) cooled Micro Pin Fin Heat Sink (MPFHS) is presented. The governing equations of flow and heat transfer are solved using a finite volume method based on collocated grid and the results are validated with the available data reported in the literature. The effect of nanoparticles volume fraction (C = 0.1, 0.2, 0.3), inlet velocity (Vin = 0.015, 0.030, 0.045 m/s), and bottom wall temperature (Twall = 299.15, 303.15, 315.15, 350.15 K) are studied on Nusselt and Euler numbers as well as temperature contours in the system. The results indicate that significant heat transfer enhancement is achieved when using NEPCM slurry as an advanced coolant. The maximum Nusselt number when NEPCM slurry (C = 0.3) with Vin = 0.015, 0.030, 0.045 (m/s) is employed, are 2.27, 1.81, 1.56 times higher than the ones with base fluid, respectively. However, with increasing bottom wall temperature, the Nusselt number first increases then decreases. The former is due to higher heat transfer capability of coolant at temperatures over the melting range of PCM particles due to partial melting of nanoparticles in this range. While, the latter phenomena is due to the lower capability of NEPCM particles and consequently coolant in absorbing heat at coolant temperatures higher than the temperature correspond to fully melted NEPCM. It was observed that NEPCM slurry has a drastic effect on Euler number, and with increasing volume fraction and decreasing inlet velocity, the Euler number increases accordingly.

Provisioning is a technique for avoiding repeated expensive computations in what-if analysis. Given a query, an analyst formulates $k$ hypotheticals, each retaining some of the tuples of a database instance, possibly overlapping, and she wishes to answer the query under scenarios, where a scenario is defined by a subset of the hypotheticals that are "turned on". We say that a query admits compact provisioning if given any database instance and any $k$ hypotheticals, one can create a poly-size (in $k$) sketch that can then be used to answer the query under any of the $2^{k}$ possible scenarios without accessing the original instance. In this paper, we focus on provisioning complex queries that combine relational algebra (the logical component), grouping, and statistics/analytics (the numerical component). We first show that queries that compute quantiles or linear regression (as well as simpler queries that compute count and sum/average of positive values) can be compactly provisioned to provide (multiplicative) approximate answers to an arbitrary precision. In contrast, exact provisioning for each of these statistics requires the sketch size to be exponential in $k$. We then establish that for any complex query whose logical component is a positive relational algebra query, as long as the numerical component can be compactly provisioned, the complex query itself can be compactly provisioned. On the other hand, introducing negation or recursion in the logical component again requires the sketch size to be exponential in $k$. While our positive results use algorithms that do not access the original instance after a scenario is known, we prove our lower bounds even for the case when, knowing the scenario, limited access to the instance is allowed.

Strategic network formation arises where agents receive benefit from connections to other agents, but also incur costs for forming links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization against attack. An agent's benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework is a stylized model of settings where reachability rather than centrality is the primary concern and vertices vulnerable to attacks may reduce risk via costly measures. In the reachability benefit model without attack or immunization, the set of equilibria is the empty graph and any tree. The introduction of attack and immunization changes the game dramatically; new equilibrium topologies emerge, some more sparse and some more dense than trees. We show that, under a mild assumption on the adversary, every equilibrium network with $n$ agents contains at most $2n-4$ edges for $n\geq 4$. So despite permitting topologies denser than trees, the amount of overbuilding is limited. We also show that attack and immunization don't significantly erode social welfare: every non-trivial equilibrium with respect to several adversaries has welfare at least as that of any equilibrium in the attack-free model. We complement our theory with simulations demonstrating fast convergence of a new bounded rationality dynamic which generalizes linkstable best response but is considerably more powerful in our game. The simulations further elucidate the wide variety of asymmetric equilibria and demonstrate topological consequences of the dynamics e.g. heavy-tailed degree distributions. Finally, we report on a behavioral experiment on our game with over 100 participants, where despite the complexity of the game, the resulting network was surprisingly close to equilibrium., Comment: The short version of this paper appears in the proceedings of WINE-16

In this paper, we introduce a new model for sublinear algorithms called \emph{dynamic sketching}. In this model, the underlying data is partitioned into a large \emph{static} part and a small \emph{dynamic} part and the goal is to compute a summary of the static part (i.e, a \emph{sketch}) such that given any \emph{update} for the dynamic part, one can combine it with the sketch to compute a given function. We say that a sketch is \emph{compact} if its size is bounded by a polynomial function of the length of the dynamic data, (essentially) independent of the size of the static part. A graph optimization problem $P$ in this model is defined as follows. The input is a graph $G(V,E)$ and a set $T \subseteq V$ of $k$ terminals; the edges between the terminals are the dynamic part and the other edges in $G$ are the static part. The goal is to summarize the graph $G$ into a compact sketch (of size poly$(k)$) such that given any set $Q$ of edges between the terminals, one can answer the problem $P$ for the graph obtained by inserting all edges in $Q$ to $G$, using only the sketch. We study the fundamental problem of computing a maximum matching and prove tight bounds on the sketch size. In particular, we show that there exists a (compact) dynamic sketch of size $O(k^2)$ for the matching problem and any such sketch has to be of size $\Omega(k^2)$. Our sketch for matchings can be further used to derive compact dynamic sketches for other fundamental graph problems involving cuts and connectivities. Interestingly, our sketch for matchings can also be used to give an elementary construction of a \emph{cut-preserving vertex sparsifier} with space $O(kC^2)$ for $k$-terminal graphs; here $C$ is the total capacity of the edges incident on the terminals. Additionally, we give an improved lower bound (in terms of $C$) of $\Omega(C/\log{C})$ on size of cut-preserving vertex sparsifiers.

A classical trading experiment consists of a set of unit demand buyers and unit supply sellers with identical items. Each agent's value or opportunity cost for the item is their private information and preferences are quasi-linear. Trade between agents employs a double oral auction (DOA) in which both buyers and sellers call out bids or offers which an auctioneer recognizes. Transactions resulting from accepted bids and offers are recorded. This continues until there are no more acceptable bids or offers. Remarkably, the experiment consistently terminates in a Walrasian price. The main result of this paper is a mechanism in the spirit of the DOA that converges to a Walrasian equilibrium in a polynomial number of steps, thus providing a theoretical basis for the above-described empirical phenomenon. It is well-known that computation of a Walrasian equilibrium for this market corresponds to solving a maximum weight bipartite matching problem. The uncoordinated but rational responses of agents thus solve in a distributed fashion a maximum weight bipartite matching problem that is encoded by their private valuations. We show, furthermore, that every Walrasian equilibrium is reachable by some sequence of responses. This is in contrast to the well known auction algorithms for this problem which only allow one side to make offers and thus essentially choose an equilibrium that maximizes the surplus for the side making offers. Our results extend to the setting where not every agent pair is allowed to trade with each other.

We resolve the space complexity of linear sketches for approximating the maximum matching problem in dynamic graph streams where the stream may include both edge insertion and deletion. Specifically, we show that for any $\epsilon > 0$, there exists a one-pass streaming algorithm, which only maintains a linear sketch of size $\tilde{O}(n^{2-3\epsilon})$ bits and recovers an $n^\epsilon$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. In contrast to the extensively studied insertion-only model, to the best of our knowledge, no non-trivial single-pass streaming algorithms were previously known for approximating the maximum matching problem on general dynamic graph streams. Furthermore, we show that our upper bound is essentially tight. Namely, any linear sketch for approximating the maximum matching to within a factor of $O(n^\epsilon)$ has to be of size $n^{2-3\epsilon -o(1)}$ bits. We establish this lower bound by analyzing the corresponding simultaneous number-in-hand communication model, with a combinatorial construction based on Ruzsa-Szemer\'{e}di graphs.