Embeddings of graph distances into trees is one of the most important techniques in algorithms. Often, applications of these embeddings require that the trees into which distances are embedded are spanning trees of the input graph. However, this requirement makes embedding much more difficult. This is largely due to the fact that embedding graph distances into hierarchical partitions of the graph is well-understood and, in turn, embedding hierarchical partitions into non-spanning trees is trivial. On the other hand, no such embedding of hierarchical partitions into spanning trees is known.

In length-constrained minimum spanning tree (MST) we are given an $n$-node graph $G = (V,E)$ with edge weights $w : E \to \mathbb{Z}_{\geq 0}$ and edge lengths $l: E \to \mathbb{Z}_{\geq 0}$ along with a root node $r \in V$ and a length-constraint $h \in \mathbb{Z}_{\geq 0}$. Our goal is to output a spanning tree of minimum weight according to $w$ in which every node is at distance at most $h$ from $r$ according to $l$. We give a polynomial-time algorithm for planar graphs which, for any constant $ε> 0$, outputs an $O\left(\log^{1+ε} n\right)$-approximate solution with every node at distance at most $(1+ε)h$ from $r$ for any constant $ε> 0$. Our algorithm is based on new length-constrained versions of classic planar separators which may be of independent interest. Additionally, our algorithm works for length-constrained Steiner tree. Complementing this, we show that any algorithm on general graphs for length-constrained MST in which nodes are at most $2h$ from $r$ cannot achieve an approximation of $O\left(\log ^{2-ε} n\right)$ for any constant $ε> 0$ under standard complexity assumptions; as such, our results separate the approximability of length-constrained MST in planar and general graphs.

In the Steiner Path Aggregation Problem , our goal is to aggregate paths in a directed network into a single arborescence without significantly disrupting the paths. In particular, we are given a directed multigraph with colored arcs, a root, and k terminals, each of which has a monochromatic path to the root. Our goal is to find an arborescence in which every terminal has a path to the root, and its path does not switch colors too many times. We give an efficient algorithm that finds such a solution with at most 2 log 4 3 ⁡ k color switches. Up to constant factors this is the best possible universal bound, as there are graphs requiring at least log 2 ⁡ k color switches.

Expander decompositions form the basis of one of the most flexible paradigms for close-to-linear-time graph algorithms. Length-constrained expander de-compositions generalize this paradigm to better work for problems with lengths, distances and costs. Roughly, an$(h,s)$-length$\phi$-expander decomposition is a small collection of length increases to a graph so that nodes within distance$h$can route flow over paths of length$hs$with congestion at most$1/\phi$. In this work, we give a close-to-linear time algorithm for computing length-constrained expander decompositions in graphs with general lengths and capacities. Notably, and unlike previous works, our algorithm allows for one to trade off off between the size of the decomposition and the length of routing paths: for any$\epsilon > 0$not too small, our algorithm computes in close-to-linear time an$(h, s)$-length$\phi$-expander decomposition of size$m\cdot\phi\cdot n^{\epsilon}$where$s$= exp(poly$(1/\epsilon)$). The key foundations of our algorithm are: (1) a simple yet powerful structural theorem which states that the union of a sequence of sparse length-constrained cuts is itself sparse and (2) new algorithms for efficiently computing sparse length-constrained flows.

Average linkage Hierarchical Agglomerative Clustering (HAC) is an extensively studied and applied method for hierarchical clustering. Recent applications to massive datasets have driven significant interest in near-linear-time and efficient parallel algorithms for average linkage HAC. We provide hardness results that rule out such algorithms. On the sequential side, we establish a runtime lower bound of $n^{3/2-ε}$ on $n$ node graphs for sequential combinatorial algorithms under standard fine-grained complexity assumptions. This essentially matches the best-known running time for average linkage HAC. On the parallel side, we prove that average linkage HAC likely cannot be parallelized even on simple graphs by showing that it is CC-hard on trees of diameter $4$. On the possibility side, we demonstrate that average linkage HAC can be efficiently parallelized (i.e., it is in NC) on paths and can be solved in near-linear time when the height of the output cluster hierarchy is small.

We give an algorithm for Centroid-Linkage Hierarchical Agglomerative Clustering (HAC), which computes a $c$-approximate clustering in roughly $n^{1+O(1/c^2)}$ time. We obtain our result by combining a new centroid-linkage HAC algorithm with a novel fully dynamic data structure for nearest neighbor search which works under adaptive updates. We also evaluate our algorithm empirically. By leveraging a state-of-the-art nearest-neighbor search library, we obtain a fast and accurate centroid-linkage HAC algorithm. Compared to an existing state-of-the-art exact baseline, our implementation maintains the clustering quality while delivering up to a $36\times$ speedup due to performing fewer distance comparisons.

We introduce the concept of low-step multi-commodity flow emulators for any undirected, capacitated graph. At a high level, these emulators contain approximate multi-commodity flows whose paths contain a small number of edges, shattering the infamous flow decomposition barrier for multi-commodity flow.

We give a poly-time algorithm for the k-edge-connected spanning subgraph (k-ECSS) problem that returns a solution of cost no greater than the cheapest (k+10)-ECSS on the same graph. Our approach enhances the iterative relaxation framework with a new ingredient, which we call ghost values, that allows for high sparsity in intermediate problems. Our guarantees improve upon the best-known approximation factor of 2 for k-ECSS whenever the optimal value of (k+10)-ECSS is close to that of k-ECSS. This is a property that holds for the closely related problem k-edge-connected spanning multi-subgraph (k-ECSM), which is identical to k-ECSS except edges can be selected multiple times at the same cost. As a consequence, we obtain a 1+O(1/k)-approximation algorithm for k-ECSM, which resolves a conjecture of Pritchard and improves upon a recent 1+O(1/√k)-approximation algorithm of Karlin, Klein, Oveis Gharan, and Zhang. Moreover, we present a matching lower bound for k-ECSM, showing that our approximation ratio is tight up to the constant factor in O(1/k), unless P=NP.

A spanning tree T of graph G is a $\rho$-approximate universal Steiner tree (UST) for root vertex r if, for any subset of vertices S containing r, the cost of the minimal subgraph of T connecting S is within a $\rho$ factor of the minimum cost tree connecting S in G. Busch et al. (FOCS 2012) showed that every graph admits $2^{O(\sqrt{\log n})}$-approximate USTs by showing that USTs are equivalent to strong sparse partition hierarchies (up to poly-logs). Further, they posed poly-logarithmic USTs and strong sparse partition hierarchies as open questions.We settle these open questions by giving polynomial-time algorithms for computing both $O\left(\log ^{7} n\right)$-approximate USTs and poly-logarithmic strong sparse partition hierarchies. We reduce the existence of these objects to the previously studied cluster aggregation problem and a class of well-separated point sets which we call dangling nets. For graphs with constant doubling dimension or constant pathwidth we obtain improved bounds by deriving $O(\log n)$-approximate USTs and $O(1)$ strong sparse partition hierarchies. Our doubling dimension result is tight up to second order terms.

Computing routing schemes that support both high throughput and low latency is one of the core challenges of network optimization. Such routes can be formalized as h-length flows which are defined as flows whose flow paths have length at most h. Many well-studied algorithmic primitives—such as maximal and maximum length-constrained disjoint paths—are special cases of h-length flows. Likewise the optimal h-length flow is a fundamental quantity in network optimization, characterizing, up to poly-log factors, how quickly a network can accomplish numerous distributed primitives.

Participatory budgeting is a method used by city governments to select public projects to fund based on residents' votes. Many cities use participatory budgeting at a district level. Typically, a budget is divided among districts proportionally to their population, and each district holds an election over local projects and then uses its budget to fund the projects most preferred by its voters. However, district-level participatory budgeting can yield poor social welfare because it does not necessarily fund projects supported across multiple districts. On the other hand, decision making that only takes global social welfare into account can be unfair to districts: A social-welfare-maximizing solution might not fund any of the projects preferred by a district, despite the fact that its constituents pay taxes to the city. Thus, we study how to fairly maximize social welfare in a participatory budgeting setting with a single city-wide election. We propose a notion of fairness that guarantees each district at least as much welfare as it would have received in a district-level election. We show that, although optimizing social welfare subject to this notion of fairness is NP-hard, we can efficiently construct a lottery over welfare-optimal outcomes that is fair in expectation. Moreover, we show that, when we are allowed to slightly relax fairness, we can efficiently compute a fair solution that is welfare-maximizing, but which may overspend the budget.

We study the store-and-forward packet routing problem for simultaneous multicasts, in which multiple packets have to be forwarded along given trees as fast as possible. This is a natural generalization of the seminal work of Leighton, Maggs and Rao, which solved this problem for unicasts, i.e. the case where all trees are paths. They showed the existence of asymptotically optimal O(C + D)-length schedules, where the congestion C is the maximum number of packets sent over an edge and the dilation D is the maximum depth of a tree. This improves over the trivial O(CD) length schedules. We prove a lower bound for multicasts, which shows that there do not always exist schedules of non-trivial length, o(CD). On the positive side, we construct O(C+D+log² n)-length schedules in any n-node network. These schedules are near-optimal, since our lower bound shows that this length cannot be improved to O(C+D) + o(log n).

Network design problems aim to compute low-cost structures such as routes, trees and subgraphs. Often, it is natural and desirable to require that these structures have small hop length or hop diameter. Unfortunately, optimization problems with hop constraints are much harder and less well understood than their hop-unconstrained counterparts. A significant algorithmic barrier in this setting is the fact that hop-constrained distances in graphs are very far from being a metric.

In the weighted minimum strongly connected spanning subgraph (WMSCSS ) problem we must purchase a minimum-cost strongly connected spanning subgraph of a digraph. We show that half-integral linear program (LP) solutions for WMSCSS can be efficiently rounded to integral solutions at a multiplicative 1.5 cost. This rounding matches a known 1.5 integrality gap lower bound for a half-integral instance. More generally, we show that LP solutions whose non-zero entries are at least a value f>0 can be rounded at a multiplicative cost of 2−f.

Data aggregation is a fundamental primitive in distributed computing wherein a network computes a function of every nodes' input. However, while compute time is non-negligible in modern systems, standard models of distributed computing do not take compute time into account. Rather, most distributed models of computation only explicitly consider communication time. In this paper, we introduce a model of distributed computation that considers both computation and communication so as to give a theoretical treatment of data aggregation. We study both the structure of and how to compute the fastest data aggregation schedule in this model. As our first result, we give a polynomial-time algorithm that computes the optimal schedule when the input network is a complete graph. Moreover, since one may want to aggregate data over a pre-existing network, we also study data aggregation scheduling on arbitrary graphs. We demonstrate that this problem on arbitrary graphs is hard to approximate within a multiplicative 1.5 factor. Finally, we give an O(log n ⋅ log(OPT/t_m))-approximation algorithm for this problem on arbitrary graphs, where n is the number of nodes and OPT is the length of the optimal schedule.

We show the reverse greedy algorithm is between a (2k−2)- and a 2k-approximation for k-center.

In this paper we study how to optimally balance cheap inflexible resources with more expensive, reconfigurable resources despite uncertainty in the input problem. Specifically, we introduce the MinEMax model to study "build versus rent" problems. In our model different scenarios appear independently. Before knowing which scenarios appear, we may build rigid resources that cannot be changed for different scenarios. Once we know which scenarios appear, we are allowed to rent reconfigurable but expensive resources to use across scenarios. Although computing the objective in our model might seem to require enumerating exponentially-many possibilities, we show it is well-estimated by a surrogate objective which is representable by a polynomial-size LP. In this surrogate objective we pay for each scenario only to the extent that it exceeds a certain threshold. Using this objective we design algorithms that approximately-optimally balance inflexible and reconfigurable resources for several NP-hard covering problems. For example, we study minimum spanning and Steiner trees, minimum cuts and facility location variants. Up to constants our approximation guarantees match those of previous algorithms for the previously-studied demand-robust and stochastic two-stage models. Lastly, we demonstrate that our problem is sufficiently general to smoothly interpolate between previous demand-robust and stochastic two-stage problems.

We formalize the problem of selecting the optimal set of options for planning as that of computing the smallest set of options so that planning converges in less than a given maximum of value-iteration passes. We first show that the problem is $\NP$-hard, even if the task is constrained to be deterministic—the first such complexity result for option discovery. We then present the first polynomial-time boundedly suboptimal approximation algorithm for this setting, and empirically evaluate it against both the optimal options and a representative collection of heuristic approaches in simple grid-based domains.

The radio network model is a well-studied model of wireless, multi-hop networks. However, radio networks make the strong assumption that messages are delivered deterministically. The recently introduced noisy radio network model relaxes this assumption by dropping messages independently at random. In this work we quantify the relative computational power of noisy radio networks and classic radio networks. In particular, given a non-adaptive protocol for a fixed radio network we show how to reliably simulate this protocol if noise is introduced with a multiplicative cost of $\mathrm{poly}(\log Δ, \log \log n)$ rounds where $n$ is the number nodes in the network and $Δ$ is the max degree. Moreover, we demonstrate that, even if the simulated protocol is not non-adaptive, it can be simulated with a multiplicative $O(Δ\log ^2 Δ)$ cost in the number of rounds. Lastly, we argue that simulations with a multiplicative overhead of $o(\log Δ)$ are unlikely to exist by proving that an $Ω(\log Δ)$ multiplicative round overhead is necessary under certain natural assumptions.

Distributed graph algorithms that separately optimize for either the number of rounds used or the total number of messages sent have been studied extensively. However, algorithms simultaneously efficient with respect to both measures have been elusive. For example, only very recently was it shown that for Minimum Spanning Tree (MST), an optimal message and round complexity is achievable (up to polylog terms) by a single algorithm in the CONGEST model of communication.

The widely-studied radio network model [Chlamtac and Kutten, 1985] is a graph-based description that captures the inherent impact of collisions in wireless communication. In this model, the strong assumption is made that node v receives a message from a neighbor if and only if exactly one of its neighbors broadcasts. We relax this assumption by introducing a new noisy radio network model in which random faults occur at senders or receivers. Specifically, for a constant noise parameter p ∈ [0,1), either every sender has probability p of transmitting noise or every receiver of a single transmission in its neighborhood has probability p of receiving noise.

The combinatorial explosion that plagues planning and reinforcement learning (RL) algorithms can be moderated using state abstraction. Prohibitively large task representations can be condensed such that essential information is preserved, and consequently, solutions are tractably computable. However, exact abstractions, which treat only fully-identical situations as equivalent, fail to present opportunities for abstraction in environments where no two situations are exactly alike. In this work, we investigate approximate state abstractions, which treat nearly-identical situations as equivalent. We present theoretical guarantees of the quality of behaviors derived from four types of approximate abstractions. Additionally, we empirically demonstrate that approximate abstractions lead to reduction in task complexity and bounded loss of optimality of behavior in a variety of environments.