In the online non-metric variant of the facility location problem, there is a given graph consisting of a set $F$ of facilities (each with a certain opening cost), a set $C$ of potential clients, and weighted connections between them. The online part of the input is a sequence of clients from $C$, and in response to any requested client, an online algorithm may open an additional subset of facilities and must connect the given client to an open facility.
We give an online, polynomial-time deterministic algorithm for this problem, with a competitive ratio of $O(\log |F| \cdot (\log |C| + \log \log |F|))$. The result is optimal up to loglog factors. Our algorithm improves over the $O((\log |C| + \log |F|) \cdot (\log |C| + \log \log |F|))$-competitive construction that first reduces the facility location instance to a set cover one and then later solves such instance using the deterministic algorithm by Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical scenario where $|F| \ll |C|$.
We achieve this by a more direct approach: we design an algorithm for a fractional relaxation of the non-metric facility location problem with clustered facilities. To handle the constraints of such non-covering LP, we combine the dual fitting and multiplicative weight updates approach. By maintaining certain additional monotonicity properties of the created fractional solution, we can handle the dependencies between facilities and connections in a rounding routine.
Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the first deterministic algorithm for the online node-weighted Steiner tree problem. The resulting competitive ratio is $O(\log k \cdot \log^2 \ell)$ on graphs of $\ell$ nodes and $k$ terminals.
We describe the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs, and for (non-metric) fractional facility location. We also describe the first parallel algorithms requiring only near-linear total work and finishing in polylog time. The algorithms compute $(1+ε)$-approximate solutions in time (and work) $O^*(N/ε^2)$, where $N$ is the number of non-zeros in the constraint matrix. For facility location, $N$ is the number of eligible client/facility pairs.
We study the metric facility location problem with client insertions and deletions. This setting differs from the classic dynamic facility location problem, where the set of clients remains the same, but the metric space can change over time. We show a deterministic algorithm that maintains a constant factor approximation to the optimal solution in worst-case time $\tilde O(2^{O(κ^2)})$ per client insertion or deletion in metric spaces while answering queries about the cost in $O(1)$ time, where $κ$ denotes the doubling dimension of the metric. For metric spaces with bounded doubling dimension…
We study the problem of online facility location with delay. In this problem, a sequence of $n$ clients appear in the metric space, and they need to be eventually connected to some open facility. The clients do not have to be connected immediately, but such a choice comes with a penalty: each client incurs a waiting cost (the difference between its arrival and connection time). At any point in time, an algorithm may decide to open a facility and connect any subset of clients to it. This is a well-studied problem both of its own, and within the general class of network design problems with dela…
In this work, we study the classic submodular maximization problem under knapsack constraints and beyond. We first present an $(7/16-\varepsilon)$-approximate algorithm for single knapsack constraint, which requires $O(n\cdot\max\{\varepsilon^{-1},\log\log n\})$ queries, and two passes in the streaming setting. This provides improvements in approximation ratio, query complexity and number of passes on the stream. We next show that there exists an $(1/2-\varepsilon)$-approximate deterministic algorithm for constant number of binary packing constraints, which achieves a query complexity of $O_{\…
We consider a natural extension to the metric uncapacitated Facility Location Problem (FLP) in which requests ask for different commodities out of a finite set $S$ of commodities. Ravi and Sinha (SODA'04) introduced the model as the Multi-Commodity Facility Location Problem (MFLP) and considered it an offline optimization problem. The model itself is similar to the FLP: i.e., requests are located at points of a finite metric space and the task of an algorithm is to construct facilities and assign requests to facilities while minimizing the construction cost and the sum over all assignment dist…
We provide nearly optimal algorithms for online facility location (OFL) with predictions. In OFL, $n$ demand points arrive in order and the algorithm must irrevocably assign each demand point to an open facility upon its arrival. The objective is to minimize the total connection costs from demand points to assigned facilities plus the facility opening cost. We further assume the algorithm is additionally given for each demand point $x_i$ a natural prediction $f_{x_i}^{\mathrm{pred}}$ which is supposed to be the facility $f_{x_i}^{\mathrm{opt}}$ that serves $x_i$ in the offline optimal solution…
A deterministic approximation algorithm is presented for the maximization of non-monotone submodular functions over a ground set of size $n$ subject to cardinality constraint $k$; the algorithm is based upon the idea of interlacing two greedy procedures. The algorithm uses interlaced, thresholded greedy procedures to obtain tight ratio $1/4 - ε$ in $O \left( \frac{n}ε \log \left( \frac{k}ε \right) \right)$ queries of the objective function, which improves upon both the ratio and the quadratic time complexity of the previously fastest deterministic algorithm for this problem. The algorithm is v…
We consider network design problems with deadline or delay. All previous results for these models are based on randomized embedding of the graph into a tree (HST) and then solving the problem on this tree. We show that this is not necessary. In particular, we design a deterministic framework for these problems which is not based on embedding. This enables us to provide deterministic $\text{poly-log}(n)$-competitive algorithms for Steiner tree, generalized Steiner tree, node weighted Steiner tree, (non-uniform) facility location and directed Steiner tree with deadlines or with delay (where $n$ …
This paper presents a distributed O(1)-approximation algorithm, with expected-$O(\log \log n)$ running time, in the $\mathcal{CONGEST}$ model for the metric facility location problem on a size-$n$ clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric …
In this paper, we give first constant factor approximation for capacitated knapsack median problem (CKM) for hard uniform capacities, violating the budget only by an additive factor of $f_{max}$ where $f_{max}$ is the maximum cost of a facility opened by the optimal and violating capacities by $(2+ε)$ factor. Natural LP for the problem is known to have an unbounded integrality gap when any one of the two constraints is allowed to be violated by a factor less than $2$. Thus, we present a result which is very close to the best achievable from the natural LP. To the best of our knowledge, the pro…
University of Science and Technology of China（中国科学技术大学）
We study the problem of maximizing a non-monotone, non-negative submodular function subject to a matroid constraint. The prior best-known deterministic approximation ratio for this problem is $\frac{1}{4}-ε$ under $\mathcal{O}(({n^4}/ε)\log n)$ time complexity. We show that this deterministic ratio can be improved to $\frac{1}{4}$ under $\mathcal{O}(nr)$ time complexity, and then present a more practical algorithm dubbed TwinGreedyFast which achieves $\frac{1}{4}-ε$ deterministic ratio in nearly-linear running time of $\mathcal{O}(\frac{n}ε\log\frac{r}ε)$. Our approach is based on a novel algo…
In this paper, we study the two-facility location game on a line with optional preference where the acceptable set of facilities for each agent could be different and an agent's cost is his distance to the closest facility within his acceptable set. The objective is to minimize the total cost of all agents while achieving strategyproofness. We design a deterministic strategyproof mechanism for the problem with approximation ratio of 2.75, improving upon the earlier best ratio of n/2+1.
In the k-median problem we are given sets of facilities and customers, and distances between them. For a given set F of facilities, the cost of serving a customer u is the minimum distance between u and a facility in F. The goal is to find a set F of k facilities that minimizes the sum, over all customers, of their service costs. Following Mettu and Plaxton, we study the incremental medians problem, where k is not known in advance, and the algorithm produces a nested sequence of facility sets where the kth set has size k. The algorithm is c-cost-competitive if the cost of each set is at most c…
In this work, we present a combinatorial, deterministic single-pass streaming algorithm for the problem of maximizing a submodular function, not necessarily monotone, with respect to a cardinality constraint (SMCC). In the case the function is monotone, our algorithm reduces to the optimal streaming algorithm of Badanidiyuru et al. (2014). In general, our algorithm achieves ratio $α/ (1 + α) - \varepsilon$, for any $\varepsilon > 0$, where $α$ is the ratio of an offline (deterministic) algorithm for SMCC used for post-processing. Thus, if exponential computation time is allowed, our algorithm …
We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph $G$, a positive integer $r$, and a set $S$ of vertices, approximates the conductance of $S$ in the $r$-step random walk on $G$ to within a factor of $1+ε$, where $ε>0$ is an arbitrarily small constant. More generally, our algorithm computes an $ε$-spectral approximation to the normalized Laplacian of the $r$-step walk. Our algorithm combines the derandomized square graph operation (Rozenman and Vadhan, 2005), which we recently used for solving Laplacian systems in nearly logarithmic space (Murtagh, Reing…
In the $d$-dimensional cow-path problem, a cow living in $\mathbb{R}^d$ must locate a $(d - 1)$-dimensional hyperplane $H$ whose location is unknown. The only way that the cow can find $H$ is to roam $\mathbb{R}^d$ until it intersects $\mathcal{H}$. If the cow travels a total distance $s$ to locate a hyperplane $H$ whose distance from the origin was $r \ge 1$, then the cow is said to achieve competitive ratio $s / r$. It is a classic result that, in $\mathbb{R}^2$, the optimal (deterministic) competitive ratio is $9$. In $\mathbb{R}^3$, the optimal competitive ratio is known to be at most $\ap…
The Quantitative Group Testing (QGT) is about learning a (hidden) subset $K$ of some large domain $N$ using a sequence of queries, where a result of a query provides information about the size of the intersection of the query with the unknown subset $K$. Almost all previous work focused on randomized algorithms minimizing the number of queries; however, in case of large domains $N$, randomization may result in a significant deviation from the expected precision. Others assumed unlimited computational power (existential results) or adaptiveness of queries. In this work we propose efficient non-…
In an undirected graph, a $k$-cut is a set of edges whose removal breaks the graph into at least $k$ connected components. The minimum weight $k$-cut can be computed in $O(n^{O(k)})$ time, but when $k$ is treated as part of the input, computing the minimum weight $k$-cut is NP-Hard [Holdschmidt and Hochbaum 1994]. For $\operatorname{poly}(m,n,k)$-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017]. Saran and Vazirani [1995] showed that a $(2 - 2/k)$-approximately minimum weight $k$-cut can be computed by $O(k)$ min…
We consider the problem of computing the Boolean convolution (with wraparound) of $n$~vectors of dimension $m$, or, equivalently, the problem of computing the sumset $A_1+A_2+\ldots+A_n$ for $A_1,\ldots,A_n \subseteq \mathbb{Z}_m$. Boolean convolution formalizes the frequent task of combining two subproblems, where the whole problem has a solution of size $k$ if for some $i$ the first subproblem has a solution of size~$i$ and the second subproblem has a solution of size $k-i$. Our problem formalizes a natural generalization, namely combining solutions of $n$ subproblems subject to a modular co…
We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our wo…
In this paper, we present a framework used to construct and analyze algorithms for online optimization problems with deadlines or with delay over a metric space. Using this framework, we present algorithms for several different problems. We present an $O(D^{2})$-competitive deterministic algorithm for online multilevel aggregation with delay on a tree of depth $D$, an exponential improvement over the $O(D^{4}2^{D})$-competitive algorithm of Bienkowski et al. (ESA '16), where the only previously-known improvement was for the special case of deadlines by Buchbinder et al. (SODA '17). We also pre…
The extended Lüroth's Theorem says that if the transcendence degree of $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)/\KK$ is 1 then there exists $f \in \KK(\underline{X})$ such that $\KK(\mathsf{f}_1,\dots,\mathsf{f}_m)$ is equal to $\KK(f)$. In this paper we show how to compute $f$ with a probabilistic algorithm. We also describe a probabilistic and a deterministic algorithm for the decomposition of multivariate rational functions. The probabilistic algorithms proposed in this paper are softly optimal when $n$ is fixed and $d$ tends to infinity. We also give an indecomposability test based on gcd com…
A long line of research about connectivity in the Massively Parallel Computation model has culminated in the seminal works of Andoni et al. [FOCS'18] and Behnezhad et al. [FOCS'19]. They provide a randomized algorithm for low-space MPC with conjectured to be optimal round complexity $O(\log D + \log \log_{\frac m n} n)$ and $O(m)$ space, for graphs on $n$ vertices with $m$ edges and diameter $D$. Surprisingly, a recent result of Coy and Czumaj [STOC'22] shows how to achieve the same deterministically. Unfortunately, however, their algorithm suffers from large local computation time. We present…
We consider the classic Facility Location, $k$-Median, and $k$-Means problems in metric spaces of doubling dimension $d$. We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is $2^{(\log(1/\eps)/\eps)^{O(d^2)}} n \log^4 n + 2^{O(d)} n \log^9 n$, making a significant improvement over the state-of-the-art algorithms which run in time $n^{(d/\eps)^{O(d)}}$. Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting $k$-Medians and $k$-Means, and efficient bicriteria appro…
In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x \in \mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $\hat x$, the Discrete Fourier Transform (DFT) of $x$. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al.\@ (J Fourier Anal Appl 2018), which obtains $O(k^2 \log^{-1}k \cdot \log^{5.5}n)$ samples and a similar runtime with the $\ell_2/\ell_1$ guarantee. We focus on the stronger $\ell_{\infty}/\ell…
