Computer Science

We provide a deterministic CONGEST algorithm to constant factor approximate the minimum dominating set on graphs of bounded arboricity in O(logn) rounds. This improves over the well-known randomized algorithm of Lenzen and Wattenhofer[DISC2010] by making it a deterministic algorithm.

Let G=(V,E,w) be a weighted, digraph subject to a sequence of adversarial edge deletions. In the decremental single-source reachability problem (SSR), we are given a fixed source s and the goal is to maintain a data structure that can answer path-queries s↣v for any v∈V . In the more general single-source shortest paths (SSSP) problem the goal is to return an approximate shortest path to v , and in the SCC problem the goal is to maintain strongly connected components of G and to answer path queries within each component. All of these problems have been very actively studied over the past two decades, but all the fast algorithms are randomized and, more significantly, they can only answer path queries if they assume a weaker model: they assume an oblivious adversary which is not adaptive and must fix the update sequence in advance. This assumption significantly limits the use of these data structures, most notably preventing them from being used as subroutines in static algorithms. All the above problems are notoriously difficult in the adaptive setting. In fact, the state-of-the-art is still the Even and Shiloach tree, which dates back all the way to 1981 and achieves total update time O(mn) . We present the first algorithms to break through this barrier: 1) deterministic decremental SSR/SCC with total update time m n 2/3+o(1) 2) deterministic decremental SSSP with total update time n 2+2/3+o(1) . To achieve these results, we develop two general techniques of broader interest for working with dynamic graphs: 1) a generalization of expander-based tools to dynamic directed graphs, and 2) a technique that we call congestion balancing and which provides a new method for maintaining flow under adversarial deletions. Using the second technique, we provide the first near-optimal algorithm for decremental bipartite matching.

In the decremental single-source shortest paths problem, the goal is to maintain distances from a fixed source s to every vertex v in an m -edge graph undergoing edge deletions. In this paper, we conclude a long line of research on this problem by showing a near-optimal deterministic data structure that maintains (1+ϵ) -approximate distance estimates and runs in m 1+o(1) total update time. Our result, in particular, removes the oblivious adversary assumption required by the previous breakthrough result by Henzinger et al. [FOCS'14], which leads to our second result: the first almost-linear time algorithm for (1?��? -approximate min-cost flow in undirected graphs where capacities and costs can be taken over edges and vertices. Previously, algorithms for max flow with vertex capacities, or min-cost flow with any capacities required super-linear time. Our result essentially completes the picture for approximate flow in undirected graphs. The key technique of the first result is a novel framework that allows us to treat low-diameter graphs like expanders. This allows us to harness expander properties while bypassing shortcomings of expander decomposition, which almost all previous expander-based algorithms needed to deal with. For the second result, we break the notorious flow-decomposition barrier from the multiplicative-weight-update framework using randomization.

There is a recent exciting line of work in distributed graph algorithms in the CONGEST model that exploit expanders. All these algorithms so far are based on two tools: expander decomposition and expander routing. An (ϵ,ϕ) -expander decomposition removes ϵ -fraction of the edges so that the remaining connected components have conductance at least ϕ , i.e., they are ϕ -expanders, and expander routing allows each vertex v in a ϕ -expander to very quickly exchange deg(v) messages with any other vertices, not just its local neighbors. In this paper, we give the first efficient deterministic distributed algorithms for both tools. We show that an (ϵ,ϕ) -expander decomposition can be deterministically computed in poly( ϵ −1 ) n o(1) rounds for ϕ=poly(ϵ) n −o(1) , and that expander routing can be performed deterministically in poly( ϕ −1 ) n o(1) rounds. Both results match previous bounds of randomized algorithms by [Chang and Saranurak, PODC 2019] and [Ghaffari, Kuhn, and Su, PODC 2017] up to subpolynomial factors. Consequently, we derandomize existing distributed algorithms that exploit expanders. We show that a minimum spanning tree on n o(1) -expanders can be constructed deterministically in n o(1) rounds, and triangle detection and enumeration on general graphs can be solved deterministically in O( n 0.58 ) and n 2/3+o(1) rounds, respectively. We also give the first polylogarithmic-round randomized algorithm for constructing an (ϵ,ϕ) -expander decomposition in poly( ϵ −1 ,logn) rounds for ϕ=1/poly( ϵ −1 ,logn) . The previous algorithm by [Chang and Saranurak, PODC 2019] needs n Ω(1) rounds for any ϕ≥1/polylogn .

In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph G , a vertex pair (s,t)∈V(G)×V(G) , and a set of edge faults F⊆E(G) , a replacement path P(s,t,F) is an s - t shortest path in G∖F . For integer parameters L,f , a replacement path covering (RPC) is a collection of subgraphs of G , denoted by G L,f ={ G 1 ,…, G r } , such that for every set F of at most f faults (i.e., |F|≤f ) and every replacement path P(s,t,F) of at most L edges, there exists a subgraph G i ∈ G L,f that contains all the edges of P and does not contain any of the edges of F . The covering value of the RPC G L,f is then defined to be the number of subgraphs in G L,f . We present efficient deterministic constructions of (L,f) -RPCs whose covering values almost match the randomized ones, for a wide range of parameters. Our time and value bounds improve considerably over the previous construction of Parter (DISC 2019). We also provide an almost matching lower bound for the value of these coverings. A key application of our above deterministic constructions is the derandomization of the algebraic construction of the distance sensitivity oracle by Weimann and Yuster (FOCS 2010). The preprocessing and query time of the our deterministic algorithm nearly match the randomized bounds. This resolves the open problem of Alon, Chechik and Cohen (ICALP 2019).

Embeddings of graphs into distributions of trees that preserve distances in expectation are a cornerstone of many optimization algorithms. Unfortunately, online or dynamic algorithms which use these embeddings seem inherently randomized and ill-suited against adaptive adversaries. In this paper we provide a new tree embedding which addresses these issues by deterministically embedding a graph into a single tree containing O(logn) copies of each vertex while preserving the connectivity structure of every subgraph and O( log 2 n) -approximating the cost of every subgraph. Using this embedding we obtain several new algorithmic results: We reduce an open question of Alon et al. [SODA 2004] -- the existence of a deterministic poly-log-competitive algorithm for online group Steiner tree on a general graph -- to its tree case. We give a poly-log-competitive deterministic algorithm for a closely related problem -- online partial group Steiner tree -- which, roughly, is a bicriteria version of online group Steiner tree. Lastly, we give the first poly-log approximations for demand-robust Steiner forest, group Steiner tree and group Steiner forest.

This paper studies the problem of clustering in metric spaces while preserving the privacy of individual data. Specifically, we examine differentially private variants of the k-medians and Euclidean k-means problems. We present polynomial algorithms with constant multiplicative error and lower additive error than the previous state-of-the-art for each problem. Additionally, our algorithms use a clustering algorithm without differential privacy as a black-box. This allows practitioners to control the trade-off between runtime and approximation factor by choosing a suitable clustering algorithm to use.

We introduce a new ( ϵ p , δ p ) -differentially private algorithm for the k -means clustering problem. Given a dataset in Euclidean space, the k -means clustering problem requires one to find k points in that space such that the sum of squares of Euclidean distances between each data point and its closest respective point among the k returned is minimised. Although there exist privacy-preserving methods with good theoretical guarantees to solve this problem [Balcan et al., 2017; Kaplan and Stemmer, 2018], in practice it is seen that it is the additive error which dictates the practical performance of these methods. By reducing the problem to a sequence of instances of maximum coverage on a grid, we are able to derive a new method that achieves lower additive error then previous works. For input datasets with cardinality n and diameter Δ , our algorithm has an O( Δ 2 (k log 2 nlog(1/ δ p )/ ϵ p +k dlog(1/ δ p ) − − − − − − − − − √ / ϵ p )) additive error whilst maintaining constant multiplicative error. We conclude with some experiments and find an improvement over previously implemented work for this problem.

We consider a generalization of finding a homomorphism from an input digraph G to a fixed digraph H , HOM( H ). In this setting, we are given an input digraph G together with a list function from G to 2 H . The goal is to find a homomorphism from G to H with respect to the lists if one exists. We show that if the list function is a Maltsev polymorphism then deciding whether G admits a homomorphism to H is polynomial time solvable. In our approach, we only use the existence of the Maltsev polymorphism. Furthermore, we show that deciding whether a relational structure R admits a Maltsev polymorphism is a special case of finding a homormphism from a graph G to a graph H and a list function with a Maltsev polymorphism. Since the existence of Maltsev is not required in our algorithm, we can decide in polynomial time whether the relational structure R admits Maltsev or not. We also discuss forbidden obstructions for the instances admitting Maltsev list polymorphism. We have implemented our algorithm and tested on instances arising from linear equations, and other types of instances.

Observing the internal state of the whole system using a small number of sensor nodes is important in analysis of complex networks. Here, we study the problem of determining the minimum number of sensor nodes to discriminate attractors under the assumption that each attractor has at most K noisy nodes. We present exact and approximation algorithms for this minimization problem. The effectiveness of the algorithms is also demonstrated by computational experiments using both synthetic data and realistic biological data.