Computer Science

We revisit the fully online matching model (Huang et al., J.\ ACM, 2020), an extension of the classic online matching model due to Karp, Vazirani, and Vazirani (STOC 1990), which has recently received a lot of attention (Huang et al., SODA 2019 and FOCS 2020), partly due to applications in ride-sharing platforms. It has been shown that the fully online version is harder than the classic version for which the achievable competitive ratio is at most 0.6317 , rather than precisely 1??1 e ??.6321 . We introduce two new ideas to the construction. By optimizing the parameters of the modified construction numerically, we obtain an improved impossibility result of 0.6297 . Like the previous bound, the new bound even holds for fractional (rather than randomized) algorithms on bipartite graphs.

The ARRIVAL problem is to decide the fate of a train moving along the edges of a directed graph, according to a simple (deterministic) pseudorandom walk. The problem is in NP?�coNP but not known to be in P . The currently best algorithms have runtime 2 ?(n) where n is the number of vertices. This is not much better than just performing the pseudorandom walk. We develop a subexponential algorithm with runtime 2 O( n ??logn) . We also give a polynomial-time algorithm if the graph is almost acyclic. Both results are derived from a new general approach to solve ARRIVAL instances.

Blockchain is a distributed ledger that is decentralized, immutable, and transparent, which maintains a continuously growing list of transaction records ordered into blocks. As the core of blockchain, the consensus algorithm is an agreement to validate the correctness of blockchain transactions. For example, Bitcoin is a public blockchain where each node in Bitcoin uses the Proof of Work (PoW) algorithm to reach a consensus by competing to solve a puzzle. Unlike a public blockchain, a consortium blockchain is an enterprise-level blockchain that does not contend with the issues of creating a resource-saving global consensus protocol. This paper highilights several state-of-the art solutions in consensus algorithms for enterprise blockchain. For example, the HyperLedger by Linux Foundation includes implementing Practical Byzantine Fault Tolerance (PBFT) as the consensus algorithm. PBFT can tolerate a range of malicious nodes and reach consensus with quadratic complexity. Another consensus algorithm, HotStuff, implemented by Facebook Libra project, has achieved linear complexity of the authenticator. This paper presents the operational mechanisms of these and other consensus protocols, and analyzes and compares their advantages and drawbacks.

There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k -Center problem in this spirit are Colorful k -Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k -Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k -Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4 -approximation for Colorful k -Center with constantly many colors---settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan---and a 4 -approximation for Fair Robust k -Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k -Center admits no approximation algorithm with finite approximation guarantee, assuming that P≠NP . Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

We show that the Adaptive Greedy algorithm of Golovin and Krause (2011) achieves an approximation bound of (ln(Q/η)+1) for Stochastic Submodular Cover: here Q is the "goal value" and η is the smallest non-zero marginal increase in utility deliverable by an item. (For integer-valued utility functions, we show a bound of H(Q) , where H(Q) is the Q th Harmonic number.) Although this bound was claimed by Golovin and Krause in the original version of their paper, the proof was later shown to be incorrect by Nan and Saligrama (2017). The subsequent corrected proof of Golovin and Krause (2017) gives a quadratic bound of (ln(Q/η)+1 ) 2 . Other previous bounds for the problem are 56(ln(Q/η)+1) , implied by work of Im et al. (2016) on a related problem, and k(ln(Q/η)+1) , due to Deshpande et al. (2016) and Hellerstein and Kletenik (2018), where k is the number of states. Our bound generalizes the well-known (ln m+1) approximation bound on the greedy algorithm for the classical Set Cover problem, where m is the size of the ground set.

(see paper for full abstract) We show that the Edge-Disjoint Paths problem is W[1]-hard parameterized by the number k of terminal pairs, even when the input graph is a planar directed acyclic graph (DAG). This answers a question of Slivkins (ESA '03, SIDMA '10). Moreover, under the Exponential Time Hypothesis (ETH), we show that there is no f(k)??n o(k) algorithm for Edge-Disjoint Paths on planar DAGs, where k is the number of terminal pairs, n is the number of vertices and f is any computable function. Our hardness holds even if both the maximum in-degree and maximum out-degree of the graph are at most 2 . Our result shows that the n O(k) algorithm of Fortune et al. (TCS '80) for Edge-Disjoint Paths on DAGs is asymptotically tight, even if we add an extra restriction of planarity. As a special case of our result, we obtain that Edge-Disjoint Paths on planar directed graphs is W[1]-hard parameterized by the number k of terminal pairs. This answers a question of Cygan et al. (FOCS '13) and Schrijver (pp. 417-444, Building Bridges II, '19), and completes the landscape of the parameterized complexity status of edge and vertex versions of the Disjoint Paths problem on planar directed and planar undirected graphs.

Seminal works on light spanners from recent years provide near-optimal tradeoffs between the stretch and lightness of spanners in general graphs, minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a ``truly optimal'' tradeoff for Euclidean low-dimensional spaces. Some of these papers employ inherently different techniques than others. Moreover, the runtime of these constructions is rather high. In this work, we present a unified and fine-grained approach for light spanners. Besides the obvious theoretical importance of unification, we demonstrate the power of our approach in obtaining (1) stronger lightness bounds, and (2) faster construction times. Our results include: _ K r -minor-free graphs: A truly optimal spanner construction and a fast construction. _ General graphs: A truly optimal spanner -- almost and a linear-time construction with near-optimal lightness. _ Low dimensional Euclidean spaces: We demonstrate that Steiner points help in reducing the lightness of Euclidean 1+ϵ -spanners almost quadratically for d≥3 .

In this paper, we address computation of the degree deg DetA of Dieudonné determinant DetA of A= ∑ k=1 m A k x k t c k , where A k are n×n matrices over a field K , x k are noncommutative variables, t is a variable commuting with x k , c k are integers, and the degree is considered for t . This problem generalizes noncommutative Edmonds' problem and fundamental combinatorial optimization problems including the weighted linear matroid intersection problem. It was shown that deg DetA is obtained by a discrete convex optimization on a Euclidean building. We extend this framework by incorporating a cost scaling technique, and show that deg DetA can be computed in time polynomial of n,m, log 2 C , where C:= max k | c k | . We give a polyhedral interpretation of deg Det , which says that deg DetA is given by linear optimization over an integral polytope with respect to objective vector c=( c k ) . Based on it, we show that our algorithm becomes a strongly polynomial one. We apply this result to an algebraic combinatorial optimization problem arising from a symbolic matrix having 2×2 -submatrix structure.

The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs.

In this note, we describe a simple approach to obtain a differentially private algorithm for k-clustering with nearly the same multiplicative factor as any non-private counterpart at the cost of a large polynomial additive error. The approach is the combination of a simple geometric observation independent of privacy consideration and any existing private algorithm with a constant approximation.