Computer Science

We consider the generalized k -server problem on uniform metrics. We study the power of memoryless algorithms and show tight bounds of Θ(k!) on their competitive ratio. In particular we show that the \textit{Harmonic Algorithm} achieves this competitive ratio and provide matching lower bounds. This improves the ≈ 2 2 k doubly-exponential bound of Chiplunkar and Vishwanathan for the more general setting of uniform metrics with different weights.

We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) f t : A t → B t between subsets A t and B t of the metric space. To serve it, the algorithm has to go to a point a t ∈ A t , paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm's state to f t ( a t ) . Such transformations can model, e.g., changes to the environment that are outside of an algorithm's control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the k -taxi problem. We show that for α -Lipschitz transformations, the competitive ratio is Θ(α ) n−2 on n -point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the k -taxi problem, we prove a competitive ratio of O ~ ((nlogk ) 2 ) . For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists. The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M , what is the required cardinality of an extension M ^ ⊇M where each partial isometry on M extends to an automorphism? We give partial answers for special cases.

A follow-up to my previous tutorial on metric indexing, this paper walks through the classic structures, placing them all in the context of the recently proposed "sprawl of ambits" framework. The indexes are presented as configurations of a single, more general structure, all queried using the same search procedure.

Given a separation oracle SO for a convex function f that has an integral minimizer inside a box with radius R , we show how to find an exact minimizer of f using at most (a) O(n(n+log(R))) calls to SO and poly(n,log(R)) arithmetic operations, or (b) O(nlog(nR)) calls to SO and exp(n)⋅poly(log(R)) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O( n 2 (n+log(R))) for polynomial time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O( n 3 ) calls to an evaluation oracle, and an exponential time algorithm that makes at most O( n 2 log(n)) calls to an evaluation oracle. These improve upon the previously best O( n 3 log 2 (n)) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O( n 3 log(n)) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of certain lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that captures simultaneously the size of the search set and the density of the lattice, which we analyze via technical tools from convex geometry.

A directed graph G=(V,E) is strongly biconnected if G is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph G=(V,E) is called 2 -vertex-strongly biconnected if |V|≥3 and the induced subgraph on V∖{w} is strongly biconnected for every vertex w∈V . In this paper we study the following problem. Given a 2 -vertex-strongly biconnected directed graph G=(V,E) , compute an edge subset E 2sb ⊆E of minimum size such that the subgraph (V, E 2sb ) is 2 -vertex-strongly biconnected.

In this paper we provide new randomized algorithms with improved runtimes for solving linear programs with two-sided constraints. In the special case of the minimum cost flow problem on n -vertex m -edge graphs with integer polynomially-bounded costs and capacities we obtain a randomized method which solves the problem in O ~ (m+ n 1.5 ) time. This improves upon the previous best runtime of O ~ (m n ??????) (Lee-Sidford 2014) and, in the special case of unit-capacity maximum flow, improves upon the previous best runtimes of m 4/3+o(1) (Liu-Sidford 2020, Kathuria 2020) and O ~ (m n ??????) (Lee-Sidford 2014) for sufficiently dense graphs. For ??1 -regression in a matrix with n -columns and m -rows we obtain a randomized method which computes an ϵ -approximate solution in O ~ (mn+ n 2.5 ) time. This yields a randomized method which computes an ϵ -optimal policy of a discounted Markov Decision Process with S states and A actions per state in time O ~ ( S 2 A+ S 2.5 ) . These methods improve upon the previous best runtimes of methods which depend polylogarithmically on problem parameters, which were O ~ (m n 1.5 ) (Lee-Sidford 2015) and O ~ ( S 2.5 A) (Lee-Sidford 2014, Sidford-Wang-Wu-Ye 2018). To obtain this result we introduce two new algorithmic tools of independent interest. First, we design a new general interior point method for solving linear programs with two sided constraints which combines techniques from (Lee-Song-Zhang 2019, Brand et al. 2020) to obtain a robust stochastic method with iteration count nearly the square root of the smaller dimension. Second, to implement this method we provide dynamic data structures for efficiently maintaining approximations to variants of Lewis-weights, a fundamental importance measure for matrices which generalize leverage scores and effective resistances.

The Minimum Eccentricity Shortest Path Problem consists in finding a shortest path with minimum eccentricity in a given undirected graph. The problem is known to be NP-complete and W[2]-hard with respect to the desired eccentricity. We present fpt algorithms for the problem parameterized by the modular width, distance to cluster graph, the combination of distance to disjoint paths with the desired eccentricity, and maximum leaf number.

The minimum linear arrangement problem (MLA) consists of finding a mapping ? from vertices of a graph to integers that minimizes ??uv?�E |?(u)?�π(v)| . For trees, various algorithms are available to solve the problem in polynomial time; the best known runs in subquadratic time in n=|V| . There exist variants of the MLA in which the arrangements are constrained to certain classes of projectivity. Iordanskii, and later Hochberg and Stallmann (HS), put forward O(n) -time algorithms that solve the problem when arrangements are constrained to be planar. We also consider linear arrangements of rooted trees that are constrained to be projective. Gildea and Temperley (GT) sketched an algorithm for the projectivity constraint which, as they claimed, runs in O(n) but did not provide any justification of its cost. In contrast, Park and Levy claimed that GT's algorithm runs in O(nlog d max ) where d max is the maximum degree but did not provide sufficient detail. Here we correct an error in HS's algorithm for the planar case, show its relationship with the projective case, and derive an algorithm for the projective case that runs undoubtlessly in O(n) -time.

When searching for interesting structures in graphs, it is often important to take into account not only the graph connectivity, but also the metadata available, such as node and edge labels, or temporal information. In this paper we are interested in settings where such metadata is used to define a similarity between edges. We consider the problem of finding subgraphs that are dense and whose edges are similar to each other with respect to a given similarity function. Depending on the application, this function can be, for example, the Jaccard similarity between the edge label sets, or the temporal correlation of the edge occurrences in a temporal graph. We formulate a Lagrangian relaxation-based optimization problem to search for dense subgraphs with high pairwise edge similarity. We design a novel algorithm to solve the problem through parametric MinCut, and provide an efficient search scheme to iterate through the values of the Lagrangian multipliers. Our study is complemented by an evaluation on real-world datasets, which demonstrates the usefulness and efficiency of the proposed approach.

The modular subset sum problem consists of deciding, given a modulus m , a multiset S of n integers in 0..m−1 , and a target integer t , whether there exists a subset of S with elements summing to tmodm , and to report such a set if it exists. We give a simple O(mlogm) -time with high probability (w.h.p.) algorithm for the modular subset sum problem. This builds on and improves on a previous O(m log 7 m) w.h.p. algorithm from Axiotis, Backurs, Jin, Tzamos, and Wu (SODA 19). Our method utilizes the ADT of the dynamic strings structure of Gawrychowski et al. (SODA~18). However, as this structure is rather complicated we present a much simpler alternative which we call the Data Dependent Tree. As an application, we consider the computational version of a fundamental theorem in zero-sum Ramsey theory. The Erdős-Ginzburg-Ziv Theorem states that a multiset of 2n−1 integers always contains a subset of cardinality exactly n whose values sum to a multiple of n . We give an algorithm for finding such a subset in time O(nlogn) w.h.p. which improves on an O( n 2 ) algorithm due to Del Lungo, Marini, and Mori (Disc. Math. 09).