Optimal Output Sensitive Fault Tolerant Cuts

In this paper we consider two classic cut-problems, Global Min-Cut and Min k-Cut, via the lens of fault tolerant network design. In particular, given a graph G on n vertices, and a positive integer f, our objective is to compute an upper bound on the size of the sparsest subgraph H of G that preserves edge connectivity of G (denoted by ?(G)) in the case of Global Min-Cut, and ?(G,k) (denotes the minimum number of edges whose removal would partition the graph into at least k connected components) in the case of Min k-Cut, upon failure of any f edges of G. The subgraph H corresponding to Global Min-Cut and Min k-Cut is called f-FTCS and f-FT-k-CS, respectively. We obtain the following results about the sizes of f-FTCS and f-FT-k-CS. - There exists an f-FTCS with (n-1)(f+?(G)) edges. We complement this upper bound with a matching lower bound, by constructing an infinite family of graphs where any f-FTCS must have at least ((n-?(G)-1)(?(G)+f-1))/2+(n-?(G)-1)+/?(G)(?(G)+1))/2 edges. - There exists an f-FT-k-CS with min{(2f+?(G,k)-(k-1))(n-1), (f+?(G,k))(n-k)+?} edges. We complement this upper bound with a lower bound, by constructing an infinite family of graphs where any f-FT-k-CS must have at least ((n-?(G,k)-1)(?(G,k)+f-k+1))/2)+n-?(G,k)+k-3+((?(G,k)-k+3)(?(G,k)-k+2))/2 edges. Our upper bounds exploit the structural properties of k-connectivity certificates. On the other hand, for our lower bounds we construct an infinite family of graphs, such that for any graph in the family any f-FTCS (or f-FT-k-CS) must contain all its edges. We also add that our upper bounds are constructive. That is, there exist polynomial time algorithms that construct H with the aforementioned number of edges

Hardness of Approximate Nearest Neighbor Search

We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance. Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false, for every $\delta>0$ there exists a constant $\epsilon>0$ such that computing a $(1+\epsilon)$-approximation to the Bichromatic Closest Pair requires $n^{2-\delta}$ time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time. Our reduction uses the Distributed PCP framework of [ARW'17], but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before [BKKMS'16, BCGRS'17], but our construction is the first to yield new hardness results

Repetition Detection in a Dynamic String

A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings. We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting

Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ? 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ? 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem

An Empirical Analysis of Approximation Algorithms for the Unweighted Tree Augmentation Problem

In this thesis, we perform an experimental study of approximation algorithms for the tree augmentation problem (TAP). TAP is a fundamental problem in network design. The goal of TAP is to add the minimum number of edges from a given edge set to a tree so that it becomes 2-edge connected. Formally, given a tree T = (V, E), where V denotes the set of vertices and E denotes the set of edges in the tree, and a set of edges (or links) L ⊆ V × V disjoint from E, the objective is to find a set of edges to add to the tree F ⊆ L such that the augmented tree (V, E ∪ F) is 2-edge connected. Our goal is to establish a baseline performance for each approximation algorithm on actual instances rather than worst-case instances. In particular, we are interested in whether the algorithms rank on practical instances is consistent with their worst-case guarantee rankings. We are also interested in whether preprocessing times, implementation difficulties, and running times justify the use of an algorithm in practice. We profiled and analyzed five approximation algorithms, viz., the Frederickson algorithm, the Nagamochi algorithm, the Even algorithm, the Adjiashivili algorithm, and the Grandoni algorithm. Additionally, we used an integer program and a simple randomized algorithm as benchmarks. The performance of each algorithm was measured using space, time, and quality comparison metrics. We found that the simple randomized is competitive with the approximation algorithms and that the algorithms rank according to their theoretical guarantees. The randomized algorithm is simpler to implement and understand. Furthermore, the randomized algorithm runs faster and uses less space than any of the more sophisticated approximation algorithms

On Complexity of 1-Center in Various Metrics

We consider the classic 1-center problem: Given a set P of n points in a metric space find the point in P that minimizes the maximum distance to the other points of P. We study the complexity of this problem in d-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length d. Our results for the 1-center problem may be classified based on d as follows. $\bullet$ Small d: We provide the first linear-time algorithm for 1-center problem in fixed-dimensional $\ell_1$ metrics. On the other hand, assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large d. When $d=\Omega(n)$, we extend our conditional lower bound to rule out sub quartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\epsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension d, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of n strings each of length n, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set

The Unreasonable Success of Local Search: Geometric Optimization

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/\epsilon^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP