Data Structures And Algorithms
Daniel Wiebking
2019-04-24
We show that normalizers and permutational isomorphisms of permutation groups given by generating sets can be computed in time simply exponential in the degree of the groups. The result is obtained by exploiting canonical forms for permutation groups (up to permutational isomorphism).
Sepideh Aghamolaei, Mohammad Ghodsi
2019-02-05
A set of points 
in a metric space and a constant integer 
are given. The -center problem finds points as centers among , such that the maximum distance of any point of to their closest centers 
is minimized. Doubling metrics are metric spaces in which for any 
, a ball of radius can be covered using a constant number of balls of radius 
. Fixed dimensional Euclidean spaces are doubling metrics. The lower bound on the approximation factor of -center is 
in Euclidean spaces, however, 
-approximation algorithms with exponential dependency on 
and exist. For a given set of sets 
, a composable coreset independently computes subsets 
, such that 
contains an approximation of a measure of the set 
. We introduce a -approximation composable coreset for -center, which in doubling metrics has size sublinear in 
. This results in a 
-approximation algorithm for -center in MapReduce with a constant number of rounds in doubling metrics for any 
and sublinear communications, which is based on parametric pruning. We prove the exponential nature of the trade-off between the number of centers 
and the radius , and give a composable coreset for a related problem called dual clustering. Also, we give a new version of the parametric pruning algorithm with 
running time, 
space and 
approximation factor for metric -center.
Jurek Czyzowicz, Dariusz Dereniowski, Andrzej Pelc
2019-04-24
A robot modeled as a deterministic finite automaton has to build a structure from material available to it. The robot navigates in the infinite oriented grid 
. Some cells of the grid are full (contain a brick) and others are empty. The subgraph of the grid induced by full cells, called the field, is initially connected. The (Manhattan) distance between the farthest cells of the field is called its span. The robot starts at a full cell. It can carry at most one brick at a time. At each step it can pick a brick from a full cell, move to an adjacent cell and drop a brick at an empty cell. The aim of the robot is to construct the most compact possible structure composed of all bricks, i.e., a nest. That is, the robot has to move all bricks in such a way that the span of the resulting field be the smallest. Our main result is the design of a deterministic finite automaton that accomplishes this task and subsequently stops, for every initially connected field, in time 
, where 
is the span of the initial field and 
is the number of bricks. We show that this complexity is optimal.
Kaito Fujii, Shinsaku Sakaue
2019-04-24
We propose a new concept named adaptive submodularity ratio to study the greedy policy for sequential decision making. While the greedy policy is known to perform well for a wide variety of adaptive stochastic optimization problems in practice, its theoretical properties have been analyzed only for a limited class of problems. We narrow the gap between theory and practice by using adaptive submodularity ratio, which enables us to prove approximation guarantees of the greedy policy for a substantially wider class of problems. Examples of newly analyzed problems include important applications such as adaptive influence maximization and adaptive feature selection. Our adaptive submodularity ratio also provides bounds of adaptivity gaps. Experiments confirm that the greedy policy performs well with the applications being considered compared to standard heuristics.
Mehul Kumar, Amit Kumar, C. Pandu Rangan
2019-04-24
Most optimization problems are notoriously hard. Considerable efforts must be spent in obtaining an optimal solution to certain instances that we encounter in the real world scenarios. Often it turns out that input instances get modified locally in some small ways due to changes in the application world. The natural question here is, given an optimal solution for an old instance 
, can we construct an optimal solution for the new instance 
, where is the instance with some local modifications. Reoptimization of NP-hard optimization problem precisely addresses this concern. It turns out that for some reoptimization versions of the NP-hard problems, we may only hope to obtain an approximate solution to a new instance. In this paper, we specifically address the reoptimization of path vertex cover problem. The objective in -
vertex cover problem is to compute a minimum subset 
of the vertices in a graph 
such that after removal of from there is no path with vertices in the graph. We show that when a constant number of vertices are inserted, reoptimizing unweighted - vertex cover problem admits a PTAS. For weighted 
- vertex cover problem, we show that when a constant number of vertices are inserted, the reoptimization algorithm achieves an approximation factor of 
, hence an improvement from known 
-approximation algorithm for the optimization version. We provide reoptimization algorithm for weighted - vertex cover problem 
on bounded degree graphs, which is also an NP-hard problem. Given a 
-approximation algorithm for - vertex cover problem on bounded degree graphs, we show that it can be reoptimized within an approximation factor of 
under constant number of vertex insertions.
Ran Duan, Ce Jin, Hongxun Wu
2019-04-24
In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time 
. Here 
is the number of vertices, and 
is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for 
-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for -matrix product. The previous best upper bound for APNP on weighted digraphs was 
[Duan, Gu, Zhang 2018]. We also show an 
time algorithm for APNP in undirected graphs which also reaches optimal within logarithmic factors.
Vincent Cohen-Addad, Marcin Pilipczuk, Michał Pilipczuk
2019-04-24
We consider the classic Facility Location problem on planar graphs (non-uniform, uncapacitated). Given an edge-weighted planar graph , a set of clients 
, a set of facilities 
, and opening costs 
, the goal is to find a subset 
of 
that minimizes 
. The Facility Location problem remains one of the most classic and fundamental optimization problem for which it is not known whether it admits a polynomial-time approximation scheme (PTAS) on planar graphs despite significant effort for obtaining one. We solve this open problem by giving an algorithm that for any 
, computes a solution of cost at most 
times the optimum in time 
.
Aditya Bhaskara, Aidao Chen, Aidan Perreault, Aravindan Vijayaraghavan
2018-11-29
Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in unsupervised learning and high-dimensional data analysis. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decompositions and learning mixtures of Gaussians, such guarantees have been hard to obtain for several other important problems in unsupervised learning. A core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries, that are given by low-degree polynomials of a few base underlying random variables. In this work, we address this challenge by obtaining high-confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind. We then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in unsupervised learning: 1. Robust subspace recovery, when the fraction 
of inliers in the d-dimensional subspace 
is at least 
for any constant integer 
. This contrasts with the known worst-case intractability when 
, and the previous smoothed analysis result which needed 
(Hardt and Moitra, 2013). 2. Learning overcomplete hidden markov models, where the size of the state space is any polynomial in the dimension of the observations. This gives the first polynomial time guarantees for learning overcomplete HMMs in a smoothed analysis model. 3. Higher order tensor decompositions, where we generalize the so-called FOOBI algorithm of Cardoso to find order-
rank-one tensors in a subspace. This allows us to obtain polynomially robust decomposition algorithms for 
'th order tensors with rank 
.
Brahim Chaourar
2019-03-29
Given a graph 
, a connected cut 
is the set of edges of E linking all vertices of U to all vertices of 
such that the induced subgraphs 
and 
are connected. Given a positive weight function 
defined on 
, the connected maximum cut problem (CMAX CUT) is to find a connected cut 
such that 
is maximum among all connected cuts. CMAX CUT is NP-hard even for planar graphs. In this paper, we prove that CMAX CUT is polynomial for graphs without 
as a minor. We deduce a quadratic time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.
Mehrdad Ghadiri, Richard Santiago, Bruce Shepherd
2019-04-19
While there are well-developed tools for maximizing a submodular function subject to a matroid constraint, there is much less work on the corresponding supermodular maximization problems. We develop new techniques for attacking these problems inspired by the continuous greedy method applied to the multi-linear extension of a submodular function. We first adapt the continuous greedy algorithm to work for general twice-continuously differentiable functions. The performance of the adapted algorithm depends on a new smoothness parameter. If 
is one-sided 
-smooth, then the approximation factor only depends on . We apply the new algorithm to a broad class of quadratic supermodular functions arising in diversity maximization. This captures metric diversity maximization (
) and negative-type diversity (
). We also develop new methods (inspired by swap rounding and approximate integer decomposition) for rounding quadratics over a matroid polytope. Together with the adapted continuous greedy this leads to a 
-approximation. This is the best asymptotic approximation known for this class of quadratics and the evidence suggests that it may be tight. We then consider general (non-quadratic) functions. We give a broad parameterized family of monotone functions which include submodular functions and the just-discussed supermodular family of discrete quadratics. Such set functions are called 
-meta-submodular. We develop local search algorithms with approximation factors that depend only on . We show that the -meta-submodular families include well-known function classes including meta-submodular functions (
), proportionally submodular (
), and diversity functions based on negative-type distances or Jensen-Shannon divergence (both 
) and (semi-)metric diversity functions.
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