113 research outputs found
Asymmetric Traveling Salesman Path and Directed Latency Problems
We study integrality gaps and approximability of two closely related problems
on directed graphs. Given a set V of n nodes in an underlying asymmetric metric
and two specified nodes s and t, both problems ask to find an s-t path visiting
all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the
objective is to minimize the total cost of this path. In the directed latency
problem, the objective is to minimize the sum of distances on this path from s
to each node. Both of these problems are NP-hard. The best known approximation
algorithms for ATSPP had ratio O(log n) until the very recent result that
improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the
integrality gap of its linear programming relaxation has been known. For
directed latency, the best previously known approximation algorithm has a
guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new
algorithm for the ATSPP problem that has an approximation ratio of O(log n),
but whose analysis also bounds the integrality gap of the standard LP
relaxation of ATSPP by the same factor. This solves an open problem posed by
Chekuri and Pal [2007]. We then pursue a deeper study of this linear program
and its variations, which leads to an algorithm for the k-person ATSPP (where k
s-t paths of minimum total length are sought) and an O(log n)-approximation for
the directed latency problem
How to Walk Your Dog in the Mountains with No Magic Leash
We describe a -approximation algorithm for computing the
homotopic \Frechet distance between two polygonal curves that lie on the
boundary of a triangulated topological disk. Prior to this work, algorithms
were known only for curves on the Euclidean plane with polygonal obstacles.
A key technical ingredient in our analysis is a -approximation
algorithm for computing the minimum height of a homotopy between two curves. No
algorithms were previously known for approximating this parameter.
Surprisingly, it is not even known if computing either the homotopic \Frechet
distance, or the minimum height of a homotopy, is in NP
Approximation Algorithms for Generalized Path Scheduling
Scheduling problems where the machines can be represented as the edges of a network and each job needs to be processed by a sequence of machines that form a path in this network have been the subject of many research articles (e.g. flow shop is the special case where the network as well as the sequence of machines for each job is a simple path). In this paper we consider one such problem, called Generalized Path Scheduling (GPS) problem, which can be defined as follows. Given a set of non-preemptive jobs J and identical machines M ( |J| = n and |M| = m ). The machines are ordered on a path. Each job j = {P_j = {l_j, r_j}, p_j} is defined by its processing time p_j and a sub-path P_j from machine with index l_j to r_j (l_j, r_j ? M, and l_j ? r_j) specifying the order of machines it must go through. We assume each machine has a queue of infinite size where jobs can sit in the queue to resolve conflicts. Two objective functions, makespan and total completion time, are considered. Machines can be identical or unrelated. In the latter case, this problem generalizes the classical Flow shop problem (in which all jobs have to go through all machines from 1 to m in that order).
Generalized Path Scheduling has been studied (e.g. see [Ronald Koch et al., 2009; Zachary Friggstad et al., 2019]). In this paper, we present several improved approximation algorithms for both objectives. For the case of number of machines being sub-logarithmic in the number of jobs we present a PTAS for both makespan and total completion time. The PTAS holds even on unrelated machines setting and therefore, generalizes the result of Hall [Leslie A. Hall, 1998] for the classic problem of Flow shop. For the case of identical machines, we present an O((log m)/(log log m))-approximation algorithms for both objectives, which improve the previous best result of [Zachary Friggstad et al., 2019]. We also show that the GPS problem is NP-complete for both makespan and total completion time objectives
Approximating Connected Facility Location with Lower and Upper Bounds via LP Rounding
We consider a lower- and upper-bounded generalization of the classical facility location problem, where each facility has a capacity (upper bound) that limits the number of clients it can serve and a lower bound on the number of clients it must serve if it is opened. We develop an LP rounding framework that exploits a Voronoi diagram-based clustering approach to derive the first bicriteria constant approximation algorithm for this problem with non-uniform lower bounds and uniform upper bounds. This naturally leads to the the first LP-based approximation algorithm for the lower bounded facility location problem (with non-uniform lower bounds).
We also demonstrate the versatility of our framework by extending this and presenting the first constant approximation algorithm for some connected variant of the problems in which the facilities are required to be connected as well
Approximation Algorithms for Capacitated k-Travelling Repairmen Problems
We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known.
We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem
Approximation Schemes for Min-Sum k-Clustering
We consider the Min-Sum k-Clustering (k-MSC) problem. Given a set of points in a metric which is represented by an edge-weighted graph G = (V, E) and a parameter k, the goal is to partition the points V into k clusters such that the sum of distances between all pairs of the points within the same cluster is minimized.
The k-MSC problem is known to be APX-hard on general metrics. The best known approximation algorithms for the problem obtained by Behsaz, Friggstad, Salavatipour and Sivakumar [Algorithmica 2019] achieve an approximation ratio of O(log |V|) in polynomial time for general metrics and an approximation ratio 2+? in quasi-polynomial time for metrics with bounded doubling dimension. No approximation schemes for k-MSC (when k is part of the input) is known for any non-trivial metrics prior to our work. In fact, most of the previous works rely on the simple fact that there is a 2-approximate reduction from k-MSC to the balanced k-median problem and design approximation algorithms for the latter to obtain an approximation for k-MSC.
In this paper, we obtain the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem on metrics induced by graphs of bounded treewidth, graphs of bounded highway dimension, graphs of bounded doubling dimensions (including fixed dimensional Euclidean metrics), and planar and minor-free graphs. We bypass the barrier of 2 for k-MSC by introducing a new clustering problem, which we call min-hub clustering, which is a generalization of balanced k-median and is a trade off between center-based clustering problems (such as balanced k-median) and pair-wise clustering (such as Min-Sum k-clustering). We then show how one can find approximation schemes for Min-hub clustering on certain classes of metrics
Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means
We investigate the complexity of solving stable or perturbation-resilient
instances of k-Means and k-Median clustering in fixed dimension Euclidean
metrics (or more generally doubling metrics). The notion of stable or
perturbation resilient instances was introduced by Bilu and Linial [2010] and
Awasthi et al. [2012]. In our context we say a k-Means instance is
\alpha-stable if there is a unique OPT solution which remains unchanged if
distances are (non-uniformly) stretched by a factor of at most \alpha. Stable
clustering instances have been studied to explain why heuristics such as
Lloyd's algorithm perform well in practice. In this work we show that for any
fixed \epsilon>0, (1+\epsilon)-stable instances of k-Means in doubling metrics
can be solved in polynomial time. More precisely we show a natural multiswap
local search algorithm in fact finds the OPT solution for (1+\epsilon)-stable
instances of k-Means and k-Median in a polynomial number of iterations. We
complement this result by showing that under a plausible PCP hypothesis this is
essentially tight: that when the dimension d is part of the input, there is a
fixed \epsilon_0>0 s.t. there is not even a PTAS for (1+\epsilon_0)-stable
k-Means in R^d unless NP=RP. To do this, we consider a robust property of CSPs;
call an instance stable if there is a unique optimum solution x^* and for any
other solution x', the number of unsatisfied clauses is proportional to the
Hamming distance between x^* and x'. Dinur et al. have already shown stable
QSAT is hard to approximate for some constant Q, our hypothesis is simply that
stable QSAT with bounded variable occurrence is also hard. Given this
hypothesis, we consider "stability-preserving" reductions to prove our hardness
for stable k-Means. Such reductions seem to be more fragile than standard
L-reductions and may be of further use to demonstrate other stable optimization
problems are hard.Comment: 29 page
Approximations for Throughput Maximization
In this paper we study the classical problem of throughput maximization. In
this problem we have a collection of jobs, each having a release time
, deadline , and processing time . They have to be scheduled
non-preemptively on identical parallel machines. The goal is to find a
schedule which maximizes the number of jobs scheduled entirely in their
window. This problem has been studied extensively (even for the
case of ). Several special cases of the problem remain open. Bar-Noy et
al. [STOC1999] presented an algorithm with ratio for
machines, which approaches as increases. For ,
Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation
with ratio (for any ). Recently
Im-Li-Moseley [IPCO2017] presented an algorithm with ratio
for some absolute constant for any
fixed . They also presented an algorithm with ratio for general which approaches 1 as grows. The
approximability of the problem for remains a major open question. Even
for the case of and distinct processing times the problem is
open (Sgall [ESA2012]). In this paper we study the case of and show
that if there are distinct processing times, i.e. 's come from a set
of size , then there is a -approximation that runs in time
, where is the largest deadline.
Therefore, for constant and constant this yields a PTAS. Our algorithm
is based on proving structural properties for a near optimum solution that
allows one to use a dynamic programming with pruning
- …