89 research outputs found
Improving Christofides' Algorithm for the s-t Path TSP
We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t
path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices
including two prespecified endpoints, the problem is to find a shortest
Hamiltonian path between the two endpoints; Hoogeveen showed that the natural
variant of Christofides' algorithm is a 5/3-approximation algorithm for this
problem, and this asymptotically tight bound in fact has been the best
approximation ratio known until now. We modify this algorithm so that it
chooses the initial spanning tree based on an optimal solution to the Held-Karp
relaxation rather than a minimum spanning tree; we prove this simple but
crucial modification leads to an improved approximation ratio, surpassing the
20-year-old barrier set by the natural Christofides' algorithm variant. Our
algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of
the path-variant Held-Karp relaxation. The techniques devised in this paper can
be applied to other optimization problems as well: these applications include
improved approximation algorithms and improved LP integrality gap upper bounds
for the prize-collecting s-t path problem and the unit-weight graphical metric
s-t path TSP.Comment: 31 pages, 5 figure
Hitting Sets when the Shallow Cell Complexity is Small
The hitting set problem is a well-known NP-hard optimization problem in
which, given a set of elements and a collection of subsets, the goal is to find
the smallest selection of elements, such that each subset contains at least one
element in the selection. Many geometric set systems enjoy improved
approximation ratios, which have recently been shown to be tight with respect
to the shallow cell complexity of the set system. The algorithms that exploit
the cell complexity, however, tend to be involved and computationally
intensive. This paper shows that a slightly improved asymptotic approximation
ratio for the hitting set problem can be attained using a much simpler
algorithm: solve the linear programming relaxation, take one initial random
sample from the set of elements with probabilities proportional to the
LP-solution, and, while there is an unhit set, take an additional sample from
it proportional to the LP-solution. Our algorithm is a simple generalization of
the elegant net-finder algorithm by Nabil Mustafa. To analyze this algorithm
for the hitting set problem, we generalize the classic Packing Lemma, and the
more recent Shallow Packing Lemma, to the setting of weighted epsilon-nets.Comment: Accepted by WAOA202
The submodular joint replenishment problem
The joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management, logistics, and maintenance scheduling. In this problem, there are multiple item types, each having a given time-dependent sequence of demands that need to be satisfied. In order to satisfy demand, orders of the item types must be placed in advance of the due dates for each demand. Every time an order of item types is placed, there is an associated joint setup cost depending on the subset of item types ordered. This ordering cost can be due to machine, transportation, or labor costs, for example. In addition, there is a cost to holding inventory for demand that has yet to be served. The overall goal is to minimize the total ordering costs plus inventory holding costs. In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types. For this general problem, we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands. Then we consider three special cases of submodular functions which we call the laminar, tree, and cardinality cases, each of which can model real world scenarios that previously have not been captured. For each of these cases, we provide a constant factor approximation algorithm. Specifically, we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach. For the tree and cardinality cases, we provide two different linear programming based approximation algorithms that provide guarantees of three and five, respectively.National Science Foundation (U.S.) (CAREER Grant CMMI-0846554)United States. Air Force Office of Scientific Research (Award FA9550-11-1-0150)SMA GrantSolomon Buchsbaum AT&T Research Fun
GILP: An Interactive Tool for Visualizing the Simplex Algorithm
The Simplex algorithm for solving linear programs-one of Computing in Science
& Engineering's top 10 most influential algorithms of the 20th century-is an
important topic in many algorithms courses. While the Simplex algorithm relies
on intuitive geometric ideas, the computationally-involved mechanics of the
algorithm can obfuscate a geometric understanding. In this paper, we present
gilp, an easy-to-use Simplex algorithm visualization tool designed to
explicitly connect the mechanical steps of the algorithm with their geometric
interpretation. We provide an extensive library with example visualizations,
and our tool allows an instructor to quickly produce custom interactive HTML
files for students to experiment with the algorithm (without requiring students
to install anything!). The tool can also be used for interactive assignments in
Jupyter notebooks, and has been incorporated into a forthcoming Data Science
and Decision Making interactive textbook. In this paper, we first describe how
the tool fits into the existing literature on algorithm visualizations: how it
was designed to facilitate student engagement and instructor adoption, and how
it substantially extends existing algorithm visualization tools for Simplex. We
then describe the development and usage of the tool, and report feedback from
its use in a course with roughly 100 students. Student feedback was
overwhelmingly positive, with students finding the tool easy to use: it
effectively helped them link the algebraic and geometrical views of the Simplex
algorithm and understand its nuances. Finally, gilp is open-source, includes an
extension to visualizing linear programming-based branch and bound, and is
readily amenable to further extensions.Comment: ACM SIGCSE 2023 Manuscript, 13 pages, 5 figure
Prize-Collecting TSP with a Budget Constraint
We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint
Approximation algorithms for facility location problems. In:
Abstract We present new approximation algorithms for several facility location problems. In each facility location problem that we study, there is a set of locations at which w e m a y build a facility such a s a w arehouse, where the cost of building at location i is fi; furthermore, there is a set of client locations such a s stores that require to be serviced by a facility, and if a client at location j is assigned to a facility at location i, a cost of cij is incurred. The objective i s t o determine a set of locations at which to open facilities so as to minimize the total facility and assignment costs. In the uncapacitated case, each facility can service an unlimited number of clients, whereas in the capacitated case, each facility can serve, for example, at most u clients. These models and a number of closely related ones have been studied extensively in the Operations Research literature. We shall consider the case in which the assignment costs are symmetric and satisfy the triangle inequality. For the uncapacitated facility location, we give a polynomial-time algorithm that nds a solution within a factor of 3.16 of the optimal. This is the rst constant performance guarantee known for this problem. We also present approximation algorithms with constant performance guarantees for a number of capacitated models as well as a generalization in which there is a 2-level hierarchy of facilities. Our results are based on the ltering and rounding technique of Lin & Vitter. We also give a randomized variant of this technique that can then be derandomized to yield improved performance guarantees
Improved Bounds on Relaxations of a Parallel Machine Scheduling Problem
We consider the problem of scheduling n jobs with release dates on m identical parallel machines to minimize the average completion time of the jobs. We prove that the ratio of the average completion time of the optimal nonpreemptive schedule to that of the optimal preemptive schedule is at most 7}{3}, improving a bound of (3- 1}{m}) due to Phillips, Stein and Wein. We then use our technique to give an improved bound on the quality of a linear programming relaxation of the problem considered by Hall, Schulz, Shmoys and Wein
- …