72 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
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
Computing Near-Optimal Solutions to Combinatorial Optimization Problems
. In the past few years, there has been significant progress in our understanding of the extent to which near-optimal solutions can be efficiently computed for NP-hard combinatorial optimization problems. This paper surveys these recent developments, while concentrating on the advances made in the design and analysis of approximation algorithms, and in particular, on those results that rely on linear programming and its generalizations. In the past few years, there have been major advances in our understanding of performance guarantees for approximation algorithms for NP-hard combinatorial optimization problems. Most notably, after twenty-five years of essentially no progress, a new technique has been developed for proving that certain approximation algorithms are unlikely to exist. Partially in response to this development, there have also been significant recent advances in the design and analysis of approximation algorithms. In this survey, we will outline a few of the areas in whi..
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