Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm

Approximating the Regular Graphic TSP in near linear time

We present a randomized approximation algorithm for computing traveling salesperson tours in undirected regular graphs. Given an $n$-vertex, $k$-regular graph, the algorithm computes a tour of length at most $\left(1+\frac{7}{\ln k-O(1)}\right)n$, with high probability, in $O(nk \log k)$ time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS 2012) for the same problem, in terms of both approximation factor, and running time. The key ingredient of our algorithm is a technique that uses edge-coloring algorithms to sample a cycle cover with $O(n/\log k)$ cycles with high probability, in near linear time. Additionally, we also give a deterministic $\frac{3}{2}+O\left(\frac{1}{\sqrt{k}}\right)$ factor approximation algorithm running in time $O(nk)$.Comment: 12 page

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

Aplikasi Android Untuk Mencari Jalur Tercepat Pada Pengiriman Barang Dengan Algoritma Held-Karp

The total number of shipments to be sent from 2017 with an average total of 18 million shipments per month until 2018 reaches 24 million per month, while the process the shipping path is still as described and guessing, so couriers often wrong in estimating the fastest route or forgetting to order delivery based on previous couriers which cause the courier to have to turn around and every time the courier turns back then the shipping process becomes ineffective in terms of time, distance and cost. Based on the results of a 2018 study, Held-Karp produced a better route than Iterative Deepening Search in finding a shipping route. Therefore an Android-based application is created and uses the Google Maps API to determine the time, distance, and cost required at each point of delivery, as well as using the Held-Karp algorithm to search for the fastest path based on time, distance and cost. Based on functionality testing the application is estimated to run correctly and per the expected function, then in cyclometric complexity testing, the Held-karp algorithm is categorized in well-written and structured code and high testability because there is no v (G) above 10.The total number of shipments to be sent from 2017 with an average total of 18 million shipments per month until 2018 reaches 24 million per month, while the process the shipping path is still as described and guessing, so couriers often wrong in estimating the fastest route or forgetting to order delivery based on previous couriers which cause the courier to have to turn around and every time the courier turns back then the shipping process becomes ineffective in terms of time, distance and cost. Based on the results of a 2018 study, Held-Karp produced a better route than Iterative Deepening Search in finding a shipping route. Therefore an Android-based application is created and uses the Google Maps API to determine the time, distance, and cost required at each point of delivery, as well as using the Held-Karp algorithm to search for the fastest path based on time, distance and cost. Based on functionality testing the application is estimated to run correctly and per the expected function, then in cyclometric complexity testing, the Held-karp algorithm is categorized in well-written and structured code and high testability because there is no v (G) above 10