Approaching $\frac{3}{2}$ for the $s$-$t$-path TSP
نویسندگان
چکیده
We show that there is a polynomial-time algorithm with approximation guarantee 3 2 + ε for the s-t-path TSP, for any fixed ε > 0. It is well known that Wolsey’s analysis of Christofides’ algorithm also works for the s-t-path TSP with its natural LP relaxation except for the narrow cuts (in which the LP solution has value less than two). A fixed optimum tour has either a single edge in a narrow cut (then call the edge and the cut lonely) or at least three (then call the cut busy). Our algorithm “guesses” (by dynamic programming) lonely cuts and edges. Then we partition the instance into smaller instances and strengthen the LP, requiring value at least three for busy cuts. By setting up a k-stage recursive dynamic program, we can compute a spanning tree (V, S) and an LP solution y such that ( 2 +O(2))y is in the T -join polyhedron, where T is the set of vertices whose degree in S has the wrong parity.
منابع مشابه
An LP-based 3/2-approximation algorithm for the graphic s-t path TSP
We design a new LP-based algorithm for the graphic s-t path Traveling Salesman Problem (TSP), which achieves the best approximation factor of 1.5. The algorithm is based on the idea of narrow cuts due to An, Kleinberg, and Shmoys. It partly answers an open question of Sebő.
متن کاملApproximation Algorithms for Path TSP, ATSP, and TAP via Relaxations
Linear programming (LP) relaxations provide a powerful technique to design approximation algorithms for combinatorial optimization problems. In the first part of the thesis, we study the metric s-t path Traveling Salesman Problem (TSP) via LP relaxations. We first consider the s-t path graph-TSP, a critical special case of the metric s-t path TSP. We design a new simple LP-based algorithm for t...
متن کاملShorter Tours by Nicer Ears: 7/5-approximation for graphic TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs
We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a generalization, the connected-T -join problem, we obtain the first nontrivial approximation algorithm, with ratio 3/2. This contains the graphic s-t-path-TSP as ...
متن کاملThe Borsuk-Ulam Theorem
For simplicity, we adopt the following rules: a, b, x, y, z, X, Y , Z denote sets, n denotes a natural number, i denotes an integer, r, r1, r2, r3, s denote real numbers, c, c1, c2 denote complex numbers, and p denotes a point of En T. Let us observe that every element of IQ is irrational. Next we state a number of propositions: (1) If 0 ≤ r and 0 ≤ s and r2 = s2, then r = s. (2) If frac r ≥ fr...
متن کاملBeating the integrality ratio for s-t-tours in graphs
Among various variants of the traveling salesman problem, the s-t-path graph TSP has the special feature that we know the exact integrality ratio, 3 2 , and an approximation algorithm matching this ratio. In this paper, we go below this threshold: we devise a polynomial-time algorithm for the s-t-path graph TSP with approximation ratio 1.497. Our algorithm can be viewed as a refinement of the 3...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1707.03992 شماره
صفحات -
تاریخ انتشار 2017