The Asymmetric Traveling Salesperson Path Problem (ATSPP) is one where, given an asymmetric metric space @math with specified vertices s and t, the goal is to find an s-t path of minimum length that passes through all the vertices in V. This problem is closely related to the Asymmetric TSP (ATSP), which seeks to find a tour (instead of an @math path) visiting all the nodes: for ATSP, a @math -approximation guarantee implies an @math -approximation for ATSPP. However, no such connection is known for the integrality gaps of the linear programming relaxations for these problems: the current-best approximation algorithm for ATSPP is @math , whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elimination LP) for ATSPP is @math . In this paper, we close this gap, and improve the current best bound on the integrality gap from @math to @math . The resulting algorithm uses the structure of narrow @math - @math cuts in the LP solution to construct a (random) tree spanning tree that can be cheaply augmented to contain an Eulerian @math - @math walk. We also build on a result of Oveis Gharan and Saberi and show a strong form of Goddyn's conjecture about thin spanning trees implies the integrality gap of the subtour elimination LP relaxation for ATSPP is bounded by a constant. Finally, we give a simpler family of instances showing the integrality gap of this LP is at least 2.
The first non-trivial approximation for ATSPP was an @math -approximation by Lam and Newman @cite_2 . This was improved to @math by Chekuri and P 'al @cite_6 , and the constant was further improved in @cite_1 . The paper @cite_1 also showed that ATSP and ATSPP had approximability within a constant factor of each other. All these results are combinatorial and do not bound integrality gap of ATSPP. A bound of @math on the integrality gap of ATSPP was given by Nagarajan and Ravi @cite_10 , and was improved to @math by Friggstad, Salavatipour and Svitkina @cite_3 . Note that there is still no result known that relates the integrality gaps of the ATSP and ATSPP problems in a black-box fashion.
@cite_1 @cite_3 @cite_6 @cite_2 @cite_10
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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. We consider the maximum independent set problem on graphs with maximum degree @math . We show that the integrality gap of the Lov 'asz @math -function based SDP is @math . This improves on the previous best result of @math , and almost matches the integrality gap of @math recently shown for stronger SDPs, namely those obtained using poly- @math levels of the @math semidefinite hierarchy. The improvement comes from an improved Ramsey-theoretic bound on the independence number of @math -free graphs for large values of @math . We also show how to obtain an algorithmic version of the above-mentioned @math -based integrality gap result, via a coloring algorithm of Johansson. The resulting approximation guarantee of @math matches the best unique-games-based hardness result up to lower-order poly- @math factors. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows.