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.
In contrast to TSP, no constant-factor approximation for its asymmetric version is known. The current best approximation for ATSP is the very recent result of @cite_21 , which gives an @math -approximation algorithm. It also upper-bounds the integrality gap of the asymmetric Held-Karp LP relaxation by the same factor. Previous algorithms guarantee a solution of cost within @math factor of optimum @cite_26 @cite_19 @cite_5 @cite_14 . The algorithm of @cite_26 is shown to upper-bound the Held-Karp integrality gap by @math in @cite_12 , and a different proof that bounds the integrality gap of a slightly weaker LP is obtained in @cite_27 . The best known lower bound on the Held-Karp integrality gap is essentially 2 @cite_2 , and tightening these bounds remains an important open problem. ATSP is NP-hard to approximate within @math @cite_28 .
@cite_14 @cite_26 @cite_28 @cite_21 @cite_19 @cite_27 @cite_2 @cite_5 @cite_12
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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 present a deterministic (1+√5 2)-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on @math 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 had 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+√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. In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal. The Held-Karp heuristic for the Traveling Salesman Problem (TSP) has in practice provided near-optimal lower bounds on the cost of solutions to the TSP. We analyze the structure of Held-Karp solutions in order to shed light on their quality. In the symmetric case with triangle inequality, we show that a class of instances has planar solutions. We also show that Held-Karp solutions have a certain monotonicity property. This leads to an alternate proof of a result of Wolsey, which shows that the value of Held-Karp heuristic is always at least 2 3 OPT, where OPT is the cost of the optimum TSP tour. Additionally, we show that the value of the Held-Karp heuristic is equal to that of the linear relaxation of the biconnected-graph problem when edge costs are non-negative. In the asymmetric case with triangle inequality, we show that there are many equivalent definitions of the Held-Karp heuristic, which include finding optimally weighted 1-arborescences, 1-antiarborescences, asymmetric 1-trees, and assignment problems. We prove that monotonicity holds in the asymmetric case as well. These theorems imply that the value of the Held-Karp heuristic is no less than OPT and no less than the value of the Balas-Christofides heuristic for the asymmetric TSP. For the 1,2-TSP, we show that the Held-Karp heuristic cannot do any better than 9 10 OPT, even as the number of nodes tends to infinity. Portions of this thesis are joint work with David Shmoys.