113 research outputs found
On Approximating Multi-Criteria TSP
We present approximation algorithms for almost all variants of the
multi-criteria traveling salesman problem (TSP).
First, we devise randomized approximation algorithms for multi-criteria
maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP,
where the edge weights have to be symmetric, we devise an algorithm with an
approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge
weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps.
Our algorithms work for any fixed number k of objectives. Furthermore, we
present a deterministic algorithm for bi-criteria Max-STSP that achieves an
approximation ratio of 7/27.
Finally, we present a randomized approximation algorithm for the asymmetric
multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm
achieves a ratio of log n + eps.Comment: Preliminary version at STACS 2009. This paper is a revised full
version, where some proofs are simplifie
New Inapproximability Bounds for TSP
In this paper, we study the approximability of the metric Traveling Salesman
Problem (TSP) and prove new explicit inapproximability bounds for that problem.
The best up to now known hardness of approximation bounds were 185/184 for the
symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to
Papadimitriou and Vempala). We construct here two new bounded occurrence CSP
reductions which improve these bounds to 123/122 and 75/74, respectively. The
latter bound is the first improvement in more than a decade for the case of the
asymmetric TSP. One of our main tools, which may be of independent interest, is
a new construction of a bounded degree wheel amplifier used in the proof of our
results
The Maximum Traveling Salesman Problem with Submodular Rewards
In this paper, we look at the problem of finding the tour of maximum reward
on an undirected graph where the reward is a submodular function, that has a
curvature of , of the edges in the tour. This problem is known to be
NP-hard. We analyze two simple algorithms for finding an approximate solution.
Both algorithms require oracle calls to the submodular function. The
approximation factors are shown to be and
, respectively; so the second
method has better bounds for low values of . We also look at how these
algorithms perform for a directed graph and investigate a method to consider
edge costs in addition to rewards. The problem has direct applications in
monitoring an environment using autonomous mobile sensors where the sensing
reward depends on the path taken. We provide simulation results to empirically
evaluate the performance of the algorithms.Comment: Extended version of ACC 2013 submission (including p-system greedy
bound with curvature
On Approximating Restricted Cycle Covers
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L. The weight of a cycle cover of an edge-weighted
graph is the sum of the weights of its edges.
We come close to settling the complexity and approximability of computing
L-cycle covers. On the one hand, we show that for almost all L, computing
L-cycle covers of maximum weight in directed and undirected graphs is APX-hard
and NP-hard. Most of our hardness results hold even if the edge weights are
restricted to zero and one.
On the other hand, we show that the problem of computing L-cycle covers of
maximum weight can be approximated within a factor of 2 for undirected graphs
and within a factor of 8/3 in the case of directed graphs. This holds for
arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change
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