31 research outputs found
Approximating the Minimum Equivalent Digraph
The MEG (minimum equivalent graph) problem is, given a directed graph, to
find a small subset of the edges that maintains all reachability relations
between nodes. The problem is NP-hard. This paper gives an approximation
algorithm with performance guarantee of pi^2/6 ~ 1.64. The algorithm and its
analysis are based on the simple idea of contracting long cycles. (This result
is strengthened slightly in ``On strongly connected digraphs with bounded cycle
length'' (1996).) The analysis applies directly to 2-Exchange, a simple ``local
improvement'' algorithm, showing that its performance guarantee is 1.75.Comment: conference version in ACM-SIAM Symposium on Discrete Algorithms
(1994
Shorter tours and longer detours: Uniform covers and a bit beyond
Motivated by the well known four-thirds conjecture for the traveling salesman
problem (TSP), we study the problem of {\em uniform covers}. A graph
has an -uniform cover for TSP (2EC, respectively) if the everywhere
vector (i.e. ) dominates a convex combination of
incidence vectors of tours (2-edge-connected spanning multigraphs,
respectively). The polyhedral analysis of Christofides' algorithm directly
implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP.
Seb\H{o} asked if such graphs have -uniform covers for TSP for
some . Indeed, the four-thirds conjecture implies that such
graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform
covers for TSP. We also study uniform covers for 2EC and show that the
everywhere 15/17 vector can be efficiently written as a convex combination of
2-edge-connected spanning multigraphs.
For a weighted, 3-edge-connected, cubic graph, our results show that if the
everywhere 2/3 vector is an optimal solution for the subtour linear programming
relaxation, then a tour with weight at most 27/19 times that of an optimal tour
can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall
into this category. In this special case, we can apply our tools to obtain an
even better approximation guarantee.
To extend our approach to input graphs that are 2-edge-connected, we present
a procedure to decompose an optimal solution for the subtour relaxation for TSP
into spanning, connected multigraphs that cover each 2-edge cut an even number
of times. Using this decomposition, we obtain a 17/12-approximation algorithm
for minimum weight 2-edge-connected spanning subgraphs on subcubic,
node-weighted graphs