79 research outputs found
Improving on Best-of-Many-Christofides for -tours
The -tour problem is a natural generalization of TSP and Path TSP. Given a
graph , edge cost , and an even
cardinality set , we want to compute a minimum-cost -join
connecting all vertices of (and possibly containing parallel edges).
In this paper we give an -approximation for the -tour
problem and show that the integrality ratio of the standard LP relaxation is at
most . Despite much progress for the special case Path TSP, for
general -tours this is the first improvement on Seb\H{o}'s analysis of the
Best-of-Many-Christofides algorithm (Seb\H{o} [2013])
Approximation Algorithms for Traveling Salesman Problems
The traveling salesman problem is the probably most famous problem in combinatorial optimization. Given a graph G and nonnegative edge costs, we want to find a closed walk in G that visits every vertex at least once and has minimum cost. We consider both the symmetric traveling salesman problem (TSP) where G is an undirected graph and the asymmetric traveling salesman problem (ATSP) where G is a directed graph. We also investigate the unit-weight special cases and the more general path versions, where we do not require the walk to be closed, but to start and end in prescribed vertices s and t. In this thesis we give improved approximation algorithms and better upper bounds on the integrality ratio of the classical linear programming relaxations for several of these traveling salesman problems. For this we use techniques arising from various parts of combinatorial optimization such as linear programming, network flows, ear-decompositions, matroids, and T-joins. Our results include a (22 + &epsilon)-approximation algorithm for ATSP (for any &epsilon > 0), the first constant upper bound on the integrality ratio for s-t-path ATSP, a new upper bound on the integrality ratio for s-t-path TSP, and a black-box reduction from s-t-path TSP to TSP
Layers and Matroids for the Traveling Salesman's Paths
Gottschalk and Vygen proved that every solution of the subtour elimination
linear program for traveling salesman paths is a convex combination of more and
more restrictive "generalized Gao-trees". We give a short proof of this fact,
as a layered convex combination of bases of a sequence of increasingly
restrictive matroids. A strongly polynomial, combinatorial algorithm follows
for finding this convex combination, which is a new tool offering polyhedral
insight, already instrumental in recent results for the path TSP
A -Approximation Algorithm for Weighted Connectivity Augmentation
Connectivity augmentation problems are among the most elementary questions in
Network Design. Many of these problems admit natural -approximation
algorithms, often through various classic techniques, whereas it remains open
whether approximation factors below can be achieved. One of the most basic
examples thereof is the Weighted Connectivity Augmentation Problem (WCAP). In
WCAP, one is given an undirected graph together with a set of additional
weighted candidate edges, and the task is to find a cheapest set of candidate
edges whose addition to the graph increases its edge-connectivity. We present a
-approximation algorithm for WCAP, showing for the first
time that factors below are achievable.
On a high level, we design a well-chosen local search algorithm, inspired by
recent advances for Weighted Tree Augmentation. To measure progress, we
consider a directed weakening of WCAP and show that it has highly structured
planar solutions. Interpreting a solution of the original problem as one of
this directed weakening allows us to describe local exchange steps in a clean
and algorithmically amenable way. Leveraging these insights, we show that we
can efficiently search for good exchange steps within a component class for
link sets that is closely related to bounded treewidth subgraphs of circle
graphs. Moreover, we prove that an optimum solution can be decomposed into
smaller components, at least one of which leads to a good local search step as
long as we did not yet achieve the claimed approximation guarantee
Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches
We consider the Connectivity Augmentation Problem (CAP), a classical problem
in the area of Survivable Network Design. It is about increasing the
edge-connectivity of a graph by one unit in the cheapest possible way. More
precisely, given a -edge-connected graph and a set of extra edges,
the task is to find a minimum cardinality subset of extra edges whose addition
to makes the graph -edge-connected. If is odd, the problem is
known to reduce to the Tree Augmentation Problem (TAP) -- i.e., is a
spanning tree -- for which significant progress has been achieved recently,
leading to approximation factors below (the currently best factor is
). However, advances on TAP did not carry over to CAP so far. Indeed,
only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain
the first approximation factor below for CAP by presenting a
-approximation algorithm based on a method that is disjoint from recent
advances for TAP.
We first bridge the gap between TAP and CAP, by presenting techniques that
allow for leveraging insights and methods from TAP to approach CAP. We then
introduce a new way to get approximation factors below , based on a new
analysis technique. Through these ingredients, we obtain a
-approximation algorithm for CAP, and therefore also TAP. This leads to
the currently best approximation result for both problems in a unified way, by
significantly improving on the above-mentioned -approximation for CAP and
also the previously best approximation factor of for TAP by Grandoni,
Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from
recent TAP advances is that our approach can deal with the weighted setting
when the ratio between the largest to smallest cost on extra links is bounded,
in which case we obtain approximation factors below
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