13 research outputs found
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
Matching, matroid, and traveling salesman problem
研究成果の概要 (和文) : 巡回セールスマン問題 (TSP) は,おそらくもっとも有名な NP 困難な問題であり,TSPに対して提案された数々の手法は,離散最適化の分野全体の発展に大いに寄与してきた.特に近年,TSPに対する理論的なブレイクスルーといえる研究が数多く発表されている.本研究は,TSPへの応用を念頭に置き,離散最適化問題の効率的な解法の基礎をなす理論であるマッチング理論およびマトロイド理論の深化と拡大を行った.本研究で発表した 20篇の論文はすべて,最適化分野のトップジャーナル・トップカンファレンスを含む,定評のある査読付き国際論文誌に採録,または査読付き国際会議に採択されている.研究成果の概要 (英文) : The traveling salesman problem (TSP) is perhaps the most famous NP-hard problem, and has enhanced developments of many methods in the field of discrete optimization. In particular, TSP attracts recent intensive attention: several theoretical breakthrough papers have been published in this past decade.Our research has intended to be applied in theoretical improvement in solving TSP. Specifically, our research has achieved deepening and extending of matching theory and matroid theory, which form bases of efficient solutions to discrete optimization problems. All of our 20 papers has been accepted to reputable, international, peer-reviewed journals or conferences, including top journals and conferences in the field of optimization
Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours
We present a new approach for gluing tours over certain tight, 3-edge cuts.
Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles
in special graph classes and in proving bounds for 2-edge-connected subgraph
problem, but not much was known in this direction for gluing connected
multigraphs. We apply this approach to the traveling salesman problem (TSP) in
the case when the objective function of the subtour elimination relaxation is
minimized by a -cyclic point: ,
where the support graph is subcubic and each vertex is incident to at least one
edge with -value 1. Such points are sufficient to resolve TSP in general.
For these points, we construct a convex combination of tours in which we can
reduce the usage of edges with -value 1 from the of
Christofides algorithm to while keeping the
usage of edges with fractional -value the same as Christofides algorithm. A
direct consequence of this result is for the Uniform Cover Problem for TSP: In
the case when the objective function of the subtour elimination relaxation is
minimized by a -uniform point: , we
give a -approximation algorithm for TSP. For such points, this
lands us halfway between the approximation ratios of of
Christofides algorithm and implied by the famous "four-thirds
conjecture"
A Systematic Review of Approximability Results for Traveling Salesman Problems leveraging the TSP-T3CO Definition Scheme
The traveling salesman (or salesperson) problem, short TSP, is a problem of
strong interest to many researchers from mathematics, economics, and computer
science. Manifold TSP variants occur in nearly every scientific field and
application domain: engineering, physics, biology, life sciences, and
manufacturing just to name a few. Several thousand papers are published on
theoretical research or application-oriented results each year. This paper
provides the first systematic survey on the best currently known
approximability and inapproximability results for well-known TSP variants such
as the "standard" TSP, Path TSP, Bottleneck TSP, Maximum Scatter TSP,
Generalized TSP, Clustered TSP, Traveling Purchaser Problem, Profitable Tour
Problem, Quota TSP, Prize-Collecting TSP, Orienteering Problem, Time-dependent
TSP, TSP with Time Windows, and the Orienteering Problem with Time Windows. The
foundation of our survey is the definition scheme T3CO, which we propose as a
uniform, easy-to-use and extensible means for the formal and precise definition
of TSP variants. Applying T3CO to formally define the variant studied by a
paper reveals subtle differences within the same named variant and also brings
out the differences between the variants more clearly. We achieve the first
comprehensive, concise, and compact representation of approximability results
by using T3CO definitions. This makes it easier to understand the
approximability landscape and the assumptions under which certain results hold.
Open gaps become more evident and results can be compared more easily
Removing and adding edges for the traveling salesman problem
We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost.
For the TSP on graphic metrics (graph-TSP), we show that the approach gives a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted either to half-integral solutions to the Held-Karp relaxation or to a class of graphs that contains subcubic and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework also allows for generalizations in a natural way and leads to analogous results for the
s
,
t
-path traveling salesman problem on graphic metrics where the start and end vertices are prespecified.
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