1,203 research outputs found
The Salesman's Improved Tours for Fundamental Classes
Finding the exact integrality gap for the LP relaxation of the
metric Travelling Salesman Problem (TSP) has been an open problem for over
thirty years, with little progress made. It is known that , and a famous conjecture states . For this problem,
essentially two "fundamental" classes of instances have been proposed. This
fundamental property means that in order to show that the integrality gap is at
most for all instances of metric TSP, it is sufficient to show it only
for the instances in the fundamental class. However, despite the importance and
the simplicity of such classes, no apparent effort has been deployed for
improving the integrality gap bounds for them. In this paper we take a natural
first step in this endeavour, and consider the -integer points of one such
class. We successfully improve the upper bound for the integrality gap from
to for a superclass of these points, as well as prove a lower
bound of for the superclass. Our methods involve innovative applications
of tools from combinatorial optimization which have the potential to be more
broadly applied
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms
An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both.
Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results
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