507 research outputs found
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
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
Hybrid tractability of soft constraint problems
The constraint satisfaction problem (CSP) is a central generic problem in
computer science and artificial intelligence: it provides a common framework
for many theoretical problems as well as for many real-life applications. Soft
constraint problems are a generalisation of the CSP which allow the user to
model optimisation problems. Considerable effort has been made in identifying
properties which ensure tractability in such problems. In this work, we
initiate the study of hybrid tractability of soft constraint problems; that is,
properties which guarantee tractability of the given soft constraint problem,
but which do not depend only on the underlying structure of the instance (such
as being tree-structured) or only on the types of soft constraints in the
instance (such as submodularity). We present several novel hybrid classes of
soft constraint problems, which include a machine scheduling problem,
constraint problems of arbitrary arities with no overlapping nogoods, and the
SoftAllDiff constraint with arbitrary unary soft constraints. An important tool
in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page
Recommended from our members
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
On recognition algorithms and structure of graphs with restricted induced cycles
This is my PhD thesis which was defended in May 2021.
We call an induced cycle of length at least four a hole. The parity of a hole
is the parity of its length. Forbidding holes of certain types in a graph has
deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and
Thomas famously proved that a graph is perfect if and only if it does not
contain an odd hole or a complement of an odd hole. In 2002, Conforti,
Cornu\'{e}jols, Kapoor and Vu\v{s}kov\'{i}c provided a structural description
of the class of even-hole-free graphs. In Chapter 3, we provide a structural
description of all graphs that contain only holes of length for every
.
Analysis of how holes interact with graph structure has yielded detection
algorithms for holes of various lengths and parities. In 1991, Bienstock showed
it is NP-Hard to test whether a graph G has an even (or odd) hole containing a
specified vertex . In 2002, Conforti, Cornu\'{e}jols, Kapoor and
Vu\v{s}kov\'{i}c gave a polynomial-time algorithm to recognize even-hole-free
graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi and
Seymour provided a simpler and slightly faster algorithm to test whether a
graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour and Spirkl
provided a polynomial-time algorithm to test whether a graph contains an odd
hole. Later that year, Chudnovsky, Scott and Seymour strengthened this result
by providing a polynomial-time algorithm to test whether a graph contains an
odd hole of length at least for any fixed integer . In
Chapter 2, we provide a polynomial-time algorithm to test whether a graph
contains an even hole of length at least for any fixed integer .Comment: PhD Thesis, May 2021, Princeton University, Advisor: Paul Seymou
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