60 research outputs found
Parameterized TSP: Beating the Average
In the Travelling Salesman Problem (TSP), we are given a complete graph
together with an integer weighting on the edges of , and we are asked
to find a Hamilton cycle of of minimum weight. Let denote the
average weight of a Hamilton cycle of for the weighting . Vizing
(1973) asked whether there is a polynomial-time algorithm which always finds a
Hamilton cycle of weight at most . He answered this question in the
affirmative and subsequently Rublineckii (1973) and others described several
other TSP heuristics satisfying this property. In this paper, we prove a
considerable generalisation of Vizing's result: for each fixed , we give an
algorithm that decides whether, for any input edge weighting of ,
there is a Hamilton cycle of of weight at most (and constructs
such a cycle if it exists). For fixed, the running time of the algorithm is
polynomial in , where the degree of the polynomial does not depend on
(i.e., the generalised Vizing problem is fixed-parameter tractable with respect
to the parameter )
Hamilton cycles in sparse robustly expanding digraphs
The notion of robust expansion has played a central role in the solution of
several conjectures involving the packing of Hamilton cycles in graphs and
directed graphs. These and other results usually rely on the fact that every
robustly expanding (di)graph with suitably large minimum degree contains a
Hamilton cycle. Previous proofs of this require Szemer\'edi's Regularity Lemma
and so this fact can only be applied to dense, sufficiently large robust
expanders. We give a proof that does not use the Regularity Lemma and, indeed,
we can apply our result to suitable sparse robustly expanding digraphs.Comment: Accepted for publication in The Electronic Journal of Combinatoric
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
A domination algorithm for -instances of the travelling salesman problem
We present an approximation algorithm for -instances of the
travelling salesman problem which performs well with respect to combinatorial
dominance. More precisely, we give a polynomial-time algorithm which has
domination ratio . In other words, given a
-edge-weighting of the complete graph on vertices, our
algorithm outputs a Hamilton cycle of with the following property:
the proportion of Hamilton cycles of whose weight is smaller than that of
is at most . Our analysis is based on a martingale approach.
Previously, the best result in this direction was a polynomial-time algorithm
with domination ratio for arbitrary edge-weights. We also prove a
hardness result showing that, if the Exponential Time Hypothesis holds, there
exists a constant such that cannot be replaced by in the result above.Comment: 29 pages (final version to appear in Random Structures and
Algorithms
Partitions of combinatorial structures.
In this thesis we explore extremal, structural, and algorithmic problems involving the partitioning of combinatorial structures. We begin by considering problems from the theory of graph cuts. It is well known that every graph has a cut containing at least half its edges. We conjecture that (except for one example), given any two graphs on the same vertex set, we can partition the vertices so that at least half the edges of each graph go across the partition. We give a simple algorithm that comes close to proving this conjecture. We also prove, using probabilistic methods, that the conjecture holds for certain classes of graphs. We consider an analogue of the graph cut problem for posets and determine which graph cut results carry over to posets. We consider both extremal and algorithmic questions, and in particular, we show that the analogous maxcut problem for posets is polynomial-time solvable in contrast to the maxcut problem for graphs, which is NP-complete. Another partitioning problem we consider is that of obtaining a regular partition (in the sense of the Szemeredi Regularity Lemma) for posets, where the partition respects the order of the poset. We prove the existence of such order-preserving, regular partitions for both the comparability graph and the covering graph of a poset, and go on to derive further properties of such partitions. We give a new proof of an old result of Frankl and Furedi, which characterises all 3-uniform hypergraphs for which every set of 4 vertices spans exactly 0 or 2 edges. We use our new proof to derive a corresponding stability result. We also look at questions concerning an analogue of the graph linear extension problem for posets
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