208 research outputs found
Optimal Vertex Cover for the Small-World Hanoi Networks
The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with
an exact renormalization group and parallel-tempering Monte Carlo simulations.
The grand canonical partition function of the equivalent hard-core repulsive
lattice-gas problem is recast first as an Ising-like canonical partition
function, which allows for a closed set of renormalization group equations. The
flow of these equations is analyzed for the limit of infinite chemical
potential, at which the vertex-cover problem is attained. The relevant fixed
point and its neighborhood are analyzed, and non-trivial results are obtained
both, for the coverage as well as for the ground state entropy density, which
indicates the complex structure of the solution space. Using special
hierarchy-dependent operators in the renormalization group and Monte-Carlo
simulations, structural details of optimal configurations are revealed. These
studies indicate that the optimal coverages (or packings) are not related by a
simple symmetry. Using a clustering analysis of the solutions obtained in the
Monte Carlo simulations, a complex solution space structure is revealed for
each system size. Nevertheless, in the thermodynamic limit, the solution
landscape is dominated by one huge set of very similar solutions.Comment: RevTex, 24 pages; many corrections in text and figures; final
version; for related information, see
http://www.physics.emory.edu/faculty/boettcher
A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection
Given an undirected graph with edge costs and node weights, the minimum
bisection problem asks for a partition of the nodes into two parts of equal
weight such that the sum of edge costs between the parts is minimized. We give
a polynomial time bicriteria approximation scheme for bisection on planar
graphs.
Specifically, let be the total weight of all nodes in a planar graph .
For any constant , our algorithm outputs a bipartition of the
nodes such that each part weighs at most and the total cost
of edges crossing the partition is at most times the total
cost of the optimal bisection. The previously best known approximation for
planar minimum bisection, even with unit node weights, was . Our
algorithm actually solves a more general problem where the input may include a
target weight for the smaller side of the bipartition.Comment: To appear in STOC 201
On packing dijoins in digraphs and weighted digraphs
In this paper, we make some progress in addressing Woodall's Conjecture, and
the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and
weighted digraphs. Let be a digraph, and let . Suppose every dicut has weight at least , for some integer . Let , where each is
the integer in equal to
mod . In this paper, we prove the following results, amongst others: (1)
If , then can be partitioned into a dijoin and a
-dijoin. (2) If , then there is an
equitable -weighted packing of dijoins of size . (3) If
, then there is a -weighted packing of dijoins of size
. (4) If , , and , then can be
partitioned into three dijoins.
Each result is best possible: (1) and (4) do not hold for general , (2)
does not hold for even if , and (3) does not hold
for . The results are rendered possible by a \emph{Decompose,
Lift, and Reduce procedure}, which turns into a set of
\emph{sink-regular weighted -bipartite digraphs}, each of which
is a weighted digraph where every vertex is a sink of weighted degree or
a source of weighted degree , and every dicut has weight at least
. Our results give rise to a number of approaches for resolving Woodall's
Conjecture, fixing the refuted Edmonds-Giles Conjecture, and the
Conjecture for the clutter of minimal dijoins. They also show an intriguing
connection to Barnette's Conjecture.Comment: 71 page
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Discrete Geometry
The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (JoÌzsef Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) âą Forbidden patterns. (JaÌnos Pach) âą Projected polytopes, Gale diagrams, and polyhedral surfaces. (GuÌnter M. Ziegler) âą What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by JesuÌs De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (JuÌrgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants
Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization
International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National dâArts et MĂ©tiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry
Distributed Edge Packing
In this work we study a graph problem called edge packing in a distributed setting. An edge packing p is a function that associates a packing weight p(e) with each edge e of a graph such that the sum of the weights of the edges incident to each node is at most one. The task is to maximise the total weight of p over all edges. We are interested in approximating a maximum edge packing and in finding maximal edge packings, that is, edge packings such that the weight of no edge can be increased.
We use the model of distributed computing known as the LOCAL model. A communication network is modelled as a graph, where nodes correspond to computers and edges correspond to direct communication links. All nodes start at the same time and they run the same algorithm. Computation proceeds in synchronous communication rounds, during each of which each node can send a message through each of its communication links, receive a message from each of its communication links, and then do unbounded local computation. When a node terminates the algorithm, it must produce a local output â in this case a packing weight for each incident edge. The local outputs of the nodes must together form a feasible global solution.
The running time of an algorithm is the number of steps it takes until all nodes have terminated and announced their outputs. In a typical distributed algorithm, the running time of an algorithm is a function of n, the size of the communication graph, and â, the maximum degree of the communication graph. In this work we are interested in deterministic algorithms that have a running time that is a function of â, but not of n.
In this work we will review an O(log â)-time constant-approximation algorithm for maximum edge packing, and an O(â)-time algorithm for maximal edge packing. Maximal edge packing is an example of a problem where the best known algorithm has a running time that is linear-in-â. Other such problems include maximal matching and (â + 1)-colouring. However, few matching lower bounds exist for these problems: by prior work it is known that finding a maximal edge packing requires time Ω(log â), leaving an exponential gap between the best known lower and upper bounds. Recently Hirvonen and Suomela (PODC 2012) showed a linear-in-â lower bound for maximal matching. This lower bound, however, applies only in weaker, anonymous models of computation. In this work we show a linear-in-â lower bound for maximal edge packing. It applies also in the stronger port numbering model with orientation.
Recently Göös et al. (PODC 2012) showed that for a large class of optimisation problems, the port numbering with orientation model is as powerful as a stronger, so called unique identifier model. An open question is if this result can applied to extend our lower bound to the unique identifier model
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