14 research outputs found
Large Cuts with Local Algorithms on Triangle-Free Graphs
We study the problem of finding large cuts in -regular triangle-free
graphs. In prior work, Shearer (1992) gives a randomised algorithm that finds a
cut of expected size , where is the number of
edges. We give a simpler algorithm that does much better: it finds a cut of
expected size . As a corollary, this shows that in
any -regular triangle-free graph there exists a cut of at least this size.
Our algorithm can be interpreted as a very efficient randomised distributed
algorithm: each node needs to produce only one random bit, and the algorithm
runs in one synchronous communication round. This work is also a case study of
applying computational techniques in the design of distributed algorithms: our
algorithm was designed by a computer program that searched for optimal
algorithms for small values of .Comment: 1+17 pages, 8 figure
Large Cuts with Local Algorithms on Triangle-Free Graphs
Let G be a d-regular triangle-free graph with in edges. We present an algorithm which finds a cut in G with at least (1/2 + 0.28125/root d)rn edges in expectation, improving upon Shearer's classic result. In particular, this implies that any d-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of G. Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed d, there exists a weighted neighbourhood graph N-d such that there is a one-to-one correspondence between heavy cuts of N-d and randomised local algorithms that find large cuts in any d-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in d-regular graphs: we can compute the optimal cut of N-d to attain a lower bound on the maximum cut size of any d-regular triangle-free graph.Peer reviewe
Kernels for the Max Cut problem
Tato práce shrnuje dosavadní výzkum problému Max Cut a představuje nový kernelizační algoritmus pro Simple Max Cut problém parametrizovaný velikostí minimálního vrcholového pokrytí vstupního grafu G omezující velikost jádra funkcí O(vc(G)^4), kde vc(G) označuje velikost minimálního vrcholového pokrytí grafu G.This thesis summarizes known results of the Max Cut problem and introduces a new kernelization algorithm for the Simple Max Cut problem parameterized by vertex cover number of the input graph G bounding the size of the kernel by O(vc(G)^4), where vc(G) denotes vertex cover number of G
MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems
Let be a set of vertices, a set of labels, and let
be an matrix of independent Bernoulli random
variables with success probability . A random instance
of the weighted random intersection graph model
is constructed by drawing an edge with weight
between any two vertices for which this weight is larger than 0. In this
paper we study the average case analysis of Weighted Max Cut, assuming the
input is a weighted random intersection graph, i.e. given
we wish to find a partition of into two
sets so that the total weight of the edges having one endpoint in each set is
maximized. We initially prove concentration of the weight of a maximum cut of
around its expected value, and then show that,
when the number of labels is much smaller than the number of vertices, a random
partition of the vertices achieves asymptotically optimal cut weight with high
probability (whp). Furthermore, in the case and constant average degree,
we show that whp, a majority type algorithm outputs a cut with weight that is
larger than the weight of a random cut by a multiplicative constant strictly
larger than 1. Then, we highlight a connection between the computational
problem of finding a weighted maximum cut in
and the problem of finding a 2-coloring with minimum discrepancy for a set
system with incidence matrix . We exploit this connection
by proposing a (weak) bipartization algorithm for the case that, when it terminates, its output can be used to find
a 2-coloring with minimum discrepancy in . Finally, we prove that, whp
this 2-coloring corresponds to a bipartition with maximum cut-weight in
.Comment: 18 page