18,971 research outputs found
On Approximating Restricted Cycle Covers
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L. The weight of a cycle cover of an edge-weighted
graph is the sum of the weights of its edges.
We come close to settling the complexity and approximability of computing
L-cycle covers. On the one hand, we show that for almost all L, computing
L-cycle covers of maximum weight in directed and undirected graphs is APX-hard
and NP-hard. Most of our hardness results hold even if the edge weights are
restricted to zero and one.
On the other hand, we show that the problem of computing L-cycle covers of
maximum weight can be approximated within a factor of 2 for undirected graphs
and within a factor of 8/3 in the case of directed graphs. This holds for
arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change
Deterministic Graph Exploration with Advice
We consider the task of graph exploration. An -node graph has unlabeled
nodes, and all ports at any node of degree are arbitrarily numbered
. A mobile agent has to visit all nodes and stop. The exploration
time is the number of edge traversals. We consider the problem of how much
knowledge the agent has to have a priori, in order to explore the graph in a
given time, using a deterministic algorithm. This a priori information (advice)
is provided to the agent by an oracle, in the form of a binary string, whose
length is called the size of advice. We consider two types of oracles. The
instance oracle knows the entire instance of the exploration problem, i.e., the
port-numbered map of the graph and the starting node of the agent in this map.
The map oracle knows the port-numbered map of the graph but does not know the
starting node of the agent.
We first consider exploration in polynomial time, and determine the exact
minimum size of advice to achieve it. This size is ,
for both types of oracles.
When advice is large, there are two natural time thresholds:
for a map oracle, and for an instance oracle, that can be achieved
with sufficiently large advice. We show that, with a map oracle, time
cannot be improved in general, regardless of the size of advice.
We also show that the smallest size of advice to achieve this time is larger
than , for any .
For an instance oracle, advice of size is enough to achieve time
. We show that, with any advice of size , the time of
exploration must be at least , for any , and with any
advice of size , the time must be .
We also investigate minimum advice sufficient for fast exploration of
hamiltonian graphs
Preprocessing under uncertainty
In this work we study preprocessing for tractable problems when part of the
input is unknown or uncertain. This comes up naturally if, e.g., the load of
some machines or the congestion of some roads is not known far enough in
advance, or if we have to regularly solve a problem over instances that are
largely similar, e.g., daily airport scheduling with few charter flights.
Unlike robust optimization, which also studies settings like this, our goal
lies not in computing solutions that are (approximately) good for every
instantiation. Rather, we seek to preprocess the known parts of the input, to
speed up finding an optimal solution once the missing data is known.
We present efficient algorithms that given an instance with partially
uncertain input generate an instance of size polynomial in the amount of
uncertain data that is equivalent for every instantiation of the unknown part.
Concretely, we obtain such algorithms for Minimum Spanning Tree, Minimum Weight
Matroid Basis, and Maximum Cardinality Bipartite Maxing, where respectively the
weight of edges, weight of elements, and the availability of vertices is
unknown for part of the input. Furthermore, we show that there are tractable
problems, such as Small Connected Vertex Cover, for which one cannot hope to
obtain similar results.Comment: 18 page
Directed paths with few or many colors in colored directed graphs
Given a graph and a coloring of , not necessarily a proper coloring of either the arcs or the vertices of , we consider the complexity of finding a path of from a given vertex to another given vertex with as few different colors as possible, and of finding one with as many different colors as possible. We show that the first problem is polynomial-time solvable, and that the second problem is NP-hard. \u
Minimum cycle and homology bases of surface embedded graphs
We study the problems of finding a minimum cycle basis (a minimum weight set
of cycles that form a basis for the cycle space) and a minimum homology basis
(a minimum weight set of cycles that generates the -dimensional
()-homology classes) of an undirected graph embedded on a
surface. The problems are closely related, because the minimum cycle basis of a
graph contains its minimum homology basis, and the minimum homology basis of
the -skeleton of any graph is exactly its minimum cycle basis.
For the minimum cycle basis problem, we give a deterministic
-time algorithm for graphs embedded on an orientable
surface of genus . The best known existing algorithms for surface embedded
graphs are those for general graphs: an time Monte Carlo
algorithm and a deterministic time algorithm. For the
minimum homology basis problem, we give a deterministic -time algorithm for graphs embedded on an orientable or non-orientable
surface of genus with boundary components, assuming shortest paths are
unique, improving on existing algorithms for many values of and . The
assumption of unique shortest paths can be avoided with high probability using
randomization or deterministically by increasing the running time of the
homology basis algorithm by a factor of .Comment: A preliminary version of this work was presented at the 32nd Annual
International Symposium on Computational Geometr
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