52,115 research outputs found
The hardness of routing two pairs on one face
We prove the NP-completeness of the integer multiflow problem in planar
graphs, with the following restrictions: there are only two demand edges, both
lying on the infinite face of the routing graph. This was one of the open
challenges concerning disjoint paths, explicitly asked by M\"uller. It also
strengthens Schw\"arzler's recent proof of one of the open problems of
Schrijver's book, about the complexity of the edge-disjoint paths problem with
terminals on the outer boundary of a planar graph. We also give a directed
acyclic reduction. This proves that the arc-disjoint paths problem is
NP-complete in directed acyclic graphs, even with only two demand arcs
On disjoint paths in acyclic planar graphs
We give an algorithm with complexity for the integer
multiflow problem on instances with an acyclic planar digraph
and Eulerian. Here, is a polynomial function, , and is the maximum request . When is
fixed, this gives a polynomial algorithm for the arc-disjoint paths problem
under the same hypothesis
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
On Routing Disjoint Paths in Bounded Treewidth Graphs
We study the problem of routing on disjoint paths in bounded treewidth graphs
with both edge and node capacities. The input consists of a capacitated graph
and a collection of source-destination pairs . The goal is to maximize the number of pairs that
can be routed subject to the capacities in the graph. A routing of a subset
of the pairs is a collection of paths such that,
for each pair , there is a path in
connecting to . In the Maximum Edge Disjoint Paths (MaxEDP) problem,
the graph has capacities on the edges and a routing
is feasible if each edge is in at most of
the paths of . The Maximum Node Disjoint Paths (MaxNDP) problem is
the node-capacitated counterpart of MaxEDP.
In this paper we obtain an approximation for MaxEDP on graphs of
treewidth at most and a matching approximation for MaxNDP on graphs of
pathwidth at most . Our results build on and significantly improve the work
by Chekuri et al. [ICALP 2013] who obtained an approximation
for MaxEDP
Finding Disjoint Paths on Edge-Colored Graphs: More Tractability Results
The problem of finding the maximum number of vertex-disjoint uni-color paths
in an edge-colored graph (called MaxCDP) has been recently introduced in
literature, motivated by applications in social network analysis. In this paper
we investigate how the complexity of the problem depends on graph parameters
(namely the number of vertices to remove to make the graph a collection of
disjoint paths and the size of the vertex cover of the graph), which makes
sense since graphs in social networks are not random and have structure. The
problem was known to be hard to approximate in polynomial time and not
fixed-parameter tractable (FPT) for the natural parameter. Here, we show that
it is still hard to approximate, even in FPT-time. Finally, we introduce a new
variant of the problem, called MaxCDDP, whose goal is to find the maximum
number of vertex-disjoint and color-disjoint uni-color paths. We extend some of
the results of MaxCDP to this new variant, and we prove that unlike MaxCDP,
MaxCDDP is already hard on graphs at distance two from disjoint paths.Comment: Journal version in JOC
Complexity and Approximation Results for the Min-Sum and Min-Max Disjoint Paths Problems
Given a graph G=(V, E) and k source-sink pairs (s1, t1), …, (sk, tk) with each si, ti V, the Min-Sum Disjoint Paths problem asks to find k disjoint paths connecting all the source-sink pairs with minimized total length, while the Min-Max Disjoint Paths problem asks for k disjoint paths connecting all the source-sink pairs with minimized length of the longest path. We show that the weighted Min-Sum Disjoint Paths problem is FPNP-complete in general graphs, and the unweighted Min-Sum Disjoint Paths problem and the unweighted Min-Max Disjoint Paths problem cannot be approximated within m(m1-1) for any constant > 0 even in planar graphs, assuming P P NP, where m is the number of edges in G. We give for the first time a simple bicriteria approximation algorithm for the unweighted Min-Max Edge-Disjoint Paths problem and the weighted Min-Sum Edge-Disjoint Paths problem, w
On Hamilton decompositions of infinite circulant graphs
The natural infinite analogue of a (finite) Hamilton cycle is a two-way-infinite Hamilton path (connected spanning 2-valent subgraph).
Although it is known that every connected 2k-valent infinite circulant graph has a two-way-infinite Hamilton path, there exist many such graphs that do not have a decomposition into k edge-disjoint two-way-infinite Hamilton paths. This contrasts with the finite case where it is conjectured that every 2k-valent connected circulant graph has a decomposition into k edge-disjoint Hamilton cycles. We settle the problem of decomposing 2k-valent infinite circulant graphs into k edge-disjoint two-way-infinite Hamilton paths for k=2, in many cases when k=3, and in many other cases including where the connection set is ±{1,2,...,k} or ±{1,2,...,k - 1, 1,2,...,k + 1}
Disjoint paths in geometric graphs
Construction of the shortest paths connecting two given nodes in a geometric graph is the quintessential problem in computational geometry. We consider several variations of this problem with applications that include robotics, Geographic Information Systems, and sensor networks. The first problem we address is the development of an efficient algorithm for constructing a pair of short node-disjoint paths connecting start and target nodes. The second problem investigated is the development of efficient algorithms for constructing narrow and in-range broadcast corridors in triangulated geometric graphs. Finally, we consider the development of an approximation algorithm for constructing reduced overlap trees in three-colored geometric graphs. Theoretical analysis and a detailed experimental investigation of the proposed algorithms are also presented
Topological infinite gammoids, and a new Menger-type theorem for infinite graphs
Answering a question of Diestel, we develop a topological notion of gammoids
in infinite graphs which, unlike traditional infinite gammoids, always define a
matroid. As our main tool, we prove for any infinite graph with vertex sets
and that if every finite subset of is linked to by disjoint
paths, then the whole of can be linked to the closure of by disjoint
paths or rays in a natural topology on and its ends. This latter theorem
re-proves and strengthens the infinite Menger theorem of Aharoni and Berger for
`well-separated' sets and . It also implies the topological Menger
theorem of Diestel for locally finite graphs
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