3,775 research outputs found
A Partitioning Algorithm for Maximum Common Subgraph Problems
We introduce a new branch and bound algorithm for the maximum common subgraph and maximum common connected subgraph problems which is based around vertex labelling and partitioning. Our method in some ways resembles a traditional constraint programming approach, but uses a novel compact domain store and supporting inference algorithms which dramatically reduce the memory and computation requirements during search, and allow better dual viewpoint ordering heuristics to be calculated cheaply. Experiments show a speedup of more than an order of magnitude over the state of the art, and demonstrate that we can operate on much larger graphs without running out of memory
Between Subgraph Isomorphism and Maximum Common Subgraph
When a small pattern graph does not occur inside a larger target graph, we can ask how to find "as much of the pattern as possible" inside the target graph. In general, this is known as the maximum common subgraph problem, which is much more computationally challenging in practice than subgraph isomorphism. We introduce a restricted alternative, where we ask if all but k vertices from the pattern can be found in the target graph. This allows for the development of slightly weakened forms of certain invariants from subgraph isomorphism which are based upon degree and number of paths. We show that when k is small, weakening the invariants still retains much of their effectiveness. We are then able to solve this problem on the standard problem instances used to benchmark subgraph isomorphism algorithms, despite these instances being too large for current maximum common subgraph algorithms to handle. Finally, by iteratively increasing k, we obtain an algorithm which is also competitive for the maximum common subgraph
Exploring Communities in Large Profiled Graphs
Given a graph and a vertex , the community search (CS) problem
aims to efficiently find a subgraph of whose vertices are closely related
to . Communities are prevalent in social and biological networks, and can be
used in product advertisement and social event recommendation. In this paper,
we study profiled community search (PCS), where CS is performed on a profiled
graph. This is a graph in which each vertex has labels arranged in a
hierarchical manner. Extensive experiments show that PCS can identify
communities with themes that are common to their vertices, and is more
effective than existing CS approaches. As a naive solution for PCS is highly
expensive, we have also developed a tree index, which facilitate efficient and
online solutions for PCS
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe
The Network Improvement Problem for Equilibrium Routing
In routing games, agents pick their routes through a network to minimize
their own delay. A primary concern for the network designer in routing games is
the average agent delay at equilibrium. A number of methods to control this
average delay have received substantial attention, including network tolls,
Stackelberg routing, and edge removal.
A related approach with arguably greater practical relevance is that of
making investments in improvements to the edges of the network, so that, for a
given investment budget, the average delay at equilibrium in the improved
network is minimized. This problem has received considerable attention in the
literature on transportation research and a number of different algorithms have
been studied. To our knowledge, none of this work gives guarantees on the
output quality of any polynomial-time algorithm. We study a model for this
problem introduced in transportation research literature, and present both
hardness results and algorithms that obtain nearly optimal performance
guarantees.
- We first show that a simple algorithm obtains good approximation guarantees
for the problem. Despite its simplicity, we show that for affine delays the
approximation ratio of 4/3 obtained by the algorithm cannot be improved.
- To obtain better results, we then consider restricted topologies. For
graphs consisting of parallel paths with affine delay functions we give an
optimal algorithm. However, for graphs that consist of a series of parallel
links, we show the problem is weakly NP-hard.
- Finally, we consider the problem in series-parallel graphs, and give an
FPTAS for this case.
Our work thus formalizes the intuition held by transportation researchers
that the network improvement problem is hard, and presents topology-dependent
algorithms that have provably tight approximation guarantees.Comment: 27 pages (including abstract), 3 figure
Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs
Drawings of highly connected (dense) graphs can be very difficult to read.
Power Graph Analysis offers an alternate way to draw a graph in which sets of
nodes with common neighbours are shown grouped into modules. An edge connected
to the module then implies a connection to each member of the module. Thus, the
entire graph may be represented with much less clutter and without loss of
detail. A recent experimental study has shown that such lossless compression of
dense graphs makes it easier to follow paths. However, computing optimal power
graphs is difficult. In this paper, we show that computing the optimal
power-graph with only one module is NP-hard and therefore likely NP-hard in the
general case. We give an ILP model for power graph computation and discuss why
ILP and CP techniques are poorly suited to the problem. Instead, we are able to
find optimal solutions much more quickly using a custom search method. We also
show how to restrict this type of search to allow only limited back-tracking to
provide a heuristic that has better speed and better results than previously
known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of
the same nam
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