9,113 research outputs found
Counting approximately-shortest paths in directed acyclic graphs
Given a directed acyclic graph with positive edge-weights, two vertices s and
t, and a threshold-weight L, we present a fully-polynomial time
approximation-scheme for the problem of counting the s-t paths of length at
most L. We extend the algorithm for the case of two (or more) instances of the
same problem. That is, given two graphs that have the same vertices and edges
and differ only in edge-weights, and given two threshold-weights L_1 and L_2,
we show how to approximately count the s-t paths that have length at most L_1
in the first graph and length at most L_2 in the second graph. We believe that
our algorithms should find application in counting approximate solutions of
related optimization problems, where finding an (optimum) solution can be
reduced to the computation of a shortest path in a purpose-built auxiliary
graph
Log-space Algorithms for Paths and Matchings in k-trees
Reachability and shortest path problems are NL-complete for general graphs.
They are known to be in L for graphs of tree-width 2 [JT07]. However, for
graphs of tree-width larger than 2, no bound better than NL is known. In this
paper, we improve these bounds for k-trees, where k is a constant. In
particular, the main results of our paper are log-space algorithms for
reachability in directed k-trees, and for computation of shortest and longest
paths in directed acyclic k-trees.
Besides the path problems mentioned above, we also consider the problem of
deciding whether a k-tree has a perfect macthing (decision version), and if so,
finding a perfect match- ing (search version), and prove that these two
problems are L-complete. These problems are known to be in P and in RNC for
general graphs, and in SPL for planar bipartite graphs [DKR08].
Our results settle the complexity of these problems for the class of k-trees.
The results are also applicable for bounded tree-width graphs, when a
tree-decomposition is given as input. The technique central to our algorithms
is a careful implementation of divide-and-conquer approach in log-space, along
with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201
Hardness of Graph Pricing through Generalized Max-Dicut
The Graph Pricing problem is among the fundamental problems whose
approximability is not well-understood. While there is a simple combinatorial
1/4-approximation algorithm, the best hardness result remains at 1/2 assuming
the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate
within a factor better than 1/4 under the UGC, so that the simple combinatorial
algorithm might be the best possible. We also prove that for any , there exists such that the integrality gap of
-rounds of the Sherali-Adams hierarchy of linear programming for
Graph Pricing is at most 1/2 + .
This work is based on the effort to view the Graph Pricing problem as a
Constraint Satisfaction Problem (CSP) simpler than the standard and complicated
formulation. We propose the problem called Generalized Max-Dicut(), which
has a domain size for every . Generalized Max-Dicut(1) is
well-known Max-Dicut. There is an approximation-preserving reduction from
Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and
both our results are achieved through this reduction. Besides its connection to
Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own
right since in most arity two CSPs studied in the literature, SDP-based
algorithms perform better than LP-based or combinatorial algorithms --- for
this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page
Replacement Paths via Row Minima of Concise Matrices
Matrix is {\em -concise} if the finite entries of each column of
consist of or less intervals of identical numbers. We give an -time
algorithm to compute the row minima of any -concise matrix.
Our algorithm yields the first -time reductions from the
replacement-paths problem on an -node -edge undirected graph
(respectively, directed acyclic graph) to the single-source shortest-paths
problem on an -node -edge undirected graph (respectively, directed
acyclic graph). That is, we prove that the replacement-paths problem is no
harder than the single-source shortest-paths problem on undirected graphs and
directed acyclic graphs. Moreover, our linear-time reductions lead to the first
-time algorithms for the replacement-paths problem on the following
classes of -node -edge graphs (1) undirected graphs in the word-RAM model
of computation, (2) undirected planar graphs, (3) undirected minor-closed
graphs, and (4) directed acyclic graphs.Comment: 23 pages, 1 table, 9 figures, accepted to SIAM Journal on Discrete
Mathematic
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