2,516 research outputs found
Approximating Longest Directed Paths and Cycles
We investigate the hardness of approximating the longest path and the longest cycle in directed graphs on n vertices. We show that neither of these two problems can be polynomial time approximated within n1-ε for any ε \u3e 0 unless P = NP. In particular, the result holds for digraphs of constant bounded outdegree that contain a Hamiltonian cycle.
Assuming the stronger complexity conjecture that Satisfiability cannot be solved in subexponential time, we show that there is no polynomial time algorithm that finds a directed path of length Ω(f(n) log2n), or a directed cycle of length Ω(f(n) log n), for any nondecreasing, polynomial time computable function f in Ω(1). With a recent algorithm for undirected graphs by Gabow, this shows that long paths and cycles are harder to find in directed graphs than in undirected graphs.
We also find a directed path of length Ω(log2 n/ log log n) in Hamiltonian digraphs with bounded outdegree. With our hardness results, this shows that long directed cycles are harder to find than a long directed paths. Furthermore, we present a simple polynomial time algorithm that finds paths of length Ω(n) in directed expanders of constant bounded outdegree
Approximating Longest Path
We investigate the computational hardness of approximating the longest path and the longest cycle in undirected and directed graphs on n vertices. We show that * in any expander graph, we can find (n) long paths in polynomial time. * there is an algorithm that finds a path of length (log2 L/ log log L) in any undirected graph having a path of length L, in polynomial time. * there is an algorithm that finds a path of length (log2 n/ log log n) in any Hamiltonian directed graph of constant bounded outdegree, in polynomial time. * there cannot be an algorithm finding paths of length (n ) for any constant > 0 in a Hamiltonian directed graph of bounded outdegree in polynomial time, unless P = NP. * there cannot be an algorithm finding paths of length (log2+ n), or cycles of length (log1+ n) for any constant > 0 in a Hamiltonian directed graph of constant bounded outdegree in polynomial time, unless 3-Sat can be solved in subexponential time
Pre-Reduction Graph Products: Hardnesses of Properly Learning DFAs and Approximating EDP on DAGs
The study of graph products is a major research topic and typically concerns
the term , e.g., to show that . In this paper, we
study graph products in a non-standard form where is a
"reduction", a transformation of any graph into an instance of an intended
optimization problem. We resolve some open problems as applications.
(1) A tight -approximation hardness for the minimum
consistent deterministic finite automaton (DFA) problem, where is the
sample size. Due to Board and Pitt [Theoretical Computer Science 1992], this
implies the hardness of properly learning DFAs assuming (the
weakest possible assumption).
(2) A tight hardness for the edge-disjoint paths (EDP)
problem on directed acyclic graphs (DAGs), where denotes the number of
vertices.
(3) A tight hardness of packing vertex-disjoint -cycles for large .
(4) An alternative (and perhaps simpler) proof for the hardness of properly
learning DNF, CNF and intersection of halfspaces [Alekhnovich et al., FOCS 2004
and J. Comput.Syst.Sci. 2008]
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
Efficiently listing bounded length st-paths
The problem of listing the shortest simple (loopless) -paths in a
graph has been studied since the early 1960s. For a non-negatively weighted
graph with vertices and edges, the most efficient solution is an
algorithm for directed graphs by Yen and Lawler
[Management Science, 1971 and 1972], and an algorithm for
the undirected version by Katoh et al. [Networks, 1982], both using
space. In this work, we consider a different parameterization for this problem:
instead of bounding the number of -paths output, we bound their length. For
the bounded length parameterization, we propose new non-trivial algorithms
matching the time complexity of the classic algorithms but using only
space. Moreover, we provide a unified framework such that the solutions to both
parameterizations -- the classic -shortest and the new length-bounded paths
-- can be seen as two different traversals of a same tree, a Dijkstra-like and
a DFS-like traversal, respectively.Comment: 12 pages, accepted to IWOCA 201
Network Interdiction Using Adversarial Traffic Flows
Traditional network interdiction refers to the problem of an interdictor
trying to reduce the throughput of network users by removing network edges. In
this paper, we propose a new paradigm for network interdiction that models
scenarios, such as stealth DoS attack, where the interdiction is performed
through injecting adversarial traffic flows. Under this paradigm, we first
study the deterministic flow interdiction problem, where the interdictor has
perfect knowledge of the operation of network users. We show that the problem
is highly inapproximable on general networks and is NP-hard even when the
network is acyclic. We then propose an algorithm that achieves a logarithmic
approximation ratio and quasi-polynomial time complexity for acyclic networks
through harnessing the submodularity of the problem. Next, we investigate the
robust flow interdiction problem, which adopts the robust optimization
framework to capture the case where definitive knowledge of the operation of
network users is not available. We design an approximation framework that
integrates the aforementioned algorithm, yielding a quasi-polynomial time
procedure with poly-logarithmic approximation ratio for the more challenging
robust flow interdiction. Finally, we evaluate the performance of the proposed
algorithms through simulations, showing that they can be efficiently
implemented and yield near-optimal solutions
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
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