27,832 research outputs found
Fully dynamic cycle-equivalence in graphs
Two edges e_1 and e_2 of an undirected graph are cycle-equivalent iff all cycles that contain e_1 also contain e_2, i.e., iff e_1 and e_2 are a cut-edge pair. The cycle-equivalence classes of the control-flow graph are used in optimizing compilers to speed up existing control-flow and data-flow algorithms. While the cycle-equivalence classes can be computed in linear time, we present the first fully dynamic algorithm for maintaining the cycle-equivalence relation. In an n-node graph our data structure executes an edge insertion or deletion in O(sqrt(n.log n)) time and answers the query whether two given edges are cycle-equivalent in O(pow2(log(n))) time. We also present an algorithm for plane graphs with O(log n) update and query time and for planar graphs with O(log n) insertion time and O(log2 n) query and deletion time. Additionally, we show a lower bound of Ω(log n/log log n) for the amortized time per operation for the dynamic cycle-equivalence problem in the cell probe mode
A unified view on bipartite species-reaction and interaction graphs for chemical reaction networks
The Jacobian matrix of a dynamic system and its principal minors play a
prominent role in the study of qualitative dynamics and bifurcation analysis.
When interpreting the Jacobian as an adjacency matrix of an interaction graph,
its principal minors correspond to sets of disjoint cycles in this graph and
conditions for various dynamic behaviors can be inferred from its cycle
structure. For chemical reaction systems, more fine-grained analyses are
possible by studying a bipartite species-reaction graph. Several results on
injectivity, multistationarity, and bifurcations of a chemical reaction system
have been derived by using various definitions of such bipartite graph. Here,
we present a new definition of the species-reaction graph that more directly
connects the cycle structure with determinant expansion terms, principal
minors, and the coefficients of the characteristic polynomial and encompasses
previous graph constructions as special cases. This graph has a direct relation
to the interaction graph, and properties of cycles and sub-graphs can be
translated in both directions. A simple equivalence relation enables to
decompose determinant expansions more directly and allows simpler and more
direct proofs of previous results.Comment: 27 pages. submitted to J. Math. Bio
Minimum Equivalent Precedence Relation Systems
In this paper two related simplification problems for systems of linear
inequalities describing precedence relation systems are considered. Given a
precedence relation system, the first problem seeks a minimum subset of the
precedence relations (i.e., inequalities) which has the same solution set as
that of the original system. The second problem is the same as the first one
except that the ``subset restriction'' in the first problem is removed. This
paper establishes that the first problem is NP-hard. However, a sufficient
condition is provided under which the first problem is solvable in
polynomial-time. In addition, a decomposition of the first problem into
independent tractable and intractable subproblems is derived. The second
problem is shown to be solvable in polynomial-time, with a full
parameterization of all solutions described. The results in this paper
generalize those in [Moyles and Thompson 1969, Aho, Garey, and Ullman 1972] for
the minimum equivalent graph problem and transitive reduction problem, which
are applicable to unweighted directed graphs
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
We propose polynomial-time algorithms that sparsify planar and bounded-genus
graphs while preserving optimal or near-optimal solutions to Steiner problems.
Our main contribution is a polynomial-time algorithm that, given an unweighted
graph embedded on a surface of genus and a designated face bounded
by a simple cycle of length , uncovers a set of size
polynomial in and that contains an optimal Steiner tree for any set of
terminals that is a subset of the vertices of .
We apply this general theorem to prove that: * given an unweighted graph
embedded on a surface of genus and a terminal set , one
can in polynomial time find a set that contains an optimal
Steiner tree for and that has size polynomial in and ; * an
analogous result holds for an optimal Steiner forest for a set of terminal
pairs; * given an unweighted planar graph and a terminal set , one can in polynomial time find a set that contains
an optimal (edge) multiway cut separating and that has size polynomial
in .
In the language of parameterized complexity, these results imply the first
polynomial kernels for Steiner Tree and Steiner Forest on planar and
bounded-genus graphs (parameterized by the size of the tree and forest,
respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by
the size of the cutset). Additionally, we obtain a weighted variant of our main
contribution
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