274 research outputs found
Optimal Embedding of Functions for In-Network Computation: Complexity Analysis and Algorithms
We consider optimal distributed computation of a given function of
distributed data. The input (data) nodes and the sink node that receives the
function form a connected network that is described by an undirected weighted
network graph. The algorithm to compute the given function is described by a
weighted directed acyclic graph and is called the computation graph. An
embedding defines the computation communication sequence that obtains the
function at the sink. Two kinds of optimal embeddings are sought, the embedding
that---(1)~minimizes delay in obtaining function at sink, and (2)~minimizes
cost of one instance of computation of function. This abstraction is motivated
by three applications---in-network computation over sensor networks, operator
placement in distributed databases, and module placement in distributed
computing.
We first show that obtaining minimum-delay and minimum-cost embeddings are
both NP-complete problems and that cost minimization is actually MAX SNP-hard.
Next, we consider specific forms of the computation graph for which polynomial
time solutions are possible. When the computation graph is a tree, a polynomial
time algorithm to obtain the minimum delay embedding is described. Next, for
the case when the function is described by a layered graph we describe an
algorithm that obtains the minimum cost embedding in polynomial time. This
algorithm can also be used to obtain an approximation for delay minimization.
We then consider bounded treewidth computation graphs and give an algorithm to
obtain the minimum cost embedding in polynomial time
Packing Odd Walks and Trails in Multiterminal Networks
Let G be an undirected network with a distinguished set of terminals T ? V(G) and edge capacities cap: E(G) ? ?_+. By an odd T-walk we mean a walk in G (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd T-walks subject to capacities cap.
First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd T-walks. For even integer capacities, our algorithm constructs a packing that is half-integer. Additionally, if cap(?(v)) is divisible by 4 for any v ? V(G)-T, our algorithm constructs an integer packing.
Second, we establish and prove the corresponding min-max relation.
Third, if G is inner Eulerian (i.e. degrees of all nodes in V(G)-T are even) and cap(e) = 2 for all e ? E, we show that there exists an integer packing of odd T-trails (i.e. odd T-walks with no repeated edges) of the same value as in case of odd T-walks, and this packing can be found in polynomial time.
To achieve the above goals, we establish a connection between packings of odd T-walks and T-trails and certain multiflow problems in undirected and bidirected graphs
Parameterized Complexity Dichotomy for Steiner Multicut
The Steiner Multicut problem asks, given an undirected graph G, terminals
sets T1,...,Tt V(G) of size at most p, and an integer k, whether
there is a set S of at most k edges or nodes s.t. of each set Ti at least one
pair of terminals is in different connected components of G \ S. This problem
generalizes several graph cut problems, in particular the Multicut problem (the
case p = 2), which is fixed-parameter tractable for the parameter k [Marx and
Razgon, Bousquet et al., STOC 2011].
We provide a dichotomy of the parameterized complexity of Steiner Multicut.
That is, for any combination of k, t, p, and the treewidth tw(G) as constant,
parameter, or unbounded, and for all versions of the problem (edge deletion and
node deletion with and without deletable terminals), we prove either that the
problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or
even (para-)NP-complete). We highlight that:
- The edge deletion version of Steiner Multicut is fixed-parameter tractable
for the parameter k+t on general graphs (but has no polynomial kernel, even on
trees). We present two proofs: one using the randomized contractions technique
of Chitnis et al, and one relying on new structural lemmas that decompose the
Steiner cut into important separators and minimal s-t cuts.
- In contrast, both node deletion versions of Steiner Multicut are W[1]-hard
for the parameter k+t on general graphs.
- All versions of Steiner Multicut are W[1]-hard for the parameter k, even
when p=3 and the graph is a tree plus one node. Hence, the results of Marx and
Razgon, and Bousquet et al. do not generalize to Steiner Multicut.
Since we allow k, t, p, and tw(G) to be any constants, our characterization
includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a
polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to
constant or unbounded).Comment: As submitted to journal. This version also adds a proof of
fixed-parameter tractability for parameter k+t using the technique of
randomized contraction
Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Given a directed graph , a set of terminals and an integer , the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set of at
most (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where is a set of at most edges. These two problems indeed are
known to be equivalent. A natural generalization of the multiway cut is the
\emph{multicut} problem, in which we want to disconnect only a set of given
pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in
undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized
by . Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by . We complete the picture
here by our main result which is that both \textsc{Directed Vertex Multiway
Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time
, i.e., FPT parameterized by size of the cutset of
the solution. This answers an open question raised by Marx (Theor. Comp. Sci.
2006) and Marx and Razgon (STOC 2011). It follows from our result that
\textsc{Directed Multicut} is FPT for the case of terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)
Two-stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms
We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP-hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max-flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max-flow structure of the subproblems
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