164,380 research outputs found
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.Comment: An extended abstract of this paper will be published at the 44th
Symposium on Theory of Computing (STOC 2012
Odd Paths, Cycles and -joins: Connections and Algorithms
Minimizing the weight of an edge set satisfying parity constraints is a
challenging branch of combinatorial optimization as witnessed by the binary
hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization''
(Chapter 80). This area contains relevant graph theory problems including open
cases of the Max Cut problem, or some multiflow problems. We clarify the
interconnections of some problems and establish three levels of difficulties.
On the one hand, we prove that the Shortest Odd Path problem in an undirected
graph without cycles of negative total weight and several related problems are
NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem
27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we
provide a polynomial-time algorithm to the closely related and well-studied
Minimum-weight Odd -Join problem for non-negative weights, whose
complexity, however, was not known; more generally, we solve the Minimum-weight
Odd -Join problem in FPT time when parameterized by . If negative
weights are also allowed, then finding a minimum-weight odd -join is
equivalent to the Minimum-weight Odd -Join problem for arbitrary weights,
whose complexity is only conjectured to be polynomially solvable. The analogous
problems for digraphs are also considered.Comment: 24 pages, 2 figure
Dynamic programming on bipartite tree decompositions
We revisit a graph width parameter that we dub bipartite treewidth, along
with its associated graph decomposition that we call bipartite tree
decomposition. Bipartite treewidth can be seen as a common generalization of
treewidth and the odd cycle transversal number. Intuitively, a bipartite tree
decomposition is a tree decomposition whose bags induce almost bipartite graphs
and whose adhesions contain at most one vertex from the bipartite part of any
other bag, while the width of such decomposition measures how far the bags are
from being bipartite. Adapted from a tree decomposition originally defined by
Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by
Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial
role for solving problems related to odd-minors, which have recently attracted
considerable attention. As a first step toward a theory for solving these
problems efficiently, the main goal of this paper is to develop dynamic
programming techniques to solve problems on graphs of small bipartite
treewidth. For such graphs, we provide a number of para-NP-completeness
results, FPT-algorithms, and XP-algorithms, as well as several open problems.
In particular, we show that -Subgraph-Cover, Weighted Vertex
Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are
parameterized by bipartite treewidth. We provide the following complexity
dichotomy when is a 2-connected graph, for each of -Subgraph-Packing,
-Induced-Packing, -Scattered-Packing, and -Odd-Minor-Packing problem:
if is bipartite, then the problem is para-NP-complete parameterized by
bipartite treewidth while, if is non-bipartite, then it is solvable in
XP-time. We define 1--treewidth by replacing the bipartite graph
class by any class . Most of the technology developed here works for
this more general parameter.Comment: Presented in IPEC 202
Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication
We study nondeterministic multiparty quantum communication with a quantum
generalization of broadcasts. We show that, with number-in-hand classical
inputs, the communication complexity of a Boolean function in this
communication model equals the logarithm of the support rank of the
corresponding tensor, whereas the approximation complexity in this model equals
the logarithm of the border support rank. This characterisation allows us to
prove a log-rank conjecture posed by Villagra et al. for nondeterministic
multiparty quantum communication with message-passing.
The support rank characterization of the communication model connects quantum
communication complexity intimately to the theory of asymptotic entanglement
transformation and algebraic complexity theory. In this context, we introduce
the graphwise equality problem. For a cycle graph, the complexity of this
communication problem is closely related to the complexity of the computational
problem of multiplying matrices, or more precisely, it equals the logarithm of
the asymptotic support rank of the iterated matrix multiplication tensor. We
employ Strassen's laser method to show that asymptotically there exist
nontrivial protocols for every odd-player cyclic equality problem. We exhibit
an efficient protocol for the 5-player problem for small inputs, and we show
how Young flattenings yield nontrivial complexity lower bounds
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