68 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
On one maximum multiflow problem and related metrics
AbstractWe consider the undirected maximum multiflow (multicommodity flow) problem in the case when the commodity graph is the disjoint union of K3 and K2. We prove that if the supply graph satisfies a certain Eulerian-type condition, then the problem has an integer optimal solution. To obtain this result, we first study the corresponding dual problem on metrics and show that an optimal solution to the latter is achieved on some (2,3)-metric or some 3-cut metric
A tight relation between series--parallel graphs and bipartite distance hereditary graphs
Bandelt and Mulderâs structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by re17 peatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffinâs structural characterization of 2âconnected seriesâparallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and seriesâparallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum valu
Node-Connectivity Terminal Backup, Separately-Capacitated Multiflow, and Discrete Convexity
The terminal backup problems (Anshelevich and Karagiozova (2011)) form a
class of network design problems: Given an undirected graph with a requirement
on terminals, the goal is to find a minimum cost subgraph satisfying the
connectivity requirement. The node-connectivity terminal backup problem
requires a terminal to connect other terminals with a number of node-disjoint
paths. This problem is not known whether is NP-hard or tractable. Fukunaga
(2016) gave a -approximation algorithm based on LP-rounding scheme using a
general LP-solver. In this paper, we develop a combinatorial algorithm for the
relaxed LP to find a half-integral optimal solution in time, where is the number of nodes, is
the number of edges, is the number of terminals, is the maximum
edge-cost, is the maximum edge-capacity, and is
the time complexity of a max-flow algorithm in a network with nodes and
edges. The algorithm implies that the -approximation algorithm for
the node-connectivity terminal backup problem is also efficiently implemented.
For the design of algorithm, we explore a connection between the
node-connectivity terminal backup problem and a new type of a multiflow, called
a separately-capacitated multiflow. We show a min-max theorem which extends
Lov\'{a}sz-Cherkassky theorem to the node-capacity setting. Our results build
on discrete convexity in the node-connectivity terminal backup problem.Comment: A preliminary version of this paper was appeared in the proceedings
of the 47th International Colloquium on Automata, Languages and Programming
(ICALP 2020
HJ Inequalities Involving Lie Brackets and Feedback Stabilizability with Cost Regulation
With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k = 1, and we call any of its (suitably defined) solutions a degree -k Minimum Restraint Function. We prove that the existence of a degree -k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost
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