5 research outputs found
A tight quasi-polynomial bound for Global Label Min-Cut
We study a generalization of the classic Global Min-Cut problem, called
Global Label Min-Cut (or sometimes Global Hedge Min-Cut): the edges of the
input (multi)graph are labeled (or partitioned into color classes or hedges),
and removing all edges of the same label (color or from the same hedge) costs
one. The problem asks to disconnect the graph at minimum cost.
While the -cut version of the problem is known to be NP-hard, the above
global cut version is known to admit a quasi-polynomial randomized -time algorithm due to Ghaffari, Karger, and Panigrahi [SODA
2017]. They consider this as ``strong evidence that this problem is in P''. We
show that this is actually not the case. We complete the study of the
complexity of the Global Label Min-Cut problem by showing that the
quasi-polynomial running time is probably optimal: We show that the existence
of an algorithm with running time would
contradict the Exponential Time Hypothesis, where is the number of
vertices, and is the number of labels in the input. The key step for the
lower bound is a proof that Global Label Min-Cut is W[1]-hard when
parameterized by the number of uncut labels. In other words, the problem is
difficult in the regime where almost all labels need to be cut to disconnect
the graph. To turn this lower bound into a quasi-polynomial-time lower bound,
we also needed to revisit the framework due to Marx [Theory Comput. 2010] of
proving lower bounds assuming Exponential Time Hypothesis through the Subgraph
Isomorphism problem parameterized by the number of edges of the pattern. Here,
we provide an alternative simplified proof of the hardness of this problem that
is more versatile with respect to the choice of the regimes of the parameters
Combinatorial optimization in networks with Shared Risk Link Groups
International audienceThe notion of Shared Risk Link Groups (SRLG) captures survivability issues when a set of links of a network may fail simultaneously. The theory of survivable network design relies on basic combinatorial objects that are rather easy to compute in the classical graph models: shortest paths, minimum cuts, or pairs of disjoint paths. In the SRLG context, the optimization criterion for these objects is no longer the number of edges they use, but the number of SRLGs involved. Unfortunately, computing these combinatorial objects is NP-hard and hard to approximate with this objective in general. Nevertheless some objects can be computed in polynomial time when the SRLGs satisfy certain structural properties of locality which correspond to practical ones, namely the star property (all links affected by a given SRLG are incident to a unique node) and the span 1 property (the links affected by a given SRLG form a connected component of the network). The star property is defined in a multi-colored model where a link can be affected by several SRLGs while the span property is defined only in a mono-colored model where a link can be affected by at most one SRLG. In this paper, we extend these notions to characterize new cases in which these optimization problems can be solved in polynomial time. We also investigate the computational impact of the transformation from the multi-colored model to the mono-colored one. Experimental results are presented to validate the proposed algorithms and principles
approximating minimum label s-t cut via linear programming
Yahoo! Research; Microsoft Researc