494 research outputs found
The Dual Polynomial of Bipartite Perfect Matching
We obtain a description of the Boolean dual function of the Bipartite Perfect
Matching decision problem, as a multilinear polynomial over the Reals. We show
that in this polynomial, both the number of monomials and the magnitude of
their coefficients are at most exponential in . As an
application, we obtain a new upper bound of on the approximate degree of the bipartite perfect matching function,
improving the previous best known bound of . We deduce
that, beyond a factor, the polynomial method
cannot be used to improve the lower bound on the bounded-error quantum query
complexity of bipartite perfect matching
Robust Assignments via Ear Decompositions and Randomized Rounding
Many real-life planning problems require making a priori decisions before all
parameters of the problem have been revealed. An important special case of such
problem arises in scheduling problems, where a set of tasks needs to be
assigned to the available set of machines or personnel (resources), in a way
that all tasks have assigned resources, and no two tasks share the same
resource. In its nominal form, the resulting computational problem becomes the
\emph{assignment problem} on general bipartite graphs.
This paper deals with a robust variant of the assignment problem modeling
situations where certain edges in the corresponding graph are \emph{vulnerable}
and may become unavailable after a solution has been chosen. The goal is to
choose a minimum-cost collection of edges such that if any vulnerable edge
becomes unavailable, the remaining part of the solution contains an assignment
of all tasks.
We present approximation results and hardness proofs for this type of
problems, and establish several connections to well-known concepts from
matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape
Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction
In this work, we initiate the complexity study of Biclique Contraction and
Balanced Biclique Contraction. In these problems, given as input a graph G and
an integer k, the objective is to determine whether one can contract at most k
edges in G to obtain a biclique and a balanced biclique, respectively. We first
prove that these problems are NP-complete even when the input graph is
bipartite. Next, we study the parameterized complexity of these problems and
show that they admit single exponential-time FPT algorithms when parameterized
by the number k of edge contractions. Then, we show that Balanced Biclique
Contraction admits a quadratic vertex kernel while Biclique Contraction does
not admit any polynomial compression (or kernel) under standard
complexity-theoretic assumptions
Flows and bisections in cubic graphs
A -weak bisection of a cubic graph is a partition of the vertex-set of
into two parts and of equal size, such that each connected
component of the subgraph of induced by () is a tree of at
most vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph with a circular nowhere-zero -flow has a -weak bisection. In this paper we study problems related to the
existence of -weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.Comment: 14 pages, 6 figures - revised versio
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