652 research outputs found
Lift-and-project ranks of the stable set polytope of joined a-perfect graphs
In this paper we study lift-and-project polyhedral operators defined by
Lov?asz and Schrijver and Balas, Ceria and Cornu?ejols on the clique relaxation
of the stable set polytope of web graphs. We compute the disjunctive rank of
all webs and consequently of antiweb graphs. We also obtain the disjunctive
rank of the antiweb constraints for which the complexity of the separation
problem is still unknown. Finally, we use our results to provide bounds of the
disjunctive rank of larger classes of graphs as joined a-perfect graphs, where
near-bipartite graphs belong
Efficient enumeration of solutions produced by closure operations
In this paper we address the problem of generating all elements obtained by
the saturation of an initial set by some operations. More precisely, we prove
that we can generate the closure of a boolean relation (a set of boolean
vectors) by polymorphisms with a polynomial delay. Therefore we can compute
with polynomial delay the closure of a family of sets by any set of "set
operations": union, intersection, symmetric difference, subsets, supersets
). To do so, we study the problem: for a set
of operations , decide whether an element belongs to the closure
by of a family of elements. In the boolean case, we prove that
is in P for any set of boolean operations
. When the input vectors are over a domain larger than two
elements, we prove that the generic enumeration method fails, since
is NP-hard for some . We also study the
problem of generating minimal or maximal elements of closures and prove that
some of them are related to well known enumeration problems such as the
enumeration of the circuits of a matroid or the enumeration of maximal
independent sets of a hypergraph. This article improves on previous works of
the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of
the same name which appeared in STACS 2016. Final version for DMTCS journa
Fair allocation of indivisible goods under conflict constraints
We consider the fair allocation of indivisible items to several agents and
add a graph theoretical perspective to this classical problem. Thereby we
introduce an incompatibility relation between pairs of items described in terms
of a conflict graph. Every subset of items assigned to one agent has to form an
independent set in this graph. Thus, the allocation of items to the agents
corresponds to a partial coloring of the conflict graph. Every agent has its
own profit valuation for every item. Aiming at a fair allocation, our goal is
the maximization of the lowest total profit of items allocated to any one of
the agents. The resulting optimization problem contains, as special cases, both
{\sc Partition} and {\sc Independent Set}. In our contribution we derive
complexity and algorithmic results depending on the properties of the given
graph. We can show that the problem is strongly NP-hard for bipartite graphs
and their line graphs, and solvable in pseudo-polynomial time for the classes
of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and
graphs of bounded treewidth. Each of the pseudo-polynomial algorithms can also
be turned into a fully polynomial approximation scheme (FPTAS).Comment: A preliminary version containing some of the results presented here
appeared in the proceedings of IWOCA 2020. Version 3 contains an appendix
with a remark about biconvex bipartite graph
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