On the Lifted Multicut Polytope for Trees

Lange, Jan-Hendrik; Andres, Bjoern

doi:10.1007/978-3-030-71278-5_26

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…In fact, a number of basic and desirable properties in database theory turn out to be equivalent to acyclicity. A second example is given by the lifted multicut problem on trees, where the problem can be equivalently formulated via binary polynomial optimization [35]. The goal of the lifted multicut problem is to partition a given graph in a way that minimizes the total cost associated with having different pairs of nodes in different components.…”

Section: A Strongly Polynomial-time Algorithm For ˇ-Acyclic Hypergraphsmentioning

confidence: 99%

On the Complexity of Binary Polynomial Optimization Over Acyclic Hypergraphs

Pia

Gregorio

2022

Algorithmica

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In this work, we advance the understanding of the fundamental limits of computation for binary polynomial optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whose corresponding hypergraph is $$\beta $$ β -acyclic. We note that the $$\beta $$ β -acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs, since the problem is NP-hard on $$\alpha $$ α -acyclic instances. Our algorithm can also be applied to any general BPO problem that contains $$\beta $$ β -cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance.

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Section: A Strongly Polynomial-time Algorithm For ˇ-Acyclic Hypergraphsmentioning

confidence: 99%

On the Complexity of Binary Polynomial Optimization Over Acyclic Hypergraphs

Pia

Gregorio

2022

Algorithmica

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“…Unfortunately, this k-cut problem is sometimes also called multicut, for example in the following literature: In [5,6] multiple polytopes associated to cut problems are studied. In [17,21] the lifted multicut polytope was studied: Given a graph G with k-cut δ ⊆ E(G), and a supergraph G ⊃ G, one asks for a minimum k-cut δ ⊆ E(G ) with δ ∩ E(G) = δ. The mentioned article studies the cases of G being a tree or a path.…”

Section: Introductionmentioning

confidence: 99%

On the Dominant of the Multicut Polytope

Chimani,

Juhnke,

Nover

2024

Discrete Comput Geom

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Given a graph $$G=(V,E)$$ G = ( V , E ) and a set $$S \subseteq \left( {\begin{array}{c}V\\ 2\end{array}}\right) $$ S ⊆ V 2 of terminal pairs, the minimum multicut problem asks for a minimum edge set $$\delta \subseteq E$$ δ ⊆ E such that there is no s-t-path in $$G -\delta $$ G - δ for any $$\{s,t\}\in S$$ { s , t } ∈ S . For $$|S|=1$$ | S | = 1 this is the well known s-t-cut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope $$\textsc {MultC}^\square (G,S)$$ M U L T C □ ( G , S ) is the convex hull of all multicuts in G; the multicut dominant is given by $$\textsc {MultC}(G,S)=\textsc {MultC}^\square (G,S)+\mathbb {R}^E_{{\ge 0}}$$ M U L T C ( G , S ) = M U L T C □ ( G , S ) + R ≥ 0 E . The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut. We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path- and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.

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“…Using this notion of multicuts, in [HLA17,LA20] the lifted multicut problem was studied: Given a graph G ′ , a subgraph G ⊆ G ′ , and a multicut δ ⊆ E(G), the lifted multicut problem asks for a minimum multicut in…”

Section: Introductionmentioning

confidence: 99%

“…The polytope associated to this problem is called lifted multicut polytope. In [LA20], the lifted multicut polytope for G being a tree or a path was studied.…”

Section: Introductionmentioning

confidence: 99%

On the Dominant of the Multicut Polytope

Chimani¹,

Juhnke-Kubitzke²,

Nover³

2021

Preprint

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Given a graph G = (V, E) and a set S ⊆ V 2 of terminal pairs, the minimum multicut problem asks for a minimum edge set δ ⊆ E such that there is no s-t-path in G − δ for any {s, t} ∈ S. For |S| = 1 this is the well known s-tcut problem, but in general the minimum multicut problem is NP-complete, even if the input graph is a tree. The multicut polytope MultC (G, S) is the convex hull of all multicuts in G; the multicut dominant is given by MultC(G, S) = MultC (G, S) + R E . The latter is the relevant object for the minimization problem. While polyhedra associated to several cut problems have been studied intensively there is only little knowledge for multicut.We investigate properties of the multicut dominant and in particular derive results on liftings of facet-defining inequalities. This yields a classification of all facet-defining path-and edge inequalities. Moreover, we investigate the effect of graph operations such as node splitting, edge subdivisions, and edge contractions on the multicut-dominant and its facet-defining inequalities. In addition, we introduce facet-defining inequalities supported on stars, trees, and cycles and show that the former two can be separated in polynomial time when the input graph is a tree.

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On the Lifted Multicut Polytope for Trees

Cited by 4 publications

References 19 publications

On the Complexity of Binary Polynomial Optimization Over Acyclic Hypergraphs

On the Complexity of Binary Polynomial Optimization Over Acyclic Hypergraphs

On the Dominant of the Multicut Polytope

On the Dominant of the Multicut Polytope

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