A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs

Boyacı, Arman; Ekim, Tınaz; Shalom, Mordechai

doi:10.1016/j.ipl.2017.01.007

Cited by 11 publications

(18 citation statements)

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“…Given a graph G, the MaxCut problem is the problem of finding a partition of the vertices of G into two sets S and S such that the number of edges with one endpoint in S and the other one in S is maximum among all partitions. There were two results about polynomiality of the MaxCut problem in unit interval graphs in the past years; the first one by Bodlaender et al in 1999 [5], the second one by Boyacı et al which has been published in 2017 [7]. The result of the first paper was disproved by authors themselves a few years later [4].…”

Section: Theorem 1 a Graph Is A Mixed Unit Interval Graph If And Only If It Has A U-bubble Model Moreover Given A Mixed Unit Interval Repmentioning

confidence: 99%

“…As it was mentioned in the introduction, there is a paper A polynomial-time algorithm for the maximum cardinality cut problem in proper interval graphs by Boyacı et al from 2017 [7], claiming that the MaxCut problem is polynomial-time solvable in unit interval graphs and giving a dynamic programming algorithm based on the bubble model representation. We realized that the algorithm is incorrect; this section is devoted to it.…”

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

“…4. In Then, according to the paper [7], the size of a maximum cut in G is eight. To be more concrete, the algorithm from [7] fills the following values of dynamic table:…”

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

“…The brief idea of the algorithm in [7] is to process the columns from the biggest to the lowest column from the top bubble to the bottom one. Once we know the number of cut-vertices in the actual processed bubble B (in the column j) and the number of cut-vertices which are above B in the columns j and j + 1, we can count the exact number of edges.…”

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

“…We claim that the algorithm and its full idea from [7] are incorrect since we lose the consistency there-to obtain a maximum cut, we do not remember anything about the distribution of cut vertices within bubbles, that was used in the previously processed column. Therefore, there is no guarantee that the final outputted cut of the computed size exists.…”

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

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$${\mathcal {U}}$$-Bubble Model for Mixed Unit Interval Graphs and Its Applications: The MaxCut Problem Revisited

2021

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Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

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Section: Theorem 1 a Graph Is A Mixed Unit Interval Graph If And Only If It Has A U-bubble Model Moreover Given A Mixed Unit Interval Repmentioning

confidence: 99%

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

“…4. In Then, according to the paper [7], the size of a maximum cut in G is eight. To be more concrete, the algorithm from [7] fills the following values of dynamic table:…”

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

Section: Time Complexity Is Still Unknown On Unit Interval Graphsmentioning

confidence: 99%

See 3 more Smart Citations