Partitioning planar graphs: a fast combinatorial approach for max-cut

Liers, Frauke; Pardella, G.

doi:10.1007/s10589-010-9335-5

Cited by 18 publications

(30 citation statements)

References 37 publications

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“…Our algorithms recursively reduce a 1-planar graph to at most 3 k planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [21]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm.…”

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confidence: 99%

“…Their algorithms can be extended to work on weighted planar graphs (e.g., Mutzel [25]). The currently fastest algorithms have been suggested by Shih et al [29] and by Liers and Pardella [21]. These results have been extended to the classes of graphs not contractible to K 5 [5] and to toroidal graphs [4,12], i.e., graphs that can be embedded on the torus.A graph is 1-planar if it can be drawn on the plane, so that every edge is crossed at most once.…”

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confidence: 99%

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A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

Dahn

Kriege

Mutzel

2018

Lecture Notes in Computer Science

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We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number k of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most 3 k planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [21]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with n nodes and k edge crossings in time O(3 k · n 3/2 log n). An extended abstract of this work appeared in [8]. † Corresponding author [3]), which are of high interest in physics. Besides the theoretical merits, Ising spin glass models need to be solved in adiabatic quantum computation [23]. Other applications occur in the layout of electronic circuits [6,10].The Max-Cut problem has been shown to be NP-hard for general graphs [19]. Papadimitriou and Yannakakis [28] have shown that the Max-Cut problem is even APXhard, i.e., there does not exist a polynomial-time approximation scheme, unless P=NP. Goemans and Williamson suggested a randomized constant factor approximation algorithm [13], which has been derandomized by Mahajan and Ramesh [22] and has performance guarantee 0.87856.In the unweighted case, i.e., all weights are one, the problem is also called the simple max-cut problem or the maximum bipartite subgraph problem. It is well known that in this setting always a cut of size at least |E|/2 + (|V | − 1)/4 exists that can be computed in linear time (see, e.g., [30]). This bound is called the Edwards-Erdős bound and is tight for odd cliques. Recently, Etscheid and Mnich [11] have suggested an algorithm for the Max-Cut parameterized above Edwards-Erdős bound problem thus improving a result by Crowston et al. [7]. This problem asks if G has a cut exceeding the above bound by an amount of parameter k.There are a number of special cases, for which the weighted max-cut problem can be solved in polynomial time. The most prominent case arises if the weights of all edges are negative, since then the problem can be solved, e.g., via network flow. Other special cases are, e.g., graphs without long odd cycles [14] or weakly bipartite graphs [15].Another prominent case appears for planar input graphs. Orlova and Dorfman [26] and Hadlock [16] have shown that the Max-Cut problem can be solved in polynomial time for unweighted planar graphs. Their algorithms can be extended to work on weighted planar graphs (e.g., Mutzel [25]). The currently fastest algorithms have been suggested by Shih et al. [29] and by Liers and Pardella [21]. These results have been extended to the classes of gr...

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confidence: 99%

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confidence: 99%

A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

Dahn

Kriege

Mutzel

2018

Lecture Notes in Computer Science

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“…Thus it suffices to set c f = M instead of adding M to c f . Using any of the currently fastest Max-Cut algorithms for planar graphs [30,38] leads to O(|V | 3/2 log |V |) time in the above lemma. We could speed-up infeasibility detection by checking whether F contains a cycle of odd length prior to the transformation; while this only requires O(|V |) time via depth-first search, the overall asymptotic runtime for the lemma's claim does of course not improve.…”

Section: Algorithmmentioning

confidence: 99%

“…Moreover, the attained running time is polynomial for any k ∈ O(log |V |). Using the currently fastest Max-Cut algorithm for planar graphs [30,38], our algorithm yields a running time of O(2 k · (|V | + k) 3/2 log(|V | + k)).…”

Section: Corollarymentioning

confidence: 99%

“…The Max-Cut problem on 1-plane graphs with k edge crossings has recently been shown to be fixed-parameter tractable (FPT) with parameter k [7,8]. More precisely, it was shown that such instances can be solved in O(3 k · p(n)) time, where p(n) is the running time of a polynomial-time Max-Cut algorithm on planar graph with n nodes, e.g., [30,38]. There are no restrictions on the edge weights.Our contribution.…”

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Maximum Cut Parameterized by Crossing Number

Chimani¹,

Dahn²,

Juhnke-Kubitzke³

et al. 2020

JGAA

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Given an edge-weighted graph G on n nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number k of crossings in a given drawing of G. Our algorithm achieves a running time of O(2 k · p(n + k)), where p is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to embedded 1-planar graphs (i.e., at most one crossing per edge) and its dependency on k is of order 3 k . Finally, combining this with the fact that crossing number is fixed-parameter tractable w.r.t. itself, we see that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number. arXiv:1903.06061v2 [cs.DS] 6 Sep 2019QUBO such as multicommodity-flow problems, maximum clique, vertex cover, scheduling, and many others. Also see [10,11] for a more in-depth overview on applications.The Max-Cut problem has been shown to be NP-hard for general graphs [24]. Papadimitriou and Yannakakis [36] have shown that the Max-Cut problem is even APX-hard, i.e., there does not exist a polynomial-time approximation scheme unless P=NP. Goemans and Williamson proposed a randomized constant-factor approximation algorithm [17], which has been derandomized by Mahajan and Ramesh [31], achieving a ratio of 0.87856. Several special cases of the problem allow polynomial algorithms: If the weights of all edges are negative we obtain a Min-Cut problem, which can be solved, e.g., via network flow. Other special cases are, e.g., graphs without long odd cycles [20] or weakly bipartite graphs [19]. The case of planar input graphs is of particular interest. Orlova and Dorfman [34] and Hadlock [21] have shown how to solve Max-Cut in polynomial time for unweighted planar graphs. Those algorithms can be extended to weighted planar graphs; the currently fastest algorithms for the weighted case have been suggested by Shih et al. [38] and by Liers and Pardella [30], and achieve a running time of p(n) = O(n 3/2 log n) on planar graphs with n nodes. Barahona has shown that the planarity condition can be relaxed to graphs not contractible to K 5 [3] and to toroidal graphs (i.e., graphs that can be embedded on a genus-1 surface) with edge weights ±1 [2].Similarly, it has been shown that Max-Cut can be solved in polynomial time if the graph can be embedded on a surface of constant genus g [14,15]. However, the edge-weights have to be restricted to integers whose absolute values are bounded by a polynomial in the input. The central idea of this algorithm is to write the generating function of cuts as a linear combination of 4 g Pfaffians. This algorithm is in fact highly non-trivial to realize: In order to obtain an implementable algorithm, [15] reports on a scheme, which depends on the existence of sufficiently many prime numbers within a given interval, which cannot be guaranteed in ...

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An Improved Fixed-Parameter Algorithm for Max-Cut Parameterized by Crossing Number

Kobayashi

Miyazaki

et al. 2019

Lecture Notes in Computer Science

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The Max-Cut problem is known to be NP-hard on general graphs, while it can be solved in polynomial time on planar graphs. In this paper, we present a fixed-parameter tractable algorithm for the problem on "almost" planar graphs: Given an n-vertex graph and its drawing with k crossings, our algorithm runs in time O(2 k (n + k) 3/2 log(n + k)). Previously, Dahn, Kriege and Mutzel (IWOCA 2018) obtained an algorithm that, given an n-vertex graph and its 1-planar drawing with k crossings, runs in time O(3 k n 3/2 log n). Our result simultaneously improves the running time and removes the 1-planarity restriction.

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Partitioning planar graphs: a fast combinatorial approach for max-cut

Cited by 18 publications

References 37 publications

A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

Maximum Cut Parameterized by Crossing Number

An Improved Fixed-Parameter Algorithm for Max-Cut Parameterized by Crossing Number

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