On minimum cuts and the linear arrangement problem (original) (raw)
Related papers
Discrete Optimization, 2019
Given an undirected graph G = (V, E), a vertex k-cut of G is a vertex subset of V the removing of which disconnects the graph in at least k connected components. Given a graph G and an integer k ≥ 2, the vertex k-cut problem consists in finding a vertex k-cut of G of minimum cardinality. We first prove that the problem is NP-hard for any fixed k ≥ 3. We then present a compact formulation, and an extended formulation from which we derive a column generation and a branching scheme. Extensive computational results prove the effectiveness of the proposed methods.
The minimum size of graphs satisfying cut conditions
Discrete Applied Mathematics, 2018
A graph G of order n satisfies the cut condition (CC) if there are at least |A| edges between any set A ⊂ V (G), |A| ≤ n/2, and its complement A = V (G) \ A. For even n, G satisfies the even cut condition (ECC), if [A, A] contains at least n/2 edges, for every A ⊂ V (G), |A| = n/2. We investigate here the minimum number of edges in a graph G satisfying CC or ECC. A simple counting argument shows that for both cut conditions |E(G)| ≥ n − 1, and the star K 1,n−1 is extremal. Faudree et al. (1999) conjectured that the extremal graphs with maximum degree ∆(G) < n − 1 satisfying ECC have 3n/2 − O(1) edges. Here we prove the tight bound |E(G)| ≥ 3n/2−3, for every graph G with ∆(G) < n−1 and satisfying CC. If G is 2-connected and satisfies ECC, we prove that |E(G)| ≥ 3n/2 − 2 holds and tight, for every even n. We obtain the weaker bound |E(G)| ≥ 5n/4 − 2, for every graph of order n ≡ 0 (mod 4) with ∆(G) < n − 1 and satisfying ECC; meanwhile we conjecture that |E(G)| ≥ 3n/2 − 4 holds, for every even n.
A Branch-Price-and-Cut Algorithm for Packing Cuts in Undirected Graphs
Lecture Notes in Computer Science, 2014
The cut packing problem in an undirected graph is to find a largest cardinality collection of pairwise edge-disjoint cuts. This NPhard problem has several practical applications, and we provide the first experimental study. We propose a branch-price-and-cut algorithm to optimally solve instances from various graph classes, random and from the literature, with up to several hundred vertices. In particular we investigate how complexity results match computational experience and how combinatorial properties help improving the algorithm's performance.
2013
In this paper we study the complexity and approximability of theGc-cut problem. Given a complete undirected graph Kn = (V ; E) with |V | = n, edge weighted byw(vi, vj) ≥ 0 and an undirected cluster graph, Gc = (Vc, Ec), with |V c| = k, ak-cut is a partitionV1, . . . , Vk of V (G) such that Vi 6= ∅ for i = 1, . . . , k. TheGc-cut problem is to compute a k-cut minimizing P (i,j)∈Ec w(Vi, Vj) wherew(Vi, Vj) = P p∈Vi,q∈Vj w(p, q). DenoteGc as cluster graph and its vertices as clusters. We show that theGc-cut problem isNP-hard and even not approximable in the general case and remai ns NP-hard for cluster trees. In particular, we give a complete ch aracterization of hard cases for cluster graphs with at most four vertices by proving that the Gc-cut problem isNP-hard if and only ifGc is isomorphic to2K2. We also identify some cases where the Gc-cut problem is either polynomial or NP-hard. Finally, we propose polynomial approximation resul ts for theGc-cut problem when the edge weights ofG ...
The complexity of the matching-cut problem for planar graphs and other graph classes
Journal of Graph Theory, 2009
The Matching-Cut problem is the problem to decide whether a graph has an edge cut that is also a matching. Previously this problem was studied under the name of the Decomposable Graph Recognition problem, and proved to be N P-complete when restricted to graphs with maximum degree four. In this paper it is shown that the problem remains N P-complete for planar graphs with maximum degree four, answering a question by Patrignani and Pizzonia. It is also shown that the problem is N P-complete for planar graphs with girth five. The reduction is from planar graph 3-colorability and differs from earlier reductions. In addition, for certain graph classes polynomial time algorithms to find matching-cuts are described. These classes include claw-free graphs, co-graphs, and graphs A preliminary version of this
An algorithm for enumerating the near-minimum weight s-t cuts of a graph Ahmet Balcioglu
2000
: We provide an algorithm for enumerating near-minimum weight s-t cuts in directed and undirected graphs, with applications to network interdiction and network reliability. "Near-minimum" means within a factor of l+epilson of the minimum for some epilson > 0. The algorithm is based on recursive inclusion and exclusion of edges in locally minimum-weight cuts identified with a maximum flow algorithm. We prove a polynomial-time complexity result when epilson = 0, and for epilson > 0 we demonstrate good empirical efficiency. The algorithm is programmed in Java, run on a 733 MHz Pentium III computer with 128 megabytes of memory, and tested on a number of graphs. For example, all 274,550 near-minimum cuts within 10% of the minimum weight can be obtained in 74 seconds for a 627 vertex 2,450 edge unweighted graph. All 20,806 near-minimum cuts within 20% of minimum can be enumerated in 61 seconds on the same graph with weights being uniformly distributed integers in the range...
The complexity of multiway cuts (extended abstract)
Proceedings of the …, 1992
Absfrast. In the Multiway Cut problem we are given an edgaweighted graph and a subset of the vertices called termimds, and asked for a minimum weight set of edges that separates each terminal from all the others. when the number k of terminals is two, this is simply the min-cu~msx-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hsrd as sxrn as k = 3, but ctm be solved in polynomial time for planar graphs for any fixed k. The planar problem is NP-hsrd however, if k is not tixad. We also describe a simple approximation slgorithnt for arbkmry graphs that is guaranteed to come within a fsctor of 2-2/k of the optimal cut weight.
On disjunctive cuts for combinatorial optimization
In the successful branch-and-cut approach to combinatorial optimization, linear inequalities are used as cutting planes within a branch-and-bound framework. Although researchers often prefer to use facet-inducing inequalities as cutting planes, good computational results have recently been obtained using disjunctive cuts, which are not guaranteed to be facet-inducing in general. A partial explanation for the success of the disjunctive cuts is given in this paper. It is shown that, for six important combinatorial optimization problems (the clique partitioning, max-cut, acyclic subdigraph, linear ordering, asymmetric travelling salesman and set covering problems), certain facet-inducing inequalities can be obtained by simple disjunctive techniques. New polynomial-time separation algorithms are obtained for these inequalities as a by-product. The disjunctive approach is then compared and contrasted with some other 'general-purpose' frameworks for generating cutting planes and some conclusions are made with respect to the potential and limitations of the disjunctive approach.