Min-cut Problem Research Articles

Complexity of the min–max (regret) versions of min cut problems

This paper investigates the complexity of the min–max and min–max regret versions of the min s – t cut and min cut problems. Even if the underlying problems are closely related and both polynomial, the complexities of their min–max and min–max regret versions, for a constant number of scenarios, are quite contrasted since they are respectively strongly NP-hard and polynomial. However, for a non-constant number of scenarios, these versions become strongly NP-hard for both problems. In the interval scenario case, min–max versions are trivially polynomial. Moreover, for min–max regret versions, we obtain the same contrasted results as for a constant number of scenarios: min–max regret min s – t cut is strongly NP-hard whereas min–max regret min cut is polynomial.

Solving the Bi-Objective Maximum-Flow Network-Interdiction Problem

We describe a new algorithm for computing the efficient frontier of the “bi-objective maximum-flow network-interdiction problem.” In this problem, an “interdictor” seeks to interdict (destroy) a set of arcs in a capacitated network that are Pareto-optimal with respect to two objectives, minimizing total interdiction cost and minimizing maximum flow. The algorithm identifies these solutions through a sequence of single-objective problems solved using Lagrangian relaxation and a specialized branch-and-bound algorithm. The Lagrangian problems are simply max-flow min-cut problems, while the branch-and-bound procedure partially enumerates s-t cuts. Computational tests reveal the new algorithm to be one to two orders of magnitude faster than an algorithm that replaces the specialized branch-and-bound algorithm with a standard integer-programming solver.

Improving image segmentation quality through effective region merging using a hierarchical social metaheuristic

This paper proposes a new evolutionary region merging method in order to efficiently improve segmentation quality results. Our approach starts from an oversegmented image, which is obtained by applying a standard morphological watershed transformation on the original image. Next, each resulting region is represented by its centroid. The oversegmented image is described by a simplified undirected weighted graph, where each node represents one region and weighted edges measure the dissimilarity between pairs of regions (adjacent and non-adjacent) according to their intensities, spatial locations and original sizes. Finally, the resulting graph is iteratively partitioned in a hierarchical fashion into two subgraphs, corresponding to the two most significant components of the actual image, until a termination condition is met. This graph-partitioning task is solved by a variant of the min-cut problem (normalized cut) using a hierarchical social (HS) metaheuristic. We have efficiently applied the proposed approach to brightness segmentation on different standard test images, with good visual and objective segmentation quality results.

Railway Delay Management

We consider delay management in railway systems. Given delayed trains, we want to find a waiting policy for the connecting trains minimizing the weighted total passenger delay. If there is a single delayed train and passengers transfer at most twice along fixed routes, or if the railway network has a tree structure, the problem can be solved by reduction to min-cut problems. For delayed passenger flows on a railway network with a path structure, the problem can be solved to optimality by dynamic programming. If passengers are allowed to adapt their route to the waiting policy, the decision problem is strongly NP-complete.

Optimal capacity expansion for multi-product, multi-machine manufacturing systems with stochastic demand

We consider a discrete-time capacity expansion problem involving multiple product families, multiple machine types, and non-stationary stochastic demand. Capacity expansion decisions are made to strike an optimal balance between investment costs and lost sales costs. Motivated by current practices in the semiconductor and other high-tech industries, we assume that only minimal amounts of finished-goods inventories are held, due to the risk of obsolescence. We assume that when capacity is in short supply, management desires to ensure that a minimal service level for all product families is obtained. Our approach uses a novel assumption that demand can be approximated by a distribution whose support is a collection of rays emanating from a point and contained in real multi-dimensional space. These assumptions allow us to solve the problem as a max-flow, min-cut problem. Computational experiments show that large problems can be solved efficiently.

Parametric analysis of overall min-cuts and applications in undirected networks

The overall min-cut problem in a capacitated undirected network is well known. Recently Stoer and Wagner gave an elegant algorithm for finding such a cut. In this paper we present a parametric analysis of such a cut where the capacity of an arc { i, j} in the network is given by min{ b ij , λ}, where λ is a parameter ranging from 0 to ∞. Letting function v( λ) denote the min-cut capacity, we develop an algorithm to describe v( λ) which involves at most n applications of Stoer and Wagner scheme, where n denotes the number of nodes in the network. We use v( λ) to determine an overall min-cut for multiroute flows as defined by Kishimoto. Such multi-route flows have interesting applications in communication networks.

Min-cut partitioning with functional replication for technology-mapped circuits using minimum area overhead

Logic replication is known to be an effective technique to reduce the number of cut nets in partitioned circuits. A new replication model called functional replication is particularly useful for partitioning technology-mapped circuits. Functional replication differs from traditional replication because it considers the functional dependency of the different output signals of a logic cell on its input signals. Functional replication can lead to a higher reduction in the number of cut nets than traditional replication. In this paper, we give the first theoretical treatment of the min-cut partitioning problem with functional replication. We present a novel two-phase algorithm to compute a min-cut bipartition of a technology-mapped circuit with functional replication using a minimum amount of area overhead. Additionally, we show that our algorithm can be applied to improve the solution produced by any area-constrained functional replication partitioning heuristic.

Optimizing Component Interaction

Interactions among two or more components may consist of passing of (large) data, a continuation, or a remote method call. Optimizing or minimizing the cost of such interactions is a fundamental problem in generating efficient code for component-oriented programs. This paper lays out a foundation for modeling and optimizing the interaction between components, with the goal of optimizing the entire assembled system. Our approach consists of automatically choosing, among a number of potential candidate, implementation strategies for choice of data structures, communication mechanisms, data placement, etc., components may use during their life time. By building a high-level optimizer that has dynamic knowledge of how components will be used in a system, we may achieve performance improvements of an order of magnitude or more. In our approach, a component writer provides the alternative implementation strategies, and our analysis will recommend or use the best strategies among the alternative during execution. One way to specify different alternatives is to recognize the characterization of problem domain. A more efficient code may be generated if we recognize the dynamic nature of the component usage. For instance, at one point during execution a component may choose one alternative strategy that is different from the choice it makes at another program point. In general, information needed to decide when to use one implementation over another may not be available to a component writer, but it is available when components are put to use in a system (for instance, the knowledge about objects are allocated, say for locality analysis, is only available at runtime). Sometimes choosing among the alternative implementations can especially tricky because the different choices are not independent of each other. This means an implementation choice that works well for one component may hurt the performance of another, This problem is particularly troublesome for component writers, since they have an extremely clear view of one component but cannot see the rest. For example, programs which work on the net will generally do better with ASCII strings, whereas programs that work heavily with databases might do better with EBCDIC -- and programs which do both (or systems in which the two kinds of program coexist) have to choose between them, or pay the cost of translation. Similarly, distributed objects that interact extensively should be on the same machine, although it does not matter on which machine they reside. Many of these optimizations can be reduced to a graph-cutting problem in which nodes in the graph correspond to objects and edges and its weights correspond to their interact. To optimize the interaction, we need to find a min-cut on the graph, and in general the min-cut problem is NP hard. We give heuristics that often give an almost linear time performance on actual instances of the problem. We present a vision for how future high-level optimizations that can optimize programs as they are written today, eliminating some of the inefficiencies caused by the high level component based programming. In the full paper we present theoretical analysis and case studies that show the feasibility of this approach, however much research remains to be done to achieve the promise.

A Simple Algorithm for the Planar Multiway Cut Problem

The traditional min-cut problem involves finding a cut with minimum weight between two specified vertices. The planar multiway cut problem is a NP-hard generalization of the min-cut problem. It involves separating a weighted planar graph with k specified vertices into k components such that the total weight between the components is minimized. This problem has important applications in computer science, engineering, and management science. In this study, we developed a very simple algorithm with time complexity O((k−32)k−1·(n−k)2k−4·[nk−32k2+12k]·log(n−k)). Our algorithm is based on some simple theorems that characterize the structure of the k-way cut. It is also better than the best known algorithm with time complexity O(4kk·n2k−1·logn) for the planar multiway cut problem.

Compact Representations of Cuts

Consider the ${n}\choose{2}$ (or $O(n^{2})$) min-cut problems on a graph with n nodes and nonnegative edge weights. Gomory and Hu [J. Soc. Indust. Appl. Math., 9 (1961), pp. 551--570] showed (essentially) that there are at most n-1 different min-cuts. They also described a compact structure (the flow equivalent tree) of size O(n) with the following property: for any pair of nodes, the value of a min-cut can be obtained from this structure. Furthermore, they showed how this structure can be found by solving only n-1 min-cut problems. This paper contains generalizations of these results. For example, consider a k-terminal cut problem on a graph: for a given set of k nodes, delete a minimum weight set of edges (called a k-cut) so that each of the k nodes is in a different component. There are ${n}\choose{k}$ (or O(nk)) such problems. Hassin [Math. Oper. Res., 13 (1988), pp. 535--542] showed (essentially) that there are at most ${n-1}\choose{k-1}$ (or O(nk-1)) different min k-cuts. We describe a compact structure of size O(nk-1) with the following property: for any k nodes, the value of a min k-cut can be obtained from this structure. We also show how this structure can be found by solving only ${n-1}\choose{k-1}$ k-terminal cut problems. This work builds upon the results of Hassin [Math. Oper. Res., 13 (1988), pp. 535--542], [Oper. Res. Lett., 9 (1990), pp. 315--318], and [Lecture Notes in Comput. Sci. 450, 1990, pp. 228--299].

On the Power of Logic Resynthesis

A linear arrangement problem, called the minmax mincut problem, emerging from circuit design is investigated. Its input is a series-parallel directed hypergraph (SPDH), and the output is a linear arrangement (and a layout). The primary objective is to minimize the longest path, and the secondary objective is to minimize the cutwidth. It is shown that cutwidth D, subject to longest path minimization, is affected by two terms: pattern number k and balancing number m. Also, k and m are both lower bounds on the cutwidth. An algorithm, running in linear time, produces layouts with cutwidths $D \leq 2(k+m)$. There exist examples with $k=\Omega (N)$, where N is the number of vertices; however, m is always O(log N). We show that every SPDH, after $local\ logic\ resynthesis$ (specifically, after reordering the serial paths), can be linearly placed with cutwidth D=O(log N). Simultaneously, its dual SPDH can be linearly placed with the same vertex order and with cutwidth D=O(log N). Therefore, after local resynthesis the area can be reduced by a factor of N/log N. Application to gate-matrix layout style is demonstrated.

A Relaxation of a Min-Cut Problem in an Anisotropic Continuous Network

Strang [18] introduced optimization problems on a Euclidean domain which are closely related with problems in mechanics and noted that the problems are regarded as continuous versions of famous max-flow and min-cut problems. In [15] we generalized the problems and called the generalized problems max-flow and min-cut problems of Strang's type. In this paper we formulate a relaxed version of the min-cut problem of Strang's type and prove the existence of optimal solutions under some suitable conditions. The conditions are essential. In fact, there is an example of the relaxed version which has no optimal solutions if the conditions are not fulfilled. We give such an example in the final section.

All-Pairs Min-Cut in Sparse Networks

Algorithms are presented for the all-pairs min-cut problem in bounded tree-width, planar, and sparse networks. The approach used is to preprocess the inputn-vertex network so that afterward, the value of a min-cut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute min-cuts subsequently. In particular, after anO(nlogn) preprocessing of a bounded tree-width network, it is possible to find the value of a min-cut between any two vertices in constant time. This implies that for such networks the all-pairs min-cut problem can be solved in timeO(n2). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, γ, of the input network. The parameter γ varies between 1 and Θ(n); the algorithms perform well when γ=o(n). The value of a min-cut can be found in timeO(n+γ2logγ) and all-pairs min-cut can be solved in timeO(n2+γ4logγ) for sparse networks. The corresponding running times for planar networks areO(n+γlogγ) andO(n2+γ3logγ), respectively. The latter bounds depend on a result of independent interest; outerplanar networks have small “mimicking” networks that are also outerplanar.

Optimal min-area min-cut replication in partitioned circuits

Previous results show that node replication can be used to reduce the number of cut edges substantially in a partitioned circuit. The node replication approach is particularly useful for fully utilizing pin-limited devices such as multiple field-programmable gate array. Hwang and El Gamal [1992, 1995] formulated the min-cut replication problem, which is to determine min-cut replication sets for the components of a k-way partition such that the cut size of the partition is minimized after the replication. They gave an optimal algorithm for finding min-cut replication sets for a k-way partitioned digraph. However their optimal min-cut replication algorithm does not guarantee min-cut replication sets of minimum sizes. Furthermore, their algorithm is not optimal for hypergraphs. In this paper, we optimally solve the min-area min-cut replication problem on digraphs, which is to find min-cut replication sets with the minimum sizes. More important, we give an optimal solution to the hypergraph min-area min-cut replication problem using a much smaller flow network model. We implemented our algorithms in a package called Hyper-MAMC, and interfaced Hyper-MAMC to the TAPIR package. We compared the replication results by Hyper-MAMC with those obtained by MC-Rep in the TAPIR package on the exact same initial partitions of a set of MCNC Partition93 benchmark circuits. On average, Hyper-MAMC produces 57.3% fewer cut nets and runs much faster than MC-Rep.

Min-cut partitioning on underlying tree and graph structures

We consider two generalizations of the min-cut partitioning problem where the nodes of a circuit C are to be mapped to the vertices of an underlying graph G, and the cost function to be minimized is the cost of associating the nets of C with the edges of G. Let P be the number of pins, the the number of nodes of G, and d be the maximum number of cells on a net of C. In the first problem the graph G is a tree T. An iterative improvement heuristic is given (Vijayan, 1991) with O(P.t/sup 3/) time per pass. Our proposed heuristic guarantees identical solutions in O(P.t.min(d,t)) time per pass. The second problem is defined on any graph G. The standard iterative improvement heuristic requires O(P t/sup 4/) time per pass, but our proposed approach guarantees O(P.t.min(d,t)) time per pass. The problems find applications in VLSI physical design and in distributed systems.

Generalizing the all-pairs min cut problem

The all-pairs min cut (APMC) problem on a nonnegative edge-weighted graph is to find, for each pair of nodes, a min cut that separates the pair. Gomory and Hu (1961) presented a structural characterization of collections of cuts that solve the APMC problem. We show how the APMC problem can be generalized to matroids and we present several theorems that characterize the structure of solutions to this more general problem. The result of Gomory and Hu is a special case of one of these theorems. In particular, we find that the APMC problem is a matroid optimization problem.

BIPARTITIONING INTO OVERLAPPING SETS

We consider a generalization of the min-cut partitioning problem where we partition a graph G=(V,E) into two sets V1 and V2 such that |V1∩V2|≤d, d<|V|, and such that |{(u, v)|u∈V1−V2, v∈V2−V1}| is minimized. The problem is trivially solvable using flow techniques for any fixed d, but we show that it is NP-hard for integer values of d. It remains NP-hard if we impose restrictions on the size of V1, i.e., |V1|=k, k∈Z+. The latter problem variation may apply in VLSI layout and hypertext partitioning. We present polynomial time algorithms for the special cases of solid grids and series-parallel graphs. Series-parallel graphs find applications in hypertext partitioning whereas grid graphs model the mapping of a class of Partial Differential Equation computations into parallel machines.

The All-Pairs Min Cut Problem and the Minimum Cycle Basis Problem on Planar Graphs

The all-pairs min cut (APMC) problem on a nonnegative weighted graph is the problem of finding, for every pair of nodes, the min cut separating the pair. It is shown that on planar graphs, the APMC problem is equivalent to another problem, the minimum cycle basis (MCB)problem, on the dual graph. This is shown by characterizing the structure of MCBs on planar graphs in several ways. This leads to a new algorithm for solving both of these problems on planar graphs. The complexity of this algorithm equals that of the best algorithm for either problem, but is simpler.

Examples of max-flow and min-cut problems with duality gaps in continuous networks

Strang (Mathematical Programming 26, 1983) gave a method to establish a max-flow min-cut theorem in a domain of a Euclidean space. The method can be applied also to max-flow min-cut problems defined by Iri (Survey of Mathematical Programming, North-Holland, 1979) whenever the capacity functions of max-flow problems are bounded and continuous. This paper deals with max-flow min-cut problems of Strang and Iri with unbounded or noncontinuous capacity functions. It is proved that, in such problems, max-flow min-cut theorems may fail to hold.

Min-cut clustering

We describe a decomposition framework and a column generation scheme for solving a min-cut clustering problem. The subproblem to generate additional columns is itself an NP-hard mixed integer progr...