Aspects of a multivariate complexity analysis for Rectangle Tiling

We initiate a parameterized complexity study of the NP-hard problem to tile a positive integer matrix with rectangles, keeping the number of tiles and their maximum weight small. We show that the problem remains NP-hard even for binary matrices only using 1×1 and 2×2-squares as tiles and provide insight into the influence of naturally occurring parameters on the problem’s complexity.     

NP组合优化近似计算的难度

本文扼要介绍近二十年来在组合优化可近似性的研究方面所取得的进展,包括不可近似性的证明,对组合优化问题用逻辑描述的语法分类及其可近似性．     

The robust shortest path problem in series-parallel multidigraphs with interval data

In this paper the robust shortest path problem in edge series-parallel multidigraphs with interval costs is examined. The maximal regret criterion is applied to calculate the optimal solution. It is shown that this problem is NP-hard. A pseudopolynomial algorithm for the studied problem is constructed.     

The constrained minimum weighted sum of job completion times problem

We consider the problem of minimizing the weighted sum of job completion times on a single machine (subject to certain job weights) with an additional side constraint on the weighted sum of job completion times (with respect to different job weights). This problem is NP-hard, and we provide a polynomial time approximation scheme for this problem. Our method is based on Lagrangian relaxation mixed with carefully guessing the positions of certain jobs in the schedule. An earlier version of this paper appeared in the Proceedings of the 10th International IPCO Conference.     

The complexity of minimum difference cover

The complexity of searching minimum difference covers, both in Z⁺ and in Z_n, is studied. We prove that these two optimization problems are NP-hard. To obtain this result, we characterize those sets—called extrema—having themselves plus zero as minimum difference cover. Such a combinatorial characterization enables us to show that testing whether sets are not extrema, both in Z⁺ and in Z_n, is NP-complete. However, for these two decision problems we exhibit pseudo-polynomial time algorithms.     

The node-edge weighted 2-edge connected subgraph problem: Linear relaxation, facets and separation

Let G=(V,E) be a undirected k-edge connected graph with weights c_e on edges and w_v on nodes. The minimum 2-edge connected subgraph problem, 2ECSP for short, is to find a 2-edge connected subgraph of G, of minimum total weight. The 2ECSP generalizes the well-known Steiner 2-edge connected subgraph problem. In this paper we study the convex hull of the incidence vectors corresponding to feasible solutions of 2ECSP. First, a natural integer programming formulation is given and it is shown that its linear relaxation is not sufficient to describe the polytope associated with 2ECSP even when G is series-parallel. Then, we introduce two families of new valid inequalities and we give sufficient conditions for them to be facet-defining. Later, we concentrate on the separation problem. We find polynomial time algorithms to solve the separation of important subclasses of the introduced inequalities, concluding that the separation of the new inequalities, when G is series-parallel, is polynomially solvable.     

The complete optimal stars-clustering-tree problem

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G=(V,E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a “group broadcast” property, “group fault tolerance” and “group privacy”. We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard.     

The complexity of detecting fixed-density clusters

We study the complexity of finding a subgraph of a certain size and a certain density, where density is measured by the average degree. Let γ:N→Q₊ be any density function, i.e., γ is computable in polynomial time and satisfies γ(k)?k-1 for all k∈N. Then γ-CLUSTER is the problem of deciding, given an undirected graph G and a natural number k, whether there is a subgraph of G on k vertices that has average degree at least γ(k). For γ(k)=k-1, this problem is the same as the well-known CLIQUE problem, and thus NP-complete. In contrast to this, the problem is known to be solvable in polynomial time for γ(k)=2. We ask for the possible functions γ such that γ-CLUSTER remains NP-complete or becomes solvable in polynomial time. We show a rather sharp boundary: γ CLUSTER is NP-complete if γ=2+Ω(1/k^1-ε) for some ε>0 and has a polynomial-time algorithm for γ=2+O(1/k). The algorithm also shows that for γ=2+O(1/k^1-o(1)), γ-CLUSTER is solvable in subexponential time 2^no(1).     

The Wiener maximum quadratic assignment problem

We investigate a special case of the maximum quadratic assignment problem where one matrix is a product matrix and the other matrix is the distance matrix of a one-dimensional point set. We show that this special case, which we call the Wiener maximum quadratic assignment problem, is NP-hard in the ordinary sense and solvable in pseudo-polynomial time.Our approach also yields a polynomial time solution for the following problem from chemical graph theory: find a tree that maximizes the Wiener index among all trees with a prescribed degree sequence. This settles an open problem from the literature.     

Approximation algorithms for constructing some required structures in digraphs

We consider a new problem of constructing some required structures in digraphs, where all arcs installed in such required structures are supposed to be cut from some pieces of a specific material of length L. Formally, we consider the model: a digraph D = (V, A; w), a structure S and a specific material of length L, where w: A → R⁺, we are asked to construct a subdigraph D′ from D, having the structure S, such that each arc in D′ is constructed by a part of a piece or/and some whole pieces of such a specific material, the objective is to minimize the number of pieces of such a specific material to construct all arcs in D′.     

Inapproximability of finding maximum hidden sets on polygons and terrains

How many people can hide in a given terrain, without any two of them seeing each other? We are interested in finding the precise number and an optimal placement of people to be hidden, given a terrain with n vertices. In this paper, we show that this is not at all easy: The problem of placing a maximum number of hiding people is almost as hard to approximate as the problem, i.e., it cannot be approximated by any polynomial-time algorithm with an approximation ratio of n for some >0, unless P=NP. This is already true for a simple polygon with holes (instead of a terrain). If we do not allow holes in the polygon, we show that there is a constant >0 such that the problem cannot be approximated with an approximation ratio of 1+.     

The complexity of a minimum reload cost diameter problem

We consider the minimum diameter spanning tree problem under the reload cost model which has been introduced by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: Minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85]. In this model an undirected edge-coloured graph G is given, together with a nonnegative symmetrical integer matrix R specifying the costs of changing from a colour to another one. The reload cost of a path in G arises at its internal nodes, when passing from the colour of one incident edge to the colour of the other. We prove that, unless P=NP, the problem of finding a spanning tree of G having a minimum diameter with respect to reload costs, when restricted to graphs with maximum degree 4, cannot be approximated within any constant α<2 if the reload costs are unrestricted, and cannot be approximated within any constant β<5/3 if the reload costs satisfy the triangle inequality. This solves a problem left open by Wirth and Steffan [H.-C. Wirth, J. Steffan, Reload cost problems: minimum diameter spanning tree, Discrete Appl. Math. 113 (2001) 73-85].     

The combinatorics of pivoting for the maximum weight clique

In this paper we prove the equivalence between a pivoting-based heuristic (PBH) for the maximum weight clique problem and a combinatorial greedy heuristic. It is also proved that PBH always returns a local solution although this is not always guaranteed for Lemke's method, on which PBH is based.     

Solving the maximum edge weight clique problem via unconstrained quadratic programming

The unconstrained quadratic binary program (UQP) is proving to be a successful modeling and solution framework for a variety of combinatorial optimization problems. Experience reported in the literature with several problem classes has demonstrated that this approach works surprisingly well in terms of solution quality and computational times, often rivaling and sometimes surpassing more traditional methods. In this paper we report on the application of UQP to the maximum edge-weighted clique problem. Computational experience is reported illustrating the attractiveness of the approach.     

The complexity of recognizing linear systems with certain integrality properties

Let A be a 0 − 1 matrix with precisely two 1’s in each column and let 1 be the all-one vector. We show that the problems of deciding whether the linear system (1) defines an integral polyhedron, (2) is totally dual integral (TDI), and (3) box-totally dual integral (box-TDI) are all co-NP-complete, thereby confirming the conjecture on NP-hardness of recognizing TDI systems made by Edmonds and Giles in 1984. Supported in part by NSA grant H98230-05-1-0081 and NSF grants DMS-0556091 and ITR-0326387. Supported in part by the Research Grants Council of Hong Kong and Seed Funding for Basic Research of HKU.     

The maximum balanced subgraph of a signed graph: Applications and solution approaches

The Maximum Balanced Subgraph Problem (MBSP) is the problem of finding a subgraph of a signed graph that is balanced and maximizes the cardinality of its vertex set. This paper is the first one to discuss applications of the MBSP arising in three different research areas: the detection of embedded structures, portfolio analysis in risk management and community structure. The efficient solution of the MBSP is also in the focus of this paper. We discuss pre-processing routines and heuristic solution approaches to the problem. a GRASP metaheuristic is developed and improved versions of a greedy heuristic are discussed. Extensive computational experiments are carried out on a set of instances from the applications previously mentioned as well as on a set of random instances.     

Minimizing the weighted number of late jobs in UET open shops

We consider a two-machine open shop problem where the jobs have release dates and due dates, and where all single operations have unit processing times. The goal is to minimize the weighted number of late jobs. We derive a polynomial time algorithm for this problem, thereby answering an open question posed in a recent paper by Brucker et al.This research was supported by the Christian Doppler Laboratorium für Diskrete Optimierung.     

A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles

The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n ^2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n ^2K+3) algorithm for testing membership inG This work was supported by NSF grant ENG75-00568 to Cornell University. Some of the results of this paper have appeared in Hsu's unpublished Ph.D. dissertation [9].     

On the complexity of submodular function minimisation on diamonds

Let (L;?,?) be a finite lattice and let n be a positive integer. A function f:Lⁿ→R is said to be submodular if for all . In this article we study submodular functions when L is a diamond. Given oracle access to f we are interested in finding such that as efficiently as possible. We establish

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  a min–max theorem, which states that the minimum of the submodular function is equal to the maximum of a certain function defined over a certain polyhedron; and

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  a good characterisation of the minimisation problem, i.e., we show that given an oracle for computing a submodular f:Ln→Z and an integer m such that , there is a proof of this fact which can be verified in time polynomial in n and ; and

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  a pseudopolynomial-time algorithm for the minimisation problem, i.e., given an oracle for computing a submodular f:Ln→Z one can find in time bounded by a polynomial in n and .