On complexity of unconstrained hyperbolic 0-1 programming problems

Single- and multiple-ratio unconstrained hyperbolic 0-1 programming problems are considered. We prove that checking whether these problems have a unique solution is NP-hard. Furthermore, we show that finding the global maximizer of problems with unique solution remains NP-hard. We also discuss complexity of local search and approximability for multiple-ratio problems.     

An approximation algorithm for multidimensional assignment problems minimizing the sum of squared errors

Given a complete k-partite graph G=(V₁,V₂,…,V_k;E) satisfying |V₁|=|V₂|=?=|V_k|=n and weights of all  k-cliques of G, the  k-dimensional assignment problem finds a partition of vertices of G into a set of (pairwise disjoint) n k-cliques that minimizes the sum total of weights of the chosen cliques. In this paper, we consider a case in which the weight of a clique is defined by the sum of given weights of edges induced by the clique. Additionally, we assume that vertices of G are embedded in the d-dimensional space Q^d and a weight of an edge is defined by the square of the Euclidean distance between its two endpoints. We describe that these problem instances arise from a multidimensional Gaussian model of a data-association problem.We propose a second-order cone programming relaxation of the problem and a polynomial time randomized rounding procedure. We show that the expected objective value obtained by our algorithm is bounded by (5/2−3/k) times the optimal value. Our result improves the previously known bound (4−6/k) of the approximation ratio.     

On the complexity of computing treelength

We resolve the computational complexity of determining the treelength of a graph, thereby solving an open problem of Dourisboure and Gavoille, who introduced this parameter, and asked to determine the complexity of recognizing graphs of a bounded treelength Dourisboure and Gavoille (2007) [6]. While recognizing graphs with treelength 1 is easily seen as equivalent to recognizing chordal graphs, which can be done in linear time, the computational complexity of recognizing graphs with treelength 2 was unknown until this result. We show that the problem of determining whether a given graph has a treelength at most k is NP-complete for every fixed k≥2, and use this result to show that treelength in weighted graphs is hard to approximate within a factor smaller than . Additionally, we show that treelength can be computed in time O^∗(1.7549ⁿ) by giving an exact exponential time algorithm for the Chordal Sandwich problem and showing how this algorithm can be used to compute the treelength of a graph.     

A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem

We present a novel Lagrangian method to find good feasible solutions in theoretical and empirical aspects. After investigating the concept of Lagrangian capacity, which is the value of the capacity constraint that Lagrangian relaxation can find an optimal solution, we formally reintroduce Lagrangian capacity suitable to the 0-1 multidimensional knapsack problem and present its new geometric equivalent condition including a new associated property. Based on the property, we propose a new Lagrangian heuristic that finds high-quality feasible solutions of the 0-1 multidimensional knapsack problem. We verify the advantage of the proposed heuristic by experiments. We make comparisons with existing Lagrangian approaches on benchmark data and show that the proposed method performs well on large-scale data.     

On the Shintani Zeta Function for the Space of Pairs of Binary Hermitian Forms

In this paper we determine the principal part of the adjusted zeta function for the space of pairs of binary Hermitian forms.     

On the complexity of the continuous unbounded knapsack problem with uncertain coefficients

In this paper, a linear-time algorithm is developed for the minmax-regret version of the continuous unbounded knapsack problem with n items and uncertain objective function coefficients, where the interval estimates for these coefficients are known. This improves the previously known bound of time for this optimization problem.     

On the complexity of project scheduling to minimize exposed time

We consider project scheduling where the project manager’s objective is to minimize the time from when an adversary discovers the project until the completion of the project. We analyze the complexity of the problem identifying both polynomially solvable and NP-hard versions of the problem. The complexity of the problem is seen to be dependent on the nature of renewable resource constraints, precedence constraints, and the ability to crash activities in the project.     

On the complexity of constructing Golomb Rulers

A Golomb Ruler is a ruler with integer marks where the distances between every two marks are distinct. Golomb Rulers find diverse applications in computer science and electrical engineering. According to our knowledge the computational complexity of problems related to the construction of Golomb Rulers is unknown. We provide natural definitions for problems related to the construction of such rulers. The main contribution of this work is -completeness results for two such decision problems.     

一类(0,1)-矩阵的最大积和式

本文给出了每条线恰有n-2个1的n阶(0.1)-矩阵的最大积和式的组合表达式.     

On the Monte carlo boolean decision tree complexity of read-once formulae

In the boolean decision tree model there is at least a linear gap between the Monte Carlo and the Las Vegas complexity of a function depending on the error probability. We prove for a large class of read-once formulae that this trivial speed-up is the best that a Monte Carlo algorithm can achieve. For every formula F belonging to that class we show that the Monte Carlo complexity of F with two-sided error p is (1 ? 2p)R(F), and with one-sided error p is (1 ? p)R(F), where R(F) denotes the Las Vegas complexity of F. The result follows from a general lower bound that we derive on the Monte Carlo complexity of these formulae. This bound is analogous to the lower bound due to Saks and Wigderson on their Las Vegas complexity.     

On the complexity of some subgraph problems

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new NP-completeness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O(nlogn)) time algorithm for such problems in graphs with vertex degree bounded by 3.     

On the preservation of absolutely continuous spectrum for Schrödinger operators

We present general principles for the preservation of a.c. spectrum under weak perturbations. The Schrödinger operators on the strip and on the Caley tree (Bethe lattice) are considered.     

On the complexity of the dominating induced matching problem in hereditary classes of graphs

The dominating induced matching problem, also known as efficient edge domination, is the problem of determining whether a graph has an induced matching that dominates every edge of the graph. This problem is known to be NP-complete. We study the computational complexity of the problem in special graph classes. In the present paper, we identify a critical class for this problem (i.e., a class lying on a “boundary” separating difficult instances of the problem from polynomially solvable ones) and derive a number of polynomial-time results. In particular, we develop polynomial-time algorithms to solve the problem for claw-free graphs and convex graphs.     

The complexity of the word-problem for finite matrix rings

We analyze the so-called word-problem for , the ring of matrices over . We prove that the term-equivalence problem for the semigroup (and so for the ring) is coNP-complete.

     

On the complexity of a class of combinatorial optimization problems with uncertainty

We consider a robust (minmax-regret) version of the problem of selecting p elements of minimum total weight out of a set of m elements with uncertainty in weights of the elements. We present a polynomial algorithm with the order of complexity O((min {p,m-p})² m) for the case where uncertainty is represented by means of interval estimates for the weights. We show that the problem is NP-hard in the case of an arbitrary finite set of possible scenarios, even if there are only two possible scenarios. This is the first known example of a robust combinatorial optimization problem that is NP-hard in the case of scenario-represented uncertainty but is polynomially solvable in the case of the interval representation of uncertainty. Received: July 1998 / Accepted: May 2000?Published online March 22, 2001     

On the complexity of the independent set problem in triangle graphs

We consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G−I, there is a vertex w∈I such that {u,v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NP-complete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P=NP.     

On the Absolutely Continuous Spectrum of Dirac Operator

Abstract

We prove that the massless Dirac operator in ?³ with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.     

On the complexity of crossings in permutations

We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π^∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. This rank aggregation problem can be phrased as a one-sided two-layer crossing minimization problem for a series of bipartite graphs or for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. We contribute the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. As our results, we correct the construction from [C. Dwork, R. Kumar, M. Noar, D. Sivakumar, Rank aggregation methods for the Web, Proc. WWW10 (2001) 613-622] and prove the NP-hardness of the common crossing minimization problem for k=4 permutations. Then we establish a 2−2/k-approximation, improving the previous factor of 2. The max version is shown NP-hard for every k≥4, and there is a 2-approximation. Both approximations are optimal, if the common permutation is selected from the given ones. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.