On the complexity of recognizing a class of generalized networks

The problem of determining whether a given linear programming problem can be converted to a generalized network flow problem having no unit-weight cycles is shown to be NP-hard. The same argument also shows that the problem of determining whether a gain matroid is bicircular is NP-hard.     

On the communication complexity oft-intersection problems in generalized Boolean algebras

We consider the following game: Two players independently choose a chain in a partially ordered set. How many bits of information have to be communicated until at least one of the players knows whether the chains have exactlyt elements in common? This model generalizes thet-intersection problem for subsets of a finite set. We establish the deterministic communication complexity in general. For the special cases of generalized Boolean algebras, we present improved nondeterministic and probabilistic protocols that are of optimal order of complexity for classes with fixed widthq.     

On the Maximum Matching Graph of a Graph

1IntroductionMatchingtheory,aswellastheassignmentprobleminlinearprogramming,hasawiderangeofapplicationinthetheoryandpracticeofoperationsresearch.Bysomepracticalmotivations,e.g.,forfindingalloptimalsolutions,peoplewanttoknowthestructurepropertiesofallmaximummatchingsofagraphG.InthecasethatGhasperfectmatchings,extensiveworkhasbeendoneontheso-calledperfectmatChinggrape(or1-factorgraph),inwhichtwoperfectmatchingsMIandMZaresaidtobeadjacentifMI~MZ@E(C)whereCisanMI-alternatingcycleofG.Therewer…     

Parameterized complexity of even/odd subgraph problems

We study the parameterized complexity of the problems of determining whether a graph contains a k-edge subgraph (k-vertex induced subgraph) that is a Π-graph for Π-graphs being one of the following four classes of graphs: Eulerian graphs, even graphs, odd graphs, and connected odd graphs. We also consider the parameterized complexity of their parametric dual problems.For these sixteen problems, we show that eight of them are fixed parameter tractable and four are W[1]-hard. Our main techniques are the color-coding method of Alon, Yuster and Zwick, and the random separation method of Cai, Chan and Chan.     

Computational complexity in additive hedonic games

We investigate the computational complexity of several decision problems in hedonic coalition formation games and demonstrate that attaining stability in such games remains NP-hard even when they are additive. Precisely, we prove that when either core stability or strict core stability is under consideration, the existence problem of a stable coalition structure is NP-hard in the strong sense. Furthermore, the corresponding decision problems with respect to the existence of a Nash stable coalition structure and of an individually stable coalition structure turn out to be NP-complete in the strong sense.     

A probabilistic analysis of some greedy cardinality matching algorithms

This paper deals with the expected cardinality of greedy matchings in random graphs. Different versions of the greedy heuristic for the cardinality matching problem are considered. Experimental data and some theoretical results are reported.     

On the computational complexity of partial covers of Theta graphs

By use of elementary geometric arguments we prove the existence of a special integral solution of a certain system of linear equations. The existence of such a solution then yields the NP-hardness of the decision problem on the existence of locally injective homomorphisms to Theta graphs with three distinct odd path lengths.     

On the solution and complexity of a generalized linear complementarity problem

We introduce some sufficient conditions under which a generalized linear complementarity problem (GLCP) can be solved as a pure linear complementarity problem. We also establish that the GLCP is in general a NP-Hard problem.Support of this work has been provided by the Instituto Nacional de Investigação Cientifica de Portugal (INIC) under contract 89/EXA/5.     

On the complexity of decomposing matrices arising in satellite communication

Decomposing a square matrix into a weighted sum of permutation matrices, such that the sum of the weights becomes minimal, is NP-hard. This result justifies the heuristic approach to this problem proposed by several authors. An application of this problem arises from intercity communication via transmission satellites.     

On the Intersection Number of Matchings and Minimum Weight Perfect Matchings of Multicolored Point Sets

Let P and Q be disjoint point sets with 2r and 2s elements respectively, and M₁ and M₂ be their minimum weight perfect matchings (with respect to edge lengths). We prove that the edges of M₁ and M₂ intersect at most |M₁|+|M₂|−1 times. This bound is tight. We also prove that P and Q have perfect matchings (not necessarily of minimum weight) such that their edges intersect at most min{r,s} times. This bound is also sharp. Supported by PAPIIT(UNAM) of México, Proyecto IN110802 Supported by FAI-UASLP and by CONACYT of México, Proyecto 32168-E Supported by CONACYT of México, Proyecto 37540-A     

The computational complexity of the elimination problem in generalized sports competitions

Consider a sports competition among various teams playing against each other in pairs (matches) according to a previously determined schedule. At some stage of the competition one may ask whether a particular team still has a (theoretical) chance to win the competition. The computational complexity of this question depends on the way scores are allocated according to the outcome of a match. For competitions with at most 3 different outcomes of a match the complexity is already known. In practice there are many competitions in which more than 3 outcomes are possible. We determine the complexity of the above problem for competitions with an arbitrary number of different outcomes. Our model also includes competitions that are asymmetric in the sense that away playing teams possibly receive other scores than home playing teams.     

On the information-based complexity of stochastic programming

Existing complexity results in stochastic linear programming using the Turing model depend only on problem dimensionality. We apply techniques from the information-based complexity literature to show that the smoothness of the recourse function is just as important. We derive approximation error bounds for the recourse function of two-stage stochastic linear programs and show that their worst case is exponential and depends on the solution tolerance, the dimensionality of the uncertain parameters and the smoothness of the recourse function.     

On the complexity of approximating the maximal inscribed ellipsoid for a polytope

We give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope.Research supported by NSF Grant DMS-8706133.Research supported by NSF Grant DMS-8904406.     

On the complexity of the exchange algorithm for minimax optimization problems

We present an exchange algorithm for the solution of minimax optimization problems involving convex functions. For a certain class of functions, the complexity of this algorithm is shown to be either linear in the number of functions, or at least squared in that number.     

Expected time complexity of the auction algorithm and the push relabel algorithm for maximum bipartite matching on random graphs

In this paper we analyze the expected time complexity of the auction algorithm for the matching problem on random bipartite graphs. We first prove that if for every non‐maximum matching on graph G there exist an augmenting path with a length of at most 2l + 1 then the auction algorithm converges after N ? l iterations at most. Then, we prove that the expected time complexity of the auction algorithm for bipartite matching on random graphs with edge probability and c > 1 is w.h.p. This time complexity is equal to other augmenting path algorithms such as the HK algorithm. Furthermore, we show that the algorithm can be implemented on parallel machines with processors and shared memory with an expected time complexity of . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 384–395, 2016     

On the complexity of locating linear facilities in the plane

We consider the computational complexity of linear facility location problems in the plane, i.e., given n demand points, one wishes to find r lines so as to minimize a certain objective-function reflecting the need of the points to be close to the lines. It is shown that it is NP-hard to find r lines so as to minimize any isotone function of the distances between given points and their respective nearest lines. The proofs establish NP-hardness in the strong sense. The results also apply to the situation where the demand is represented by r lines and the facilities by n single points.     

On the complexity of cell flipping in permutation diagrams and multiprocessor scheduling problems

Permutation diagrams have been used in circuit design to model a set of single point nets crossing a channel, where the minimum number of layers needed to realize the diagram equals the clique number ω(G) of its permutation graph, the value of which can be calculated in O(nlogn) time. We consider a generalization of this model motivated by “standard cell” technology in which the numbers on each side of the channel are partitioned into consecutive subsequences, or cells, each of which can be left unchanged or flipped (i.e., reversed). We ask, for what choice of flippings will the resulting clique number be minimum or maximum. We show that when one side of the channel is fixed (no flipping), an optimal flipping for the other side can be found in O(nlogn) time for the maximum clique number, and that when both sides are free this can be solved in O(n²) time. We also prove NP-completeness of finding a flipping that gives a minimum clique number, even when one side of the channel is fixed, and even when the size of the cells is restricted to be less than a small constant. Moreover, since the complement of a permutation graph is also a permutation graph, the same complexity results hold for the stable set (independence) number. In the process of the NP-completeness proof we also prove NP-completeness of a restricted variant of a scheduling problem. This new NP-completeness result may be of independent interest.     

Confidence intervals in the solution of stochastic integer linear programming problems

A method is proposed to estimate confidence intervals for the solution of integer linear programming (ILP) problems where the technological coefficients matrix and the resource vector are made up of random variables whose distribution laws are unknown and only a sample of their values is available. This method, based on the theory of order statistics, only requires knowledge of the solution of the relaxed integer linear programming (RILP) problems which correspond to the sampled random parameters. The confidence intervals obtained in this way have proved to be more accurate than those estimated by the current methods which use the integer solutions of the sampled ILP problems.This research was partially supported by the Italian National Research Council contract no. 82.001 14.93 (P.F. Trasporti).     

On the complexity of finding paths in a two‐dimensional domain I: Shortest paths

The computational complexity of finding a shortest path in a two‐dimensional domain is studied in the Turing machine‐based computational model and in the discrete complexity theory. This problem is studied with respect to two formulations of polynomial‐time computable two‐dimensional domains: (A) domains with polynomialtime computable boundaries, and (B) polynomial‐time recognizable domains with polynomial‐time computable distance functions. It is proved that the shortest path problem has the polynomial‐space upper bound for domains of both type (A) and type (B); and it has a polynomial‐space lower bound for the domains of type (B), and has a #P lower bound for the domains of type (A). (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)