Polyhedral results for the Cardinality Constrained Multi-cycle Problem (CCMcP) and the Cardinality Constrained Cycles and Chains Problem (CCCCP)

In recent years, there has been studies on the cardinality constrained multi-cycle problems on directed graphs, some of which considered chains co-existing on the same digraph whilst others did not. These studies were inspired by the optimal matching of kidneys known as the Kidney Exchange Problem (KEP). In a KEP, a vertex on the digraph represents a donor-patient pair who are related, though the kidney of the donor is incompatible to the patient. When there are multiple such incompatible pairs in the kidney exchange pool, the kidney of the donor of one incompatible pair may in fact be compatible to the patient of another incompatible pair. If Donor A’s kidney is suitable for Patient B, and vice versa, then there will be arcs in both directions between Vertex A to Vertex B. Such exchanges form a 2-cycle. There may also be cycles involving 3 or more vertices. As all exchanges in a kidney exchange cycle must take place simultaneously, (otherwise a donor can drop out from the program once his/her partner has received a kidney from another donor), due to logistic and human resource reasons, only a limited number of kidney exchanges can occur simultaneously, hence the cardinality of these cycles are constrained. In recent years, kidney exchange programs around the world have altruistic donors in the pool. A sequence of exchanges that starts from an altruistic donor forms a chain instead of a cycle. We therefore have two underlying combinatorial optimization problems: Cardinality Constrained Multi-cycle Problem (CCMcP) and the Cardinality Constrained Cycles and Chains Problem (CCCCP). The objective of the KEP is either to maximize the number of kidney matches, or to maximize a certain weighted function of kidney matches. In a CCMcP, a vertex can be in at most one cycle whereas in a CCCCP, a vertex can be part of (but in no more than) a cycle or a chain. The cardinality of the cycles are constrained in all studies. The cardinality of the chains, however, are considered unconstrained in some studies, constrained but larger than that of cycles, or the same as that of cycles in others. Although the CCMcP has some similarities to the ATSP- and VRP-family of problems, there is a major difference: strong subtour elimination constraints are mostly invalid for the CCMcP, as we do allow smaller subtours as long as they do not exceed the size limit. The CCCCP has its distinctive feature that allows chains as well as cycles on the same directed graph. Hence, both the CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights. Most existing studies focused on solution methodologies, and as far as we aware, there is no polyhedral studies so far. In this paper, we will study the polyhedral structure of the natural arc-based integer programming models of the CCMcP and the CCCCP, both containing exponentially many constraints. We do so to pave the way for studying strong valid cuts we have found that can be applied in a Lagrangean relaxation-based branch-and-bound framework where at each node of the branch-and-bound tree, we may be able to obtain a relaxation that can be solved in polynomial time, with strong valid cuts dualized into the objective function and the dual multipliers optimised by subgradient optimisation.

Robustness in health research : do differences in health measures, techniques, and time frame matter?

Survey-based health research is in a boom phase following an increased amount of health spending in OECD countries and the interest in ageing. A general characteristic of survey-based health research is its diversity. Different studies are based on different health questions in different datasets; they use different statistical techniques; they differ in whether they approach health from an ordinal or cardinal perspective; and they differ in whether they measure short-term or long-term effects. The question in this paper is simple: do these differences matter for the findings? We investigate the effects of life-style choices (drinking, smoking, exercise) and income on six measures of health in the US Health and Retirement Study (HRS) between 1992 and 2002: (1) self-assessed general health status, (2) problems with undertaking daily tasks and chores, (3) mental health indicators, (4) BMI, (5) the presence of serious long-term health conditions, and (6) mortality. We compare ordinal models with cardinal models; we compare models with fixed effects to models without fixed-effects; and we compare short-term effects to long-term effects. We find considerable variation in the impact of different determinants on our chosen health outcome measures; we find that it matters whether ordinality or cardinality is assumed; we find substantial differences between estimates that account for fixed effects versus those that do not; and we find that short-run and long-run effects differ greatly. All this implies that health is an even more complicated notion than hitherto thought, defying generalizations from one measure to the others or one methodology to another.

A graphical notation for modelling events in business processes

Using complex event rules for capturing dependencies between business processes is an emerging trend in enterprise information systems. In previous work we have identified a set of requirements for event extensions for business process modeling languages. This paper introduces a graphical language for modeling composite events in business processes, namely BEMN, that fulfills all these requirements. These include event conjunction, disjunction and inhibition as well as cardinality of events whose graphical expression can be factored into flow-oriented process modeling and event rule modeling. Formal semantics for the language are provided.

Seed-based biclustering of gene expression data

Background Accumulated biological research outcomes show that biological functions do not depend on individual genes, but on complex gene networks. Microarray data are widely used to cluster genes according to their expression levels across experimental conditions. However, functionally related genes generally do not show coherent expression across all conditions since any given cellular process is active only under a subset of conditions. Biclustering finds gene clusters that have similar expression levels across a subset of conditions. This paper proposes a seed-based algorithm that identifies coherent genes in an exhaustive, but efficient manner. Methods In order to find the biclusters in a gene expression dataset, we exhaustively select combinations of genes and conditions as seeds to create candidate bicluster tables. The tables have two columns: (a) a gene set, and (b) the conditions on which the gene set have dissimilar expression levels to the seed. First, the genes with less than the maximum number of dissimilar conditions are identified and a table of these genes is created. Second, the rows that have the same dissimilar conditions are grouped together. Third, the table is sorted in ascending order based on the number of dissimilar conditions. Finally, beginning with the first row of the table, a test is run repeatedly to determine whether the cardinality of the gene set in the row is greater than the minimum threshold number of genes in a bicluster. If so, a bicluster is outputted and the corresponding row is removed from the table. Repeating this process, all biclusters in the table are systematically identified until the table becomes empty. Conclusions This paper presents a novel biclustering algorithm for the identification of additive biclusters. Since it involves exhaustively testing combinations of genes and conditions, the additive biclusters can be found more readily.

Unconditionally secure disjointness tests for private datasets

We present two unconditional secure protocols for private set disjointness tests. In order to provide intuition of our protocols, we give a naive example that applies Sylvester matrices. Unfortunately, this simple construction is insecure as it reveals information about the intersection cardinality. More specifically, it discloses its lower bound. By using the Lagrange interpolation, we provide a protocol for the honest-but-curious case without revealing any additional information. Finally, we describe a protocol that is secure against malicious adversaries. In this protocol, a verification test is applied to detect misbehaving participants. Both protocols require O(1) rounds of communication. Our protocols are more efficient than the previous protocols in terms of communication and computation overhead. Unlike previous protocols whose security relies on computational assumptions, our protocols provide information theoretic security. To our knowledge, our protocols are the first ones that have been designed without a generic secure function evaluation. More important, they are the most efficient protocols for private disjointness tests in the malicious adversary case.

Efficient disjointness tests for private datasets

We present efficient protocols for private set disjointness tests. We start from an intuition of our protocols that applies Sylvester matrices. Unfortunately, this simple construction is insecure as it reveals information about the cardinality of the intersection. More specifically, it discloses its lower bound. By using the Lagrange interpolation we provide a protocol for the honest-but-curious case without revealing any additional information. Finally, we describe a protocol that is secure against malicious adversaries. The protocol applies a verification test to detect misbehaving participants. Both protocols require O(1) rounds of communication. Our protocols are more efficient than the previous protocols in terms of communication and computation overhead. Unlike previous protocols whose security relies on computational assumptions, our protocols provide information theoretic security. To our knowledge, our protocols are first ones that have been designed without a generic secure function evaluation. More importantly, they are the most efficient protocols for private disjointness tests for the malicious adversary case.

Distributed private matching and set operations

Motivated by the need of private set operations in a distributed environment, we extend the two-party private matching problem proposed by Freedman, Nissim and Pinkas (FNP) at Eurocrypt’04 to the distributed setting. By using a secret sharing scheme, we provide a distributed solution of the FNP private matching called the distributed private matching. In our distributed private matching scheme, we use a polynomial to represent one party’s dataset as in FNP and then distribute the polynomial to multiple servers. We extend our solution to the distributed set intersection and the cardinality of the intersection, and further we show how to apply the distributed private matching in order to compute distributed subset relation. Our work extends the primitives of private matching and set intersection by Freedman et al. Our distributed construction might be of great value when the dataset is outsourced and its privacy is the main concern. In such cases, our distributed solutions keep the utility of those set operations while the dataset privacy is not compromised. Comparing with previous works, we achieve a more efficient solution in terms of computation. All protocols constructed in this paper are provably secure against a semi-honest adversary under the Decisional Diffie-Hellman assumption.

Democratic systems

Digital signature is a breakthrough of modern cryptographic systems. A (t, n) threshold digital signature allows every set of cardinality t or more (out-of n) co-signers to authenticate a message. In almost all existing threshold digital signatures the threshold parameter t is fixed. There are applications, however, in which the threshold parameter needs to be changed from time to time. This paper considers such a scenario, in order to discuss relevant problems, and proposes a model that solves the related problems.

Cubicity, boxicity, and vertex cover

A k-dimensional box is the cartesian product R-1 x R-2 x ... x R-k where each R-i is a closed interval on the real line. The boxicity of a graph G,denoted as box(G), is the minimum integer k such that G is the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the cartesian product R-1 x R-2 x ... x R-k where each Ri is a closed interval on the real line of the form [a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. In this paper we show that cub(G) <= t + inverted right perpendicularlog(n - t)inverted left perpendicular - 1 and box(G) <= left perpendiculart/2right perpendicular + 1, where t is the cardinality of a minimum vertex cover of G and n is the number of vertices of G. We also show the tightness of these upper bounds. F.S. Roberts in his pioneering paper on boxicity and cubicity had shown that for a graph G, box(G) <= left perpendicularn/2right perpendicular and cub(G) <= inverted right perpendicular2n/3inverted left perpendicular, where n is the number of vertices of G, and these bounds are tight. We show that if G is a bipartite graph then box(G) <= inverted right perpendicularn/4inverted left perpendicular and this bound is tight. We also show that if G is a bipartite graph then cub(G) <= n/2 + inverted right perpendicularlog n inverted left perpendicular - 1. We point out that there exist graphs of very high boxicity but with very low chromatic number. For example there exist bipartite (i.e., 2 colorable) graphs with boxicity equal to n/4. Interestingly, if boxicity is very close to n/2, then chromatic number also has to be very high. In particular, we show that if box(G) = n/2 - s, s >= 0, then chi (G) >= n/2s+2, where chi (G) is the chromatic number of G.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

Minimization of Incompletely Specified Sequential Machines

A simple yet efficient method for the minimization of incompletely specified sequential machines (ISSMs) is proposed. Precise theorems are developed, as a consequence of which several compatibles can be deleted from consideration at the very first stage in the search for a minimal closed cover. Thus, the computational work is significantly reduced. Initial cardinality of the minimal closed cover is further reduced by a consideration of the maximal compatibles (MC's) only; as a result the method converges to the solution faster than the existing procedures. "Rank" of a compatible is defined. It is shown that ordering the compatibles, in accordance with their rank, reduces the number of comparisons to be made in the search for exclusion of compatibles. The new method is simple, systematic, and programmable. It does not involve any heuristics or intuitive procedures. For small- and medium-sized machines, it canle used for hand computation as well. For one of the illustrative examples used in this paper, 30 out of 40 compatibles can be ignored in accordance with the proposed rules and the remaining 10 compatibles only need be considered for obtaining a minimal solution.

Simulation of wavelength/time multiple-pulses-per-row fiber-optic CDMA network

Passive wavelength/time fiber-optic code division multiple access (WIT FO-CDMA) network is a viable option for highspeed access networks. Constructions of 2-D codes, suitable for incoherent WIT FO-CDMA, have been proposed to reduce the time spread of the 1-D sequences. The 2-D constructions can be broadly classified as 1) hybrid codes and 2) matrix codes. In our earlier work [141, we had proposed a new family of wavelength/time multiple-pulses-per-row (W/T MPR) matrix codes which have good cardinality, spectral efficiency and at the same time have the lowest off-peak autocorrelation and cross-correlation values equal to unity. In this paper we propose an architecture for a WIT MPR FO-CDAM network designed using the presently available devices and technology. A complete FO-CDMA network of ten users is simulated, for various number of simultaneous users and shown that 0 --> 1 errors can occur only when the number of interfering users is at least equal to the threshold value.

Maximum compatible classes from compatibility matrices

A fast algorithm for the computation of maximum compatible classes (mcc) among the internal states of an incompletely specified sequential machine is presented in this paper. All the maximum compatible classes are determined by processing compatibility matrices of progressingly diminishing order, whose total number does not exceed (p + m), where p is the largest cardinality among these classes, and m is the number of such classes. Consequently the algorithm is specially suitable for the state minimization of very large sequential machines as encountered in vlsi circuits and systems.

Efficient algorithms for domination and Hamilton circuit problems on permutation graphs

The domination and Hamilton circuit problems are of interest both in algorithm design and complexity theory. The domination problem has applications in facility location and the Hamilton circuit problem has applications in routing problems in communications and operations research.The problem of deciding if G has a dominating set of cardinality at most k, and the problem of determining if G has a Hamilton circuit are NP-Complete. Polynomial time algorithms are, however, available for a large number of restricted classes. A motivation for the study of these algorithms is that they not only give insight into the characterization of these classes but also require a variety of algorithmic techniques and data structures. So the search for efficient algorithms, for these problems in many classes still continues.A class of perfect graphs which is practically important and mathematically interesting is the class of permutation graphs. The domination problem is polynomial time solvable on permutation graphs. Algorithms that are already available are of time complexity O(n2) or more, and space complexity O(n2) on these graphs. The Hamilton circuit problem is open for this class.We present a simple O(n) time and O(n) space algorithm for the domination problem on permutation graphs. Unlike the existing algorithms, we use the concept of geometric representation of permutation graphs. Further, exploiting this geometric notion, we develop an O(n2) time and O(n) space algorithm for the Hamilton circuit problem.

A note on resolving infeasibility in linear programs by constraint relaxation

We study the problem of finding a set of constraints of minimum cardinality which when relaxed in an infeasible linear program, make it feasible. We show the problem is NP-hard even when the constraint matrix is totally unimodular and prove polynomial-time solvability when the constraint matrix and the right-hand-side together form a totally unimodular matrix.