In 1994, Frieze and Teng proposed an integer linear programming formulation of the NP-Complete Exact Partition problem, whose LP-relaxation they claimed was non-degenerate. Contrary to their claim, we show how an instance of Exact Partition can produce a degenerate polytope, and study conditions for which this can happen. We then give details of one of the smallest such degenerate Frieze-Teng polytopes, along with a closely related non-degenerate Frieze-Teng polytope that encodes an equivalent problem. We note that for the purposes of the complexity results in the literature that use their formulation, these degenerate polytopes can be avoided via a simple preprocessing step.

We consider the problem of finding the monotone eccentricity of a vertex v of a polytope. This is the largest number of pivots through the graph of the polytope required to reach an optimal vertex starting from v. In particular, we study a polytope introduced by Frieze and Teng [3] derived from the exact partition problem. This polytope is simple and nearly 0/1, with at most two fractional components per vertex. We show that Frieze and Teng’s result on the complexity of computing lower bounds on diameters of exact partition polytopes can be extended to show that computing monotone eccentricity is also NP-Hard, and in fact ^ -hard, even on simple polytopes.

The problem of computing the Elementary Flux Modes (EFMs) and Minimal Cut Sets (MCSs) of metabolic network is a fundamental one in metabolic networks. A key insight is that they can be understood as a dual pair of monotone Boolean functions (MBFs). Using this insight, this computation reduces to the question of generating from an oracle a dual pair of MBFs. If one of the two sets (functions) is known, then the other can be computed through a process known as dualization. Fredman and Khachiyan provided two algorithms, which they called simply A and B that can serve as an engine for oracle-based generation or dualization of MBFs. We look at efficiencies available in implementing their algorithm B, which we will refer to as FK-B. Like their algorithm A, FK-B certifies whether two given MBFs in the form of Conjunctive Normal Form and Disjunctive Normal Form are dual or not, and in case of not being dual it returns a conflicting assignment (CA), that is, an assignment that makes one of the given Boolean functions True and the other one False. The FK-B algorithm is a recursive algorithm that searches through the tree of assignments to find a CA. If it does not find any CA, it means that the given Boolean functions are dual. In this article, we propose six techniques applicable to the FK-B and hence to the dualization process. Although these techniques do not reduce the time complexity, they considerably reduce the running time in practice. We evaluate the proposed improvements by applying them to compute the MCSs from the EFMs in the 19 small- and medium-sized models from the BioModels database along with 4 models of biomass synthesis in Escherichia coli that were used in an earlier computational survey Haus et al. (2008).

The colourful simplicial depth (CSD) of a point x ∈ℝ^2 relative to a configuration P = ( P 1 , P 2 , … , P k ) of n points in k colour classes is the number of closed simplices (triangles) with vertices from three different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a given point x , and of finding a point in ℝ^2 , called a median , that maximizes colourful simplicial depth. Our algorithm for colourful simplicial depth runs in O(n logn) time, and in O ( n ) time if the points are already sorted around x . This is optimal for sorted inputs. Our algorithm for computing the colourful median runs in O ( n 4 ) time. Both results extend known algorithms for the monochromatic versions of these problems, and match the corresponding time complexities.

Purpose The Surrey (British Columbia, Canada) fire department has an annual cycle for hiring full-time firefighters. This paper optimizes the timing of the annual hiring period. A key issue is handling workplace absences, which can be covered by overtime cost or full-time hires. Design/methodology/approach Short-term and long-term absences patterns are analyzed according to season and age cohorts of the firefighters. These are then used in both an explanatory and time series model to predict future absences. The hiring schedule is optimized based on these predictions and additional constraints. Findings The current practice fares well in the analysis. For the time period studied, moving to earlier hiring dates appears beneficial. This analysis is robust with respect to various assumptions. Originality/value This is a case study where analytic techniques and machine learning are applied to an organizational practice that is not commonly analyzed. In this case, the previous method was not much worse than the optimized solution. The techniques used are quite general and can be applied to various organizational decision problems.

We present a new scheduling approach to improve access to care at an inner-city community health centre in Vancouver, Canada, serving marginalised clients with complex biopsychosocial needs. In order to meet the specific care needs of clients, the centre provides a range of services on a booked and walk-in basis, and it is important that clients are seen in a timely manner. To align schedules with client demand, we developed a schedule optimisation model that maximises time nurses spend with clients. This new objective function allows for a simple mixed integer linear programming structure that directly incorporates carryover demand. Client-centred key performance indicators were evaluated using a discrete event simulation model. Optimisation aligns schedules to demand, leading to fewer clients who leave without being seen due to an extended wait. This increases the number of clients receiving care by up to 9 per week, without compromising wait times. Furthermore, our approach addresses service delivery concerns, including baseline nurse coverage for triage and weekly variability in total nurse hours. Strategically aligning nurse shifts to demand is an effective approach to better meet client needs without increasing total nurse staffing levels in a community health centre context. (C) 2021 Elsevier Ltd. All rights reserved.

We consider the Bipartite Boolean Quadratic Programming Problem (BQP01), which generalizes the well-known Boolean quadratic programming problem (QP01). The model has applications in graph theory, matrix factorization and bioinformatics, among others. The primary focus of this paper is on studying the structure of the Bipartite Boolean Quadric Polytope (BQPm , n) resulting from a linearization of a quadratic integer programming formulation of BQP01. We present some basic properties and partial relaxations of BQPm , n, as well as some families of facets and valid inequalities. We find facet-defining inequalities including a family of odd-cycle inequalities. We discuss various approaches to obtain a valid inequality and facets from those of the related Boolean quadric polytope. The key strategy is based on rounding coefficients, and it is applied to the families of clique and cut inequalities in BQPm , n.

MOTIVATION:Constraint-based modeling of metabolic networks helps researchers gain insight into the metabolic processes of many organisms, both prokaryotic and eukaryotic. Minimal cut sets (MCSs) are minimal sets of reactions whose inhibition blocks a target reaction in a metabolic network. Most approaches for finding the MCSs in constrained-based models require, either as an intermediate step or as a byproduct of the calculation, the computation of the set of elementary flux modes (EFMs), a convex basis for the valid flux vectors in the network. Recently, Ballerstein et al. proposed a method for computing the MCSs of a network without first computing its EFMs, by creating a dual network whose EFMs are a superset of the MCSs of the original network. However, their dual network is always larger than the original network and depends on the target reaction. Here we propose the construction of a different dual network, which is typically smaller than the original network and is independent of the target reaction, for the same purpose. We prove the correctness of our approach, minimal coordinated support (MCS2), and describe how it can be modified to compute the few smallest MCSs for a given target reaction.RESULTS:We compare MCS2 to the method of Ballerstein et al. and two other existing methods. We show that MCS2 succeeds in calculating the full set of MCSs in many models where other approaches cannot finish within a reasonable amount of time. Thus, in addition to its theoretical novelty, our approach provides a practical advantage over existing methods.AVAILABILITY AND IMPLEMENTATION:MCS2 is freely available at https://github.com/RezaMash/MCS under the GNU 3.0 license.SUPPLEMENTARY INFORMATION:Supplementary data are available at Bioinformatics online.

From the point of view of optimization, a critical issue is relating the combinatorial diameter of a polyhedron to its number of facets $f$ and dimension $d$. In the seminal paper of Klee and Walkup [KW67], the Hirsch conjecture of an upper bound of $f-d$ was shown to be equivalent to several seemingly simpler statements, and was disproved for unbounded polyhedra through the construction of a particular 4-dimensional polyhedron $U_4$ with 8 facets. The Hirsch bound for bounded polyhedra was only recently disproved by Santos. We consider analogous properties for a variant of the combinatorial diameter called the circuit diameter. In this variant, the walks are built from the circuit directions of the polyhedron, which are the minimal non-trivial solutions to the system defining the polyhedron. We are able to prove that circuit variants of the so-called non-revisiting conjecture and $d$-step conjecture both imply the circuit analogue of the Hirsch conjecture. For the equivalences in [KW67], the wedge construction was a fundamental proof technique. We exhibit why it is not available in the circuit setting, and what are the implications of losing it as a tool. Further, we show the circuit analogue of the non-revisiting conjecture implies a linear bound on the circuit diameter of all unbounded polyhedra - in contrast to what is known for the combinatorial diameter. Finally, we give two proofs of a circuit version of the $4$-step conjecture. These results offer some hope that the circuit version of the Hirsch conjecture may hold in general. A challenge in the circuit setting is that different realizations of polyhedra of the same combinatorial structure may have different diameters. We adapt the notion of simplicity to work with circuits in the form of C-simple and wedge-simple polyhedra. We show that it suffices to consider such polyhedra.

The Quadratic Travelling Salesman Problem (QTSP) is to find a least cost Hamiltonian cycle in an edge-weighted graph, where costs are defined for all pairs of edges contained in the Hamiltonian cycle. The problem is shown to be strongly NP-hard on a Halin graph. We also consider a variation of the QTSP, called the k-neighbour TSP (TSP(k)). Two edges e and f, e≠f, are k-neighbours on a tour τ if and only if a shortest path (with respect to the number of edges) between e and f along τ and containing both e and f, has exactly k edges, for k≥2. In (TSP(k)), a fixed nonzero cost is considered for a pair of distinct edges in the cost of a tour τ only when the edges are p-neighbours on τ for 2≤p≤k. We give a linear time algorithm to solve TSP(k) on a Halin graph for k=3, extending existing algorithms for the cases k=1,2. Our algorithm can be extended further to solve TSP(k) in polynomial time on a Halin graph with n nodes when k=O(logn). The possibility of extending our results to some fully reducible class of graphs is also discussed. TSP(k) can be used to model the Permuted Variable Length Markov Model in bioinformatics as well as an optimal routing problem for unmanned aerial vehicles (UAVs).

The colourful simplicial depth of a point x in the plane relative to a configuration of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point, called a median, that maximizes colourful simplicial depth. For computing the colourful simplicial depth of x, our algorithm runs in time O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For finding the colourful median, we get a time of O(n^4). For comparison, the running times of the best known algorithm for the monochrome version of these problems are O(n log(n)) in general, improving to O(n) if the points are sorted around x for monochrome depth, and O(n^4) for finding a monochrome median.

Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to minimally-dependent subsystems of the presentation of the polyhedron. These can be understood as the set of all possible edge directions, including edges that may arise from translation of the facets. It is appealing to consider a circuit analogue of the Hirsch conjecture for graph diameter, as suggested by Borgwardt et al. [BFH15]. They ask whether the known counterexamples to the Hirsch conjecture give rise to counterexamples for this relaxed notion of circuit diameter. We show that the most basic counterexample to the unbounded Hirsch conjecture, the Klee-Walkup polyhedron, does have a circuit diameter that satisfies the Hirsch bound, regardless of representation. We also examine the circuit diameter of the bounded Klee-Walkup polytope.

Consider a variant of the graph diameter of a polyhedron where each step in a walk between two vertices travels maximally in a circuit direction instead of along incident edges. Here circuit directions are non-trivial solutions to minimally-dependent subsystems of the presentation of the polyhedron. These can be understood as the set of all possible edge directions, including edges that may arise from translation of the facets.

Monotone Boolean functions (MBFs) are Boolean functions f:{0,1}n→{0,1} satisfying the monotonicity condition x≤y⇒f(x)≤f(y) for any x,y∈{0,1}n. The number of MBFs in n variables is known as the nth Dedekind number. It is a longstanding computational challenge to determine these numbers exactly: these values are only known for n at most 8. Two monotone Boolean functions are equivalent if one can be obtained from the other by permuting the variables. The number of inequivalent MBFs in n variables was known only for up to n=6. In this paper we propose a strategy to count inequivalent MBFs by breaking the calculation into parts based on the profiles of these functions. As a result we are able to compute the number of inequivalent MBFs in 7 variables. The number obtained is 490013148.

The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.

The Consecutive Ones Property is an important notion for binary matrices, both from a theoretical and applied point of view. Tucker gave in 1972 a characterization of matrices that do not satisfy the Consecutive Ones Property in terms of forbidden submatrices, the Tucker patterns. We describe here a linear time algorithm to find a Tucker pattern in a non-C1P binary matrix, which allows to extract in linear time a certificate for the non-C1P. We also describe an output-sensitive algorithm to enumerate all Tucker patterns of a non-C1P binary matrix. This paper had been withdrawn due to some missing cases in Algorithms 2 and 3.

A binary matrix has the consecutive ones property (C1P) if it is possible to order the columns so that all 1s are consecutive in every row. In [McConnell, SODA 2004 768-777] the notion of incompatibility graph of a binary matrix was introduced and it was shown that odd cycles of this graph provide a certificate that a matrix does not have the consecutive ones property. A bound of (k+2) was claimed for the smallest odd cycle of a non-C1P matrix with k columns. In this note we show that this result can be obtained simply and directly via Tucker patterns, and that the correct bound is (k+2) when k is even, but (k+3) when k is odd.

A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of d-prismatoids of width larger than d, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases of Q.

The colourful simplicial depth problem in dimension d is to find a configuration of (d +1) sets of (d +1) points such that the origin is contained in the convex hull of each set (colour) but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d 2 +1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is d (d+1)2 2 e. A promising method to improve this lower bound is to look at combinatorial octahedral systems generated by such configurations. The difficulty to employing this approach is handling the many symmetric configurations that arise. We propose a table of invariants which exclude many of partial configurations, and use this to improve the lower bound in dimension 4..