Fix a positive integer $n$ and consider the bipartite graph whose vertices are the $3$-element subsets and the $2$-element subsets of $[n]=\{1,2,\dots,n\}$, and there is an edge between $A$ and $B$ if $A\subset B$. We prove that the domination number of this graph is $\binom{n}{2}-\lfloor\frac{(n+1)^2}{8}\rfloor$, we characterize the dominating sets of minimum size, and we observe that the minimum size dominating set can e chosen as an independent set. This is an exact version of an asymptotic result from [Balogh2021]. For the corresponding bipartite graph between the $(k+1)$-element subsets and the $k$-elements subsets of $[n]$ ($k\geq 3$), we provide a new construction for small independent dominating sets. This improves on a construction from [Gerbner2012], where these independent dominating sets have been studied under the name saturating flat antichains.

This is the second of two papers investigating for which positive integers m there exists a maximal antichain of size m in the Boolean lattice B_n (the power set of [n]:={1,2,… ,n} , ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only l-sets and (l+1) -sets for some l.

For a simple graph G = (V (G), E(G)), a total labeling ∂ is called an edge irregular total k-labeling of G if ∂ : V (G) ∪ E(G) → {1, 2, . . . , k} such that for any two different edges uv and u'v' in E(G), we have wt∂(uv) not equal to wt∂(u'v') where wt∂(uv) = ∂(u) + ∂(v) + ∂(uv). The minimum k for which G has an edge irregulartotal k-labeling is called the total edge irregularity strength, denoted by tes(G). It is known that ceil((|E(G)|+2)/3) is a lower bound for the total edge irregularity strength of a graph G. In this paper we prove that if G is a bipartite graph for which this bound is tight then this is also true for Cartesian product of G with any path.

Extending a classical theorem of Sperner, we characterize the integers m such that there exists a maximal antichain of size m in the Boolean lattice Bn, that is, the power set of [n]:={1,2,… ,n} , ordered by inclusion. As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family ℱ of k-sets with |ℱ|=t such that the shadow of ℱ has size s, where the shadow of ℱ is the collection of (k − 1)-sets that are contained in at least one member of ℱ . We provide a complete answer for t⩽ k+1 . Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is √(2)k^3/2+√(8)k^5/4+O(k) .

This is the second in a sequence of three papers investigating the question for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the previous paper we characterized those $m$ between $\binom{n}{\lceil n/2\rceil}-\lceil n/2\rceil^2$ and the maximum size $\binom{n}{\lceil n/2 \rceil}$ that are not sizes of maximal antichains. In this paper we show that all smaller $m$ are sizes of maximal antichains.

Building on classical theorems of Sperner and Kruskal-Katona, we investigate antichains $\mathcal {F}$ in the Boolean lattice Bn of all subsets of $[n]:=\{1,2,\dots ,n\}$ , where $\mathcal {F}$ is flat, meaning that it contains sets of at most two consecutive sizes, say $\mathcal {F}=\mathcal {A}\cup {\mathscr{B}}$ , where $\mathcal {A}$ contains only k-subsets, while ${\mathscr{B}}$ contains only (k − 1)-subsets. Moreover, we assume $\mathcal {A}$ consists of the first m k-subsets in squashed (colexicographic) order, while ${\mathscr{B}}$ consists of all (k − 1)-subsets not contained in the subsets in $\mathcal {A}$ . Given reals α, β > 0, we say the weight of $\mathcal {F}$ is $\alpha \cdot |\mathcal {A}|+\beta \cdot |{\mathscr{B}}|$ . We characterize the minimum weight antichains $\mathcal {F}$ for any given n,k,α,β, and we do the same when in addition $\mathcal {F}$ is a maximal antichain. We can then derive asymptotic results on both the minimum size and the minimum Lubell function.

We study a capacity alignment planning problem for a coal chain. Given a set of train operators, a set of train paths and a terminal comprising of a dump station and a set of routes from the dump station to the stockyard, we seek a feasible assignment of train operators to train paths, to time slots at the dump station, and to routes. The assignment must maximize the number of system paths in the resulting schedule and the schedule should perform well with respect to various performance criteria. We model the problem as a mixed-integer conic program (MICP) with multiple objectives which we solve using a hierarchical optimization procedure. In each stage of this procedure, we solve a single objective MICP. Depending upon whether we evaluate the associated performance criteria under a 2- or 1-norm, we reformulate the MICP as either a mixed-integer second-order cone program or as a mixed-integer linear program, respectively, and can streamline the hierarchical optimization procedure by exploiting properties of the model or observed behaviour on practical instances. We compare the performance of the procedure under the different norms on a real instance of the problem and find that the quality of the solutions found by the faster 1-norm procedure compares well to the solution found under the 2-norm.

We study a certain polytope depending on a graph G and a parameter β ∈ (0,1) that arises from embedding the Hamiltonian cycle problem in a discounted Markov decision process. Literature suggests a conjecture a lower bound on the proportion of feasible bases corresponding to Hamiltonian cycles in the set of all feasible bases. We make progress toward a proof of the conjecture by proving results about the structure of feasible bases. In particular, we prove three main results: (1) the set of feasible bases is independent of the parameter β when the parameter is close to one, (2) the polytope can be interpreted as a generalized network flow polytope, and (3) we deduce a combinatorial interpretation of the feasible bases. We also provide a full characterization for a special class of feasible bases, and we apply this to provide some computational support for the conjecture.

Extending a classical theorem of Sperner, we investigate the question for which positive integers m there exists a maximal antichain of sizem in the Boolean lattice Bn, that is, the power set of [n] := {1, 2, . . . , n}, ordered by inclusion. We characterize all such integers m in the range ( n ⌈n/2⌉ ) −⌈n/2⌉2 6 m 6 ( n ⌈n/2⌉ ) . As an important ingredient in the proof, we initiate the study of an extension of the Kruskal-Katona theorem which is of independent interest. For given positive integers t and k, we ask which integers s have the property that there exists a family F of k-sets with |F| = t such that the shadow of F has size s, where the shadow of F is the collection of (k − 1)-sets that are contained in at least one member of F . We provide a complete answer for the case t 6 k + 1. Moreover, we prove that the largest integer which is not the shadow size of any family of k-sets is √ 2k3/2 + 4 √ 8k5/4 +O(k).

We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated with the input graph. This polytope is a subset of the space of discounted occupational measures. We characterize the feasible bases of the polytope for a general input graph G and determine the expected numbers of different types of feasible bases when the underlying graph is random. We utilize these results to demonstrate that augmenting certain additional constraints to reduce the polyhedral domain can eliminate a large number of feasible bases that do not correspond to Hamiltonian cycles. Finally, we develop a random walk algorithm on the feasible bases of the reduced polytope and present some numerical results. We conclude with a conjecture on the feasible bases of the reduced polytope.

We consider the problem of characterizing the convex hull of the graph of a bilinear function f on the n-dimensional unit cube [0,1](n). Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of f that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations (Zuckerberg, 2016) to prove some initial results in this direction. In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights. (C) 2020 Elsevier B.V. All rights reserved.

We study the convex hull of the graph of a quadratic function f(𝐱)=∑_ij∈ Ex_ix_j, where the sum is over the edge set of a graph G with vertex set {1,…,n}. Using an approach proposed by Gupte et al. (Discrete Optimization 36, 2020, 100569), we investigate minimal extended formulations using additional variables y_ij, 1≤ i<j≤ n, representing the products x_ix_j. The basic idea is to identify a set of facets of the Boolean Quadric Polytope which is sufficient for characterizing the convex hull for the given graph. Our main results are extended formulations for the cases that the underlying graph G is either an even wheel or a complete split graph.

We show that the vertices and edges of a $d$-dimensional grid graph $G=(V,E)$ ($d\geqslant 2$) can be labeled with the integers from $\{1,\ldots,\lvert V\rvert\}$ and $\{1,\ldots,\lvert E\rvert\}$, respectively, in such a way that for every subgraph $H$ isomorphic to a $d$-cube the sum of all the labels of $H$ is the same. As a consequence, for every $d\geqslant 2$, every $d$-dimensional grid graph is $Q_d$-supermagic where $Q_d$ is the $d$-cube.

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed.

Rail infrastructure forms a critical part of the mining supply chain in Australia due to the high weight to volume ratio of the product and the long distances between the mines and the ports. Across Australia, rail infrastructure has been steadily expanding to account for the growth in export volumes and the movement of mining operations further inland, and so the efficient and effective management of this critical infrastructure is vitally important. Maintenance plays a crucial role in this management as it ensures that the infrastructure assets are in a condition that allows safe, reliable, and efficient transport.In this paper we consider the annual planning of maintenance for Australia’s largest coal rail network, the Central Queensland Coal Network (CQCN), that is owned, operated, and managed, by Aurizon Holdings Pty Ltd. The current planning approach at Aurizon uses the concept of a maintenance access window (MAW) which provides a train-free time window across geographically contiguous track locations that define a maintenance zone. These train-free time windows facilitate the scheduling of specific maintenance tasks at specific track locations within zones closer to day of operation and forms the basis for a planning framework.A MIP model is introduced which facilitates the planning of different maintenance resources across this network to schedule MAWs. The model takes into account maintenance requirement forecasts as well as the availability of resources. Candidate solutions are compared using a proxy for network throughput capacity. Due to the long computation times required to solve the MIP model at the annual planning horizon a matheuristic is developed and two variants are tested. On average 80% less computational time is required to find a good solution (average gap of 5%) using the matheuristic compared to solving the MIP model directly (average gap of 1.5%).The MIP model and associated matheuristic provides a suitable framework for semi-automated maintenance planning and is being integrated into the current suite of decision support tools used by Aurizon.

Abstract. We investigate a novel scheduling problem which is motivated by an application in the Australian railway industry. Given a set of maintenance jobs and a set of train paths over a railway corridor with bidirectional traffic, we seek a schedule of jobs such that a minimum number of train paths are cancelled due to conflict with the job schedule. We show that the problem is NP-complete in general. In a special case of the problem when every job under any schedule just affects one train path, and the speed of trains is bounded from above and below, we show that the problem can be solved in polynomial time. Moreover, in another special case of the problem where the traffic is unidirectional, we show that the problem can be solved in time O(n).

A subset S of initially infected vertices of a graph G is called zero forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbor, infects this neighbor. The zero forcing number of G is the minimum cardinality of a zero forcing set in G. We study the zero forcing number of various classes of graphs, including graphs of large girth, H-free graphs for a fixed bipartite graph H, and random and pseudorandom graphs.

The network maintenance problem is motivated by the need to maintain infrastructure networks over time. We consider networks in which a commodity is transported between origin-destination pairs, and at the same time the infrastructure assets need to be maintained by resources moving in the network. In order to perform maintenance the assets have to be shut down thus reducing the system capacity. The objective is to maximise the total throughput by aligning the maintenance activities efficiently. In this paper, we study a special case of the network maintenance problem where the network consists of a single arc connecting an origin to a destination. Furthermore, there is no restriction on the amount of repair if the resource is to perform maintenance in a time period. This problem is of interest for the following reasons. Firstly, it generalises variants of the lot-sizing problem and the warehouse problem, both of which have been well-studied in the literature. Secondly, we hope that understanding this special case will be useful in tackling more general variants of the network maintenance problem. In this paper, we present a mixed integer linear programming formulation. We then show that a special class of feasible solutions, called Maximum Flow Order Up (MFOU) solutions, contains at least one optimal solution. Based on this result, we introduce an alternative integer linear programming formulation with only five decision variables. As a consequence, the optimal objective function value for any instance of the problem can be obtained in polynomial time.