The Weight Function Lemma for graph pebbling

Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble byt a vertex. Deciding if the pebbling number is at most k is \(\Pi _2^\mathsf{P}\)-complete. In this paper we develop a tool, called the Weight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply the Weight Function Lemma to several specific graphs, including the Petersen, Lemke, \(4\mathrm{th}\) weak Bruhat, and Lemke squared, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. In doing so we partly answer a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling.     

Fibonacci helps to evacuate from a convex region in a grid network

This study considers an evacuation problem where the evacuees try to escape to the boundary of an affected area, which is convex, and a grid network is embedded in the area. The boundary is unknown to the evacuees and we propose an online evacuation strategy based on the Fibonacci sequence. This strategy is proved to have a competitive ratio of 19.5, which is better than the best previously reported result of 21.     

Neighbor sum distinguishing index of 2-degenerate graphs

We consider proper edge colorings of a graph G using colors in \(\{1,\ldots ,k\}\). Such a coloring is called neighbor sum distinguishing if for each pair of adjacent vertices u and v, the sum of the colors of the edges incident with u is different from the sum of the colors of the edges incident with v. The smallest value of k in such a coloring of G is denoted by \({\mathrm ndi}_{\Sigma }(G)\). In this paper we show that if G is a 2-degenerate graph without isolated edges, then \({\mathrm ndi}_{\Sigma }(G)\le \max \{\Delta (G)+2,7\}\).     

Multiple facility location on a network with linear reliability order of edges

We study the problem of locating facilities on the nodes of a network to maximize the expected demand serviced. The edges of the input graph are subject to random failure due to a disruptive event. We consider a special type of failure correlation. The edge dependency model assumes that the failure of a more reliable edge implies the failure of all less reliable ones. Under this dependency model called Linear Reliability Order (LRO) we give two polynomial time exact algorithms. When two distinct LRO’s exist, we prove the total unimodularity of a linear programming formulation. In addition, we show that minimizing the sum of facility opening costs and expected cost of unserviced demand under two orderings reduces to a matching problem. We prove NP-hardness of the three orderings case and show that the problem with an arbitrary number of orderings generalizes the deterministic maximum coverage problem. When a demand point can be covered only if a facility exists within a distance limit, we show that the problem is NP-hard even for a single ordering.     

On global integer extrema of real-valued box-constrained multivariate quadratic functions

Determining global integer extrema of an real-valued box-constrained multivariate quadratic functions is a very difficult task. In this paper, we present an analytic method, which is based on a combinatorial optimization approach in order to calculate global integer extrema of a real-valued box-constrained multivariate quadratic function, whereby this problem will be proven to be as NP-hard via solving it by a Travelling Salesman instance. Instead, we solve it using eigenvalue theory, which allows us to calculate the eigenvalues of an arbitrary symmetric matrix using Newton’s method, which converges quadratically and in addition yields a Jordan normal form with \(1 \times 1\)-blocks, from which a special representation of the multivariate quadratic function based on affine linear functions can be derived. Finally, global integer minimizers can be calculated dynamically and efficiently most often in a small amount of time using the Fourier–Motzkin- and a Branch and Bound like Dijkstra-algorithm. As an application, we consider a box-constrained bivariate and multivariate quadratic function with ten arguments.     

An optimal online algorithm for the parallel-batch scheduling with job processing time compatibilities

We consider the online (over time) scheduling on a single unbounded parallel-batch machine with job processing time compatibilities to minimize makespan. In the problem, a constant \(\alpha >0\) is given in advance. Each job \(J_{j}\) has a normal processing time \(p_j\). Two jobs \(J_i\) and \(J_j\) are compatible if \(\max \{p_i, p_j\} \le (1+\alpha )\cdot \min \{p_i, p_j\}\). In the problem, mutually compatible jobs can form a batch being processed on the machine. The processing time of a batch is equal to the maximum normal processing time of the jobs in this batch. For this problem, we provide an optimal online algorithm with a competitive ratio of \(1+\beta _\alpha \), where \(\beta _\alpha \) is the positive root of the equation \((1+\alpha )x^{2}+\alpha x=1+\alpha \).     

Hamiltonian numbers in oriented graphs

A hamiltonian walk of a digraph is a closed spanning directed walk with minimum length in the digraph. The length of a hamiltonian walk in a digraph D is called the hamiltonian number of D, denoted by h(D). In Chang and Tong (J Comb Optim 25:694–701, 2013), Chang and Tong proved that for a strongly connected digraph D of order n, \(n\le h(D)\le \lfloor \frac{(n+1)^2}{4} \rfloor \), and characterized the strongly connected digraphs of order n with hamiltonian number \(\lfloor \frac{(n+1)^2}{4} \rfloor \). In the paper, we characterized the strongly connected digraphs of order n with hamiltonian number \(\lfloor \frac{(n+1)^2}{4} \rfloor -1\) and show that for any triple of integers n, k and t with \(n\ge 5\), \(n\ge k\ge 3\) and \(t\ge 0\), there is a class of nonisomorphic digraphs with order n and hamiltonian number \(n(n-k+1)-t\).     

Ex Ante Project Evaluation and the Complexity of Early Decision-Making

This paper looks beyond more traditional evaluation activities to focus primarily on evaluation up front. It suggests that the early appraisal of an investment case or a project should apply essentially the same evaluation criteria that will be used in ex post evaluation, and thus increase the likelihood of a successful project outcome. However, the initial plan might be altered as result of subsequent analysis, assessment, negotiation, positioning, and the exercise of power. The last part of this paper presents an empirical study of 23 projects, which examines the complexity of processes that occur in the idea- and decision phases.     

Strategic Planning in the Malawi Public Sector: Potential Tool for Progress or Regression?

The process of developing strategic plans in the public sector is regarded as one indicator of positive reform in the system. The assumption is that strategic planning ensures that (just like in the private sector) public sector organizations’ operations should cost less but deliver better service. Ultimately the argument is that public sector institutional governance and responsiveness to the citizenry is not only seen but felt. However, strategic planning has to be done in an appropriate manner for it to deliver. For instance an analysis of the underlying motives, the role of leadership in the process as well as the relevance or viability of the guiding strategic planning model significantly affect the nature of expected results. In the early 2000s, the Malawi public sector embarked on a serious drive towards strategic planning process as part of its reform package. In order to ascertain it’s potential to transform the Malawi public service, this paper focuses at critically analyzing the strategic planning process in the Malawi public sector by among other things asking: who initiated and led the process? What approach determined the process? What is the possible potentiality of the adopted strategic planning approach in enhancing the reform agenda in Malawi? Based on interviews with senior officers in key Malawi public sector institutions, the paper concludes that the strategic planning process is ‘transitional and fluid’ hence lacks the clout to effectively enhance reforms in this sector. Another finding is that lack of effective leadership has largely contributed to the status quo.     

‘Strong Personalities’ and ‘Strong Institutions’ Mediated by a ‘Strong Third Force’: Thinking ‘Systems’ in Corruption Control

Corruption in public sector is a complex, ‘messy’ and ‘fuzzy’ phenomenon which involves calculus machinations between actors and constantly growing in sophistication. Consequently, an attempt to prevent, control or fight it requires systems thinking that ranges from public officials with integrity and personal ethics [strong personalities]; administrative rules and procedures as well as governments adopting, enforcing and monitoring appropriate systems [strong institutions]. This paper provides a systems approach to enhancing public sector ethics through ‘a corruption control tripod’ in preventing, controlling and combating corruption in all its forms. This framework discusses three overarching pillars and argues that adopting a reductionist approach in corruption control would yield futility; there is a need for synthesis so that each one would reinforce the other.