Task Scheduling for Subsea Flexible Pipes Decommissioning

Subsea flexible pipelines removal is subject to order restrictions, mostly caused by crossings. It is proposed to create a computational algorithm to design an optimal order of vessel intervention over a field. A real field was studied, and, from it, the mathematical base model was created upon graph theory, with great correlation with the minimum feedback arc set problem. Vessel movements were discretized and reduced to removal, reposition, and cut, leading to a state search. A-star algorithm was implemented to guide the search for the solution. Then, the complete algorithm was built, tested in a minimal environment, and finally applied to the real instance. To improve performance, a beam search filtering was envisioned, using seven ranking functions. Constructed model is suspected to be NP-hard, by correlation to minimum feedback arc set problem, leading to a large space search. Instances containing under 100 crossings were solved optimally, without needing any assistance. After implementing the heuristics and beam search, solution time was lowered by about 20 times, demonstrating the effectiveness of the technique. Also, ranking functions for pipe repositioning based on crossing count led to better results than crossing density. For cutting, an approximation based on feedback arc set was used. GreedyFAS was employed and gave satisfactory results. Bigger instances containing around 3000 crossings could not be solved optimally in a reasonable time, even with the heuristics. Improvements in A-star estimation function and bound the solution branches might lead to an optimal solution for these larger instances. Model proposed simplifies the operational order decisions and helps build the scheduling of operations. As it is based on state search, other aspects in logistics, vessel capacities and steps in decommissioning processes may be added, adjusting the neighboring weights and branching, keeping the same core.

An Exact Method for the Minimum Feedback Arc Set Problem

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G . Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem . Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2 n ) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.

Tight Localizations of Feedback Sets

The classical NP–hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs ε ⊆ E or vertices ν ⊆ V whose removal G ∖ ε, G ∖ ν makes a given multi–digraph G =( V , E ) acyclic, respectively. Though both problems are known to be APX–hard, constant ratio approximations or proofs of inapproximability are unknown. We propose a new universal O (| V || E | 4 )–heuristic for the directed FASP. While a ratio of r ≈ 1.3606 is known to be a lower bound for the APX–hardness, at least by empirical validation we achieve an approximation of r ≤ 2. Most of the relevant applications, such as circuit testing , ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds with our approach.

On Succinct Encodings for the Tournament Fixing Problem

Single-elimination tournaments are a popular format in competitive environments. The Tournament Fixing Problem (TFP), which is the problem of finding a seeding of the players such that a certain player wins the resulting tournament, is known to be NP-hard in general and fixed-parameter tractable when parameterized by the feedback arc set number of the input tournament (an oriented complete graph) of expected wins/loses. However, the existence of polynomial kernelizations (efficient preprocessing) for TFP has remained open. In this paper, we present the first polynomial kernelization for TFP parameterized by the feedback arc set number of the input tournament. We achieve this by providing a polynomial-time routine that computes a SAT encoding where the number of clauses is bounded polynomially in the feedback arc set number.

When Rigging a Tournament, Let Greediness Blind You

A knockout tournament is a standard format of competition, ubiquitous in sports, elections and decision making. Such a competition consists of several rounds. In each round, all players that have not yet been eliminated are paired up into matches. Losers are eliminated, and winners are raised to the next round, until only one winner exists. Given that we can correctly predict the outcome of each potential match (modelled by a tournament D), a seeding of the tournament deterministically determines its winner. Having a favorite player v in mind, the Tournament Fixing Problem (TFP) asks whether there exists a seeding that makes v the winner. Aziz et al. [AAAI’14] showed that TFP is NP-hard. They initiated the study of the parameterized complexity of TFP with respect to the feedback arc set number k of D, and gave an XP-algorithm (which is highly inefficient). Recently, Ramanujan and Szeider [AAAI’17] showed that TFP admits an FPT algorithm, running in time 2^{ O(k^2 log k)} n ^{O(1)}. At the heart of this algorithm is a translation of TFP into an algebraic system of equations, solved in a black box fashion (by an ILP solver). We present a fresh, purely combinatorial greedy solution. We rely on new insights into TFP itself, which also results in the better running time bound of 2^{ O(k log k)} n^{ O(1)} . While our analysis is intricate, the algorithm itself is surprisingly simple.

Cutting Cycles of Conditional Preference Networks with Feedback Set Approach

As a tool of qualitative representation, conditional preference network (CP-net) has recently become a hot research topic in the field of artificial intelligence. The semantics of CP-nets does not restrict the generation of cycles, but the existence of the cycles would affect the property of CP-nets such as satisfaction and consistency. This paper attempts to use the feedback set problem theory including feedback vertex set (FVS) and feedback arc set (FAS) to cut cycles in CP-nets. Because of great time complexity of the problem in general, this paper defines a class of the parent vertices in a ring CP-nets firstly and then gives corresponding algorithm, respectively, based on FVS and FAS. Finally, the experiment shows that the running time and the expressive ability of the two methods are compared.

A Note on the Feedback Arc Set Problem and Acyclic Subdigraphs in Bipartite Tournaments

Given a digraph D a feedback arc set is a subset X of the arcs of D such that D − X is acyclic. Let β(D) denote de minimum cardinality of a feedback arc set of D. In this paper we prove that a bipartite tournament T with minimum out-degree at least r satisfies β(T) ≥ r2. A lower bound and an upper bound for β(T) are given in terms of the bipartite dichromatic number. We define the bipartite dichromatic number of a balanced bipartite tournament Tn,n and use this invariant to give an upper bound for the minimum cardinality of a feedback arc set of Tn,n. We also prove that for each positive integer k ≥ 3 there is an integer N(k) such that if n ≥ N(k), then each balanced bipartite tournament contains an acyclic bipartite tournament Tk,k.