. Exotic semirings such as the "(max, +) semiring" (R # {-#},max,+), or the "tropical semiring" (N #{+#},min,+), have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, HamiltonJacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities) . Despite this apparent profusion, there is a small set of common, non-naive, basic results and problems, in general not known outside the (max, +) community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of (max, +) results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of...

We investigate the all-pairs shortest paths problem in weighted graphs. We present an algorithm---the Hidden Paths Algorithm---that finds these paths in time O(m* n+n² log n), where m is the number of edges participating in shortest paths. Our algorithm is a practical substitute for Dijkstra's algorithm. We argue that m* is likely to be small in practice, since m* = O(n log n) with high probability for many probability distributions on edge weights. We also prove an Ω(mn) lower bound on the running time of any path-comparison based algorithm for the all-pairs shortest paths problem. Path-comparison based algorithms form a natural class containing the Hidden Paths Algorithm, as well as the algorithms of Dijkstra and Floyd. Lastly, we consider generalized forms of the shortest paths problem, and show that many of the standard shortest paths algorithms are effective in this more general setting.

We define general algebraic frameworks for shortest-distance problems based on the structure of semirings. We give a generic algorithm for finding single-source shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently classical shortest paths problems or to find the k-shortest distances in a directed graph. It can be used to solve single-source shortest-distance problems in weighted directed acyclic graphs over any semiring. We examine several semirings and describe some specific instances of our generic algorithms to illustrate their use and compare them with existing methods and algorithms. The proof of the soundness of all algorithms is given in detail, including their pseudocode and a full analysis of their running time complexity.

An m × n matrix C is called Monge matrix if c ij + c rs c is + c rj for all 1 i ! r m, 1 j ! s n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to optimization problems and (iii) recognition of Monge properties.

this paper occur at the tactical level. Strategic planning focuses on resource acquisition for the period from five to fifteen years ahead. Network planning problems may be viewed as the main strategic issues, but, in order to evaluate possible strategic alternatives, the subsequent stages including at least line planning and train schedule generation have to be considered. The disadvantages of the hierarchical planning are obvious, since the optimal output of a subtask which serves as the input of a subsequent task, will not result, in general, in an overall optimal solution.

Abstract. We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if K is a conditionally complete idempotent semifield, with completion ¯ K, a convex function K n → ¯ K which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of K n, which extends earlier results of Zimmermann, Samborski, and Shpiz.

In this paper we prove Kleene's result for tree series over a commutative and idempotent semiring A (which is not necessarily complete or continuous), i.e., the class of recognizable tree series over A and the class of rational tree series over A are equal. We show the result by direct automata-theoretic constructions and prove their correctness.

Abstract. We establish the following max-plus analogue of Minkowski’s theorem. Any point of a compact max-plus convex subset of (R ∪ {−∞}) n can be written as the max-plus convex combination of at most n + 1 of the extreme points of this subset. We establish related results for closed max-plus convex cones and closed unbounded max-plus convex sets. In particular, we show that a closed max-plus convex set can be decomposed as a max-plus sum of its recession cone and of the max-plus convex hull of its extreme points. 1.

In exploiting the potentials of highly parallel architectures to speed up the computation rate of systems enabled by VLSI, special attention has to be paid to designing algorithms such that they can be mapped onto parallel hardware. The main part of the Viterbi algorithm (VA) is a nonlinear feedback loop, the ACS recursion (add-compare-select recursion), which presents a bottleneck for high-speed implementations and cannot be circumvented by standard means. By identifying that the two operations of the loop form an algebraic structure called serniring, we show that the ACS recursion of the Viterbi algorithm can be written as a linear vector recursion. This allows us to employ the powerful techniques of parallel processing and pipelining, known for conventional linear systems, to achieve high throughput rates. Since the VA can be written as a linear vector recursion, it can be implemented by systolic arrays. For the class of shuffle exchange codes to be decoded by the Viterbi algorithm, hardware efficient code-optimized arrays are presented. In addition, it is shown that carry-save arithmetic can be used for the operations of ACS recursion, allowing each word-level operation to be pipelined and carried out by an efficient bit-level systolic array.

Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in R n +. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dependent sets and on computational bounds for the problem of finding the (essentially unique) basis of a finitely generated cone. AMS classification: 15A48, 15A03.