The Stochastic Shortest Path Problem : A polyhedral combinatorics perspective

In this paper, we give a new framework for the stochastic shortest path problem in finite state and action spaces. Our framework generalizes both the frameworks proposed by Bertsekas and Tsitsikli and by Bertsekas and Yu. We prove that the problem is well-defined and (weakly) polynomial when (i) there is a way to reach the target state from any initial state and (ii) there is no transition cycle of negative costs (a generalization of negative cost cycles). These assumptions generalize the standard assumptions for the deterministic shortest path problem and our framework encapsulates the latter problem (in contrast with prior works). In this new setting, we can show that (a) one can restrict to deterministic and stationary policies, (b) the problem is still (weakly) polynomial through linear programming, (c) Value Iteration and Policy Iteration converge, and (d) we can extend Dijkstra's algorithm

On the Recognition of Fuzzy Circular Interval Graphs

Fuzzy circular interval graphs are a generalization of proper circular arc graphs and have been recently introduced by Chudnovsky and Seymour as a fundamental subclass of claw-free graphs. In this paper, we provide a polynomial-time algorithm for recognizing such graphs, and more importantly for building a suitable representation.Comment: 12 pages, 2 figure

How many matchings cover the nodes of a graph?

Given an undirected graph, are there $k$ matchings whose union covers all of its nodes, that is, a matching-$k$-cover? A first, easy polynomial solution from matroid union is possible, as already observed by Wang, Song and Yuan (Mathematical Programming, 2014). However, it was not satisfactory neither from the algorithmic viewpoint nor for proving graphic theorems, since the corresponding matroid ignores the edges of the graph. We prove here, simply and algorithmically: all nodes of a graph can be covered with $k\ge 2$ matchings if and only if for every stable set $S$ we have $|S|\le k\cdot|N(S)|$. When $k=1$, an exception occurs: this condition is not enough to guarantee the existence of a matching-$1$-cover, that is, the existence of a perfect matching, in this case Tutte's famous matching theorem (J. London Math. Soc., 1947) provides the right `good' characterization. The condition above then guarantees only that a perfect $2$-matching exists, as known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953). Some results are then deduced as consequences with surprisingly simple proofs, using only the level of difficulty of bipartite matchings. We give some generalizations, as well as a solution for minimization if the edge-weights are non-negative, while the edge-cardinality maximization of matching-$2$-covers turns out to be already NP-hard. We have arrived at this problem as the line graph special case of a model arising for manufacturing integrated circuits with the technology called `Directed Self Assembly'.Comment: 10 page

Enhancing PGA Tour Performance: Leveraging ShotlinkTM Data for Optimization and Prediction

In this study, we demonstrate how data from the PGA Tour, combined with stochastic shortest path models (MDPs), can be employed to refine the strategies of professional golfers and predict future performances. We present a comprehensive methodology for this objective, proving its computational feasibility. This sets the stage for more in-depth exploration into leveraging data available to professional and amateurs for strategic optimization and forecasting performance in golf. For the replicability of our results, and to adapt and extend the methodology and prototype solution, we provide access to all our codes and analyses (R and C++)

Should Sports Professionals Consider Their Adversary's Strategy? A Case Study of Match Play in Golf

This study explores strategic considerations in professional golf's Match Play format, challenging the conventional focus on individual performance. Leveraging PGA Tour data, we investigate the impact of factoring in an adversary's strategy. Our findings suggest that while slight strategy adjustments can be advantageous in specific scenarios, the overall benefit of considering an opponent's strategy remains modest. This confirms the common wisdom in golf, reinforcing the recommendation to adhere to optimal stroke-play strategies due to challenges in obtaining precise opponent statistics. We believe that the methodology employed here could offer valuable insights into whether opponents' performances should also be considered in other two-player or team sports, such as tennis, darts, soccer, volleyball, etc. We hope that this research will pave the way for new avenues of study in these areas

Horizontal collaboration in forestry: game theory models and algorithms for trading demands

In this paper, we introduce a new cooperative game theory model that we call production-distribution game to address a major open problem for operations research in forestry, raised by R\"onnqvist et al. in 2015, namely, that of modelling and proposing efficient sharing principles for practical collaboration in transportation in this sector. The originality of our model lies in the fact that the value/strength of a player does not only depend on the individual cost or benefit of the objects she owns but also depends on her market shares (customers demand). We show however that the production-distribution game is an interesting special case of a market game introduced by Shapley and Shubik in 1969. As such it exhibits the nice property of having a non-empty core. We then prove that we can compute both the nucleolus and the Shapley value efficiently, in a nontrivial and interesting special case. We in particular provide two different algorithms to compute the nucleolus: a simple separation algorithm and a fast primal-dual algorithm. Our results can be used to tackle more general versions of the problem and we believe that our contribution paves the way towards solving the challenging open problem herein

On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs

We deal with non-rank facets of the stable set polytope of claw-free graphs. We extend results of Giles and Trotter [7] by (i) showing that for any nonnegative integer a there exists a circulant graph whose stable set polytope has a facet-inducing inequality with (a,a+1)-valued coefficients (rank facets have only coefficients 0, 1), and (ii) providing new facets of the stable set polytope with up to five different non-zero coefficients for claw-free graphs. We prove that coefficients have to be consecutive in any facet with exactly two different non-zero coefficients (assuming they are relatively prime). Last but not least, we present a complete description of the stable set polytope for graphs with stability number 2, already observed by Cook [3] and Shepherd [18

DSA-aware multiple patterning for the manufacturing of vias: Connections to graph coloring problems, IP formulations, and numerical experiments

In this paper, we investigate the manufacturing of vias in integrated circuits with a new technology combining lithography and Directed Self Assembly (DSA). Optimizing the production time and costs in this new process entails minimizing the number of lithography steps, which constitutes a generalization of graph coloring. We develop integer programming formulations for several variants of interest in the industry, and then study the computational performance of our formulations on true industrial instances. We show that the best integer programming formulation achieves good computational performance, and indicate potential directions to further speed-up computational time and develop exact approaches feasible for production

Debris Disks of Members of the Blanco 1 Open Cluster

We have used the Spitzer Space Telescope to obtain Multiband Imaging Photometer for Spitzer (MIPS) 24 um photometry for 37 members of the ~100 Myr old open cluster Blanco 1. For the brightest 25 of these stars (where we have 3sigma uncertainties less than 15%), we find significant mid-IR excesses for eight stars, corresponding to a debris disk detection frequency of about 32%. The stars with excesses include two A stars, four F dwarfs and two G dwarfs. The most significant linkage between 24 um excess and any other stellar property for our Blanco 1 sample of stars is with binarity. Blanco 1 members that are photometric binaries show few or no detected 24 um excesses whereas a quarter of the apparently single Blanco 1 members do have excesses. We have examined the MIPS data for two other clusters of similar age to Blanco 1 -- NGC 2547 and the Pleiades. The AFGK photometric binary star members of both of these clusters also show a much lower frequency of 24 um excesses compared to stars that lie near the single-star main sequence. We provide a new determination of the relation between V-Ks color and Ks-[24] color for main sequence photospheres based on Hyades members observed with MIPS. As a result of our analysis of the Hyades data, we identify three low mass Hyades members as candidates for having debris disks near the MIPS detection limit.Comment: Accepted to Ap