A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs

AbstractIn this paper, we consider the problem of finding a maximum weight 2-matching containing no cycle of a length of at most three in a weighted simple graph, which we call the weighted triangle-free 2-matching problem. Although the polynomial solvability of this problem is still open in general graphs, a polynomial-time algorithm is given by Hartvigsen and Li for the problem in subcubic graphs, i.e., graphs with a maximum degree of at most three. Our contribution is to provide another polynomial-time algorithm for the weighted triangle-free 2-matching problem in subcubic graphs. Our algorithm consists of two basic algorithms: a steepest ascent algorithm and a classical maximum weight2-matching algorithm, and is justified by fundamental results from the theory of discrete convex functions on jump systems

Min Cost Flow in balancierten Netzwerken mit konvexer Kostenfunktion

Standard matching problems can be stated in terms of skew symmetric networks. On skew symmetric networks matching problems can be solved using network flow techniques. We consider the problem of minimizing a separable convex objective function over a skew-symmetric network with a balanced flow. We call this problem the Convex Balanced Min Cost Flow (Convex BMCF) problem. We start with 2 examples of Convex BMCF problems. The first problem is a problem from condensed matter physics: We want to simulate a so called super-rough phase using methods from graph-theory. This problem has previously been studied by Blasum, Hochstaettler, Rieger a and Moll. The second problem is a typical example for the minconvex-problems previously studied by Apollonio and Sebo and Berger and Hochstaettler [9]. We review the results for skew-symmetric networks by Jungnickel and Fremuth-Paeger and Kocay and Stone. Using these results we present several algorithms to solve the Convex BMCF problem. We present the first complete version of the Primal-Dual algorithm previously studied by Fremuth-Paeger and Jungnickel. However, we only consider the case of positive costs. We also show how to apply this algorithm to the Convex BMCF problem. Then we extend the Shortest Admissible Path Approach of Jungnickel and Fremuth-Paeger [23, p. 12] to a complete algorithm for linear as well as convex cost problems on skew symmetric networks. In the same manner we show how to adapt the Capacity Scaling algorithm by Ahuja and Orlin to skew symmetric networks and balanced flows. The capacity scaling algorithm is weakly polynomial. Another possibility for a weakly polynomial algorithm is the Balanced Out-of-Kilter algorithm. This algorithm is based on Fulkerson’s Out-of-Kilter algorithm and Minoux’s adaptation of the algorithm for convex costs. We show that augmentation on valid paths is not always necessary and introduce the idea of slightly different networks. Using the same ideas for the Balanced Capacity Scaling we obtain an Enhanced Capacity Scaling algorithm. The Enhanced Capacity Scaling algorithm as well as the Balanced Out-of-Kilter algorithm are the fastest algorithms presented here with a complexity of roughly O(m2log2U). Finally we show how to solve the problem from condensed matter physics using the new idea of anti-balanced flows on skew-symmetric networks. Using the Balanced Successive Shortest Path algorithm we also obtain a new complexity limit for the minconvex problem. This improves the complexity bound of Berger [8] by a factor of m in the case of separable convex costs with positive slope. In the appendix of this thesis we consider dual approaches for the Convex BMCF problem. The Balanced Relaxation algorithm, based on the Relaxation algorithm by Bertsekas [13], does not determine a balanced flow as the resulting flow will not necessarily be integral. This way we only determine fractional matchings. As the algorithm is also slow this algorithm is probably of limited use. A better ansatz seems to be the Cancel and Tighten method by Karzanov and McCormick. We review their results and end with some ideas on how to implement a balanced version of this algorithm