Degree Sequence Optimization in Bounded Treewidth

We consider the problem of finding a subgraph of a given graph which minimizes the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already when all functions are the same, we show that it can be solved for arbitrary functions in polynomial time over graphs of bounded treewidth. Its complexity remains widely open, in particular over complete graphs and complete bipartite graphs

On Degree Sequence Optimization

We consider the problem of finding a subgraph of a given graph which maximizes a given function evaluated at its degree sequence. While the problem is intractable already for convex functions, we show that it can be solved in polynomial time for convex multi-criteria objectives. We next consider the problem with separable objectives, which is NP-hard already when all vertex functions are the square. We consider a colored extension of the separable problem, which includes the notorious exact matching problem as a special case, and show that it can be solved in polynomial time on graphs of bounded tree-depth for any vertex functions. We mention some of the many remaining open problems

A Note on M-convex Functions on Jump Systems

A jump system is defined as a set of integer points (vectors) with a certain exchange property, generalizing the concepts of matroids, delta-matroids, and base polyhedra of integral polymatroids (or submodular systems). A discrete convexity concept is defined for functions on constant-parity jump systems and it has been used in graph theory and algebra. In this paper we call it "jump M-convexity" and extend it to "jump M-natural-convexity" for functions defined on a larger class of jump systems. By definition, every jump M-convex function is a jump M-natural-convex function, and we show the equivalence of these concepts by establishing an (injective) embedding of jump M-natural-convex functions in n variables into the set of jump M-convex functions in n+1 variables. Using this equivalence we show further that jump M-natural-convex functions admit a number of natural operations such as aggregation, projection (partial minimization), convolution, composition, and transformation by a network.Comment: 16 page

Decreasing Minimization on M-convex Sets: Algorithms and Applications

This paper is concerned with algorithms and applications of decreasing minimization on an M-convex set, which is the set of integral elements of an integral base-polyhedron. Based on a recent characterization of decreasingly minimal (dec-min) elements, we develop a strongly polynomial algorithm for computing a dec-min element of an M-convex set. The matroidal feature of the set of dec-min elements makes it possible to compute a minimum cost dec-min element, as well. Our second goal is to exhibit various applications in matroid and network optimization, resource allocation, and (hyper)graph orientation. We extend earlier results on semi-matchings to a large degree by developing a structural description of dec-min in-degree bounded orientations of a graph. This characterization gives rise to a strongly polynomial algorithm for finding a minimum edge-cost dec-min orientation.Comment: 35 pages. This is a revised version of the second half of "A. Frank and K. Murota; Discrete decreasing minimization, PartI: Base-polyhedra with applications in network optimization" arXiv:1808.0760

Discrete Decreasing Minimization, Part I: Base-polyhedra with Applications in Network Optimization

Borradaile et al. (2017) investigated orientations of an undirected graph in which the sequence of in-degrees of the nodes is lexicographically minimal, which we call decreasingly minimal (=dec-min). They proved that an orientation is dec-min if and only if there is no dipath from $s$ to $t$ with in-degrees $\varrho (t) \geq \varrho (s)+2$. They conjectured that an analogous statement holds for strongly connected dec-min orientations, as well. We prove not only this conjecture but its extension to $k$-edge-connected orientations, as well. We also provide a solution to a discrete version of Megiddo's lexicographically optimal (fractional) network flow problem (1974, 1977). Our main goal is to integrate these cases into a single framework. Namely, we characterize dec-min elements of an M-convex set (which is nothing but the set of integral points of an integral base-polyhedron), and prove that the set of dec-min elements is a special M-convex set arising from a matroid base-polyhedron by translation. The topic of our investigations may be interpreted as a discrete counter-part of the work by Fujishige (1980) on the (unique) lexicographically optimal base of a base-polyhedron. We also exhibit a canonical chain (and partition) associated with a base-polyhedron. We also show that dec-min elements of an M-convex set are exactly those which minimize the square-sum of components, and describe a new min-max formula for the minimum square-sum. Our approach gives rise to a strongly polynomial algorithm for computing a dec-min element, as well as the canonical chain. The algorithm relies on a submodular function minimizer oracle in the general case, which can, however, be replaced by more efficient classic flow- and matroid algorithms in the relevant special cases.Comment: 63 page

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