Maximum Skew-Symmetric Flows and Matchings

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio

Hard cases of the multifacility location problem

AbstractLet μ be a rational-valued metric on a finite set T. We consider (a version of) the multifacility location problem: given a finite set V⊇T and a function c:V2→Z+, attach each element x∈V−T to an element γ(x)∈T minimizing ∑c(xy)μ(γ(x)γ(y)):xy∈V2, letting γ(t)≔t for each t∈T. Large classes of metrics μ have been known for which the problem is solvable in polynomial time. On the other hand, Dalhaus et al. (SIAM J. Comput. 23 (4) (1994) 864) showed that if T={t1,t2,t3} and μ(titj)=1 for all i≠j, then the problem (turning into the minimum 3-terminal cut problem) becomes strongly NP-hard. Extending that result and its generalization in (European J. Combin. 19 (1998) 71), we prove that for μ fixed, the problem is strongly NP-hard if the metric μ is nonmodular or if the underlying graph of μ is nonorientable (in a certain sense)

Concave cocirculations in a triangular grid

AbstractLet G be a planar digraph embedded in the plane such that each bounded face contains three edges and forms an equilateral triangle, and let the union R of these faces be a convex polygon. We consider the polyhedral cone B(G) formed by the real-valued functions σ on the set of boundary edges of G with the following property: there exists a concave function c on R which is affinely linear within each bounded face and satisfies c(v)−c(u)=σ(e) for each boundary edge e=(u,v). Knutson, Tao and Woodward obtained a result on honeycombs which implies that if the polygon R is a triangle, then the cone B(G) is described by linear inequalities of Horn’s type with respect to so-called puzzles, along with obvious linear constraints. The purpose of this paper is to give an alternative proof of that result, working in terms of discrete concave functions, rather than honeycombs. Our proof is based on a linear programming approach and a nonstandard flow model. Moreover, the result is extended to an arbitrary convex polygon R as above

On one maximum multiflow problem and related metrics

AbstractWe consider the undirected maximum multiflow (multicommodity flow) problem in the case when the commodity graph is the disjoint union of K3 and K2. We prove that if the supply graph satisfies a certain Eulerian-type condition, then the problem has an integer optimal solution. To obtain this result, we first study the corresponding dual problem on metrics and show that an optimal solution to the latter is achieved on some (2,3)-metric or some 3-cut metric

Condorcet domains of tiling type

A Condorcet domain (CD) is a collection of linear orders on a set of candidates satisfying the following property: for any choice of preferences of voters from this collection, a simple majority rule does not yield cycles. We propose a method of constructing "large" CDs by use of rhombus tiling diagrams and explain that this method unifies several constructions of CDs known earlier. Finally, we show that three conjectures on the maximal sizes of those CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic