Gyula Pap

Combinatorial algorithms for matchings, even factors and square-free 2-factors

Even factors and square-free 2-factors are restricted matching problems for which it seems to be difficult to generalize Edmonds’ matching algorithm directly. Here, we present a slight modification of Edmonds’ algorithm, which adapts to these restricted matching problems. Thus, we construct algorithms for these problems which do not use alternating forests.

On the maximum even factor in weakly symmetric graphs

As a common generalization of matchings and matroid intersection, Cunningham and Geelen introduced the notion of path-matchings, then they introduced the more general notion of even factor in weakly symmetric digraphs. Here we give a min-max formula for the maximum cardinality of an even factor. Our proof is purely combinatorial. We also provide a Gallai-Edmonds-type structure theorem for even factors.

Packing Non-Returning A -Paths*

Chudnovsky et al. gave a min-max formula for the maximum number of node-disjoint nonzero A-paths in group-labeled graphs [1], which is a generalization of Maders theorem on node-disjoint A-paths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside

Globally optimal pixel labeling algorithms for tree metrics

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Weighted restricted 2-matching

, which is given by an oracle for non-bipartite matching.

Matchings and Nonrainbow Colorings

We construct a combinatorial algorithm to find a maximum packing of fully node-disjoint non-returning A-paths.

Matchings of cycles and paths in directed graphs

We consider pixel labeling problems where the label set forms a tree, and where the observations are also labels. Such problems arise in feature-space analysis with a very large label set, for instance in color image segmentation. In this case a tree of labels can be constructed via hierarchical clustering of the observations. This leads to an obvious distance function between two labels, namely their distance within the tree; such tree metrics have been extensively studied outside of computer vision [14]. We provide fast algorithms that use graph cuts to exactly minimize the energy function for pixel labeling problems with tree metrics. Our work substantially improves a facility location algorithm of Kolen [18], which is impractical for large label sets L since it requires O(|L|) min cuts on large graphs. Our main technical contribution is a new ordering of swap moves that reduces the running time to the equivalent of O(log |L|) min cuts; as a result, we can handle realistic-sized color images in a few seconds.

Hypo-matchings in Directed Graphs

We give a TDI description for a class of polytopes, which corresponds to a restricted 2-matching problem. The perfect matching polytope, triangle-free perfect 2-matching polytope and relaxations of the travelling salesman polytope are members of this class. The paper shows that 2-matching problems for which the unweighted problem was known to be tractable, the weighted is also tractable.