Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating.Comment: Third version version. 38 page

Computational Complexity of Determining the Barriers to Interface Motion in Random Systems

The low-temperature driven or thermally activated motion of several condensed matter systems is often modeled by the dynamics of interfaces (co-dimension-1 elastic manifolds) subject to a random potential. Two characteristic quantitative features of the energy landscape of such a many-degree-of-freedom system are the ground-state energy and the magnitude of the energy barriers between given configurations. While the numerical determination of the former can be accomplished in time polynomial in the system size, it is shown here that the problem of determining the latter quantity is NP-complete. Exact computation of barriers is therefore (almost certainly) much more difficult than determining the exact ground states of interfaces.Comment: 8 pages, figures included, to appear in Phys. Rev.

New and simple algorithms for stable flow problems

Stable flows generalize the well-known concept of stable matchings to markets in which transactions may involve several agents, forwarding flow from one to another. An instance of the problem consists of a capacitated directed network, in which vertices express their preferences over their incident edges. A network flow is stable if there is no group of vertices that all could benefit from rerouting the flow along a walk. Fleiner established that a stable flow always exists by reducing it to the stable allocation problem. We present an augmenting-path algorithm for computing a stable flow, the first algorithm that achieves polynomial running time for this problem without using stable allocation as a black-box subroutine. We further consider the problem of finding a stable flow such that the flow value on every edge is within a given interval. For this problem, we present an elegant graph transformation and based on this, we devise a simple and fast algorithm, which also can be used to find a solution to the stable marriage problem with forced and forbidden edges. Finally, we study the stable multicommodity flow model introduced by Kir\'{a}ly and Pap. The original model is highly involved and allows for commodity-dependent preference lists at the vertices and commodity-specific edge capacities. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is NP-complete to decide whether an integral solution exists

Directed Multicut with linearly ordered terminals

Motivated by an application in network security, we investigate the following "linear" case of Directed Mutlicut. Let $G$ be a directed graph which includes some distinguished vertices $t_1, \ldots, t_k$. What is the size of the smallest edge cut which eliminates all paths from $t_i$ to $t_j$ for all $i < j$? We show that this problem is fixed-parameter tractable when parametrized in the cutset size $p$ via an algorithm running in $O(4^p p n^4)$ time.Comment: 12 pages, 1 figur