Packing non-returning A-paths algorithmically

In this paper we present an algorithmic approach to packing A-paths. It is regarded as a generalization of Edmonds' matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the so-called 3-way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing non-returning A-paths. It also implies an ear-decomposition of criticals, as a generalization of the odd ear-decomposition of factor-critical graph

Min-Max Theorems for Packing and Covering Odd (u,v)(u,v)-trails

We investigate the problem of packing and covering odd $(u,v)$-trails in a graph. A $(u,v)$-trail is a $(u,v)$-walk that is allowed to have repeated vertices but no repeated edges. We call a trail odd if the number of edges in the trail is odd. Let $\nu(u,v)$ denote the maximum number of edge-disjoint odd $(u,v)$-trails, and $\tau(u,v)$ denote the minimum size of an edge-set that intersects every odd $(u,v)$-trail. We prove that $\tau(u,v)\leq 2\nu(u,v)+1$. Our result is tight---there are examples showing that $\tau(u,v)=2\nu(u,v)+1$---and substantially improves upon the bound of $8$ obtained in [Churchley et al 2016] for $\tau(u,v)/\nu(u,v)$. Our proof also yields a polynomial-time algorithm for finding a cover and a collection of trails satisfying the above bounds. Our proof is simple and has two main ingredients. We show that (loosely speaking) the problem can be reduced to the problem of packing and covering odd $(uv,uv)$-trails losing a factor of 2 (either in the number of trails found, or the size of the cover). Complementing this, we show that the odd-$(uv,uv)$-trail packing and covering problems can be tackled by exploiting a powerful min-max result of [Chudnovsky et al 2006] for packing vertex-disjoint nonzero $A$-paths in group-labeled graphs

Packing Odd Walks and Trails in Multiterminal Networks

Let G be an undirected network with a distinguished set of terminals T ? V(G) and edge capacities cap: E(G) ? ?_+. By an odd T-walk we mean a walk in G (with possible vertex and edge self-intersections) connecting two distinct terminals and consisting of an odd number of edges. Inspired by the work of Schrijver and Seymour on odd path packing for two terminals, we consider packings of odd T-walks subject to capacities cap. First, we present a strongly polynomial time algorithm for constructing a maximum fractional packing of odd T-walks. For even integer capacities, our algorithm constructs a packing that is half-integer. Additionally, if cap(?(v)) is divisible by 4 for any v ? V(G)-T, our algorithm constructs an integer packing. Second, we establish and prove the corresponding min-max relation. Third, if G is inner Eulerian (i.e. degrees of all nodes in V(G)-T are even) and cap(e) = 2 for all e ? E, we show that there exists an integer packing of odd T-trails (i.e. odd T-walks with no repeated edges) of the same value as in case of odd T-walks, and this packing can be found in polynomial time. To achieve the above goals, we establish a connection between packings of odd T-walks and T-trails and certain multiflow problems in undirected and bidirected graphs

Half-integrality, LP-branching and FPT Algorithms

A recent trend in parameterized algorithms is the application of polytope tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011; Narayanaswamy et al., 2012). However, although interesting results have been achieved, the methods require the underlying polytope to have very restrictive properties (half-integrality and persistence), which are known only for few problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a relaxation of the search space from $\{0,1\}^V$ to $\{0,1/2,1\}^V$ such that the new problem admits a polynomial-time exact solution. Using tools from CSP (in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such relaxations, we provide a much broader class of half-integral polytopes with the required properties, unifying and extending previously known cases. In addition to the insight into problems with half-integral relaxations, our results yield a range of new and improved FPT algorithms, including an $O^*(|\Sigma|^{2k})$-time algorithm for node-deletion Unique Label Cover with label set $\Sigma$ and an $O^*(4^k)$-time algorithm for Group Feedback Vertex Set, including the setting where the group is only given by oracle access. All these significantly improve on previous results. The latter result also implies the first single-exponential time FPT algorithm for Subset Feedback Vertex Set, answering an open question of Cygan et al. (2012). Additionally, we propose a network flow-based approach to solve some cases of the relaxation problem. This gives the first linear-time FPT algorithm to edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA paper

Representative set statements for delta-matroids and the Mader delta-matroid

We present representative sets-style statements for linear delta-matroids, which are set systems that generalize matroids, with important connections to matching theory and graph embeddings. Furthermore, our proof uses a new approach of sieving polynomial families, which generalizes the linear algebra approach of the representative sets lemma to a setting of bounded-degree polynomials. The representative sets statements for linear delta-matroids then follow by analyzing the Pfaffian of the skew-symmetric matrix representing the delta-matroid. Applying the same framework to the determinant instead of the Pfaffian recovers the representative sets lemma for linear matroids. Altogether, this significantly extends the toolbox available for kernelization. As an application, we show an exact sparsification result for Mader networks: Let $G=(V,E)$ be a graph and $\mathcal{T}$ a partition of a set of terminals $T \subseteq V(G)$, $|T|=k$. A $\mathcal{T}$-path in $G$ is a path with endpoints in distinct parts of $\mathcal{T}$ and internal vertices disjoint from $T$. In polynomial time, we can derive a graph $G'=(V',E')$ with $T \subseteq V(G')$, such that for every subset $S \subseteq T$ there is a packing of $\mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in $G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the) cut-covering lemma, which corresponds to the case that $\mathcal{T}$ contains only two blocks. To prove the Mader network sparsification result, we furthermore define the class of Mader delta-matroids, and show that they have linear representations. This should be of independent interest

Packing non-returning A-paths algorithmically

In this paper we present an algorithmic approach to packing A-paths. It is regarded as a generalization of Edmonds' matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the so-called 3-way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing non-returning A-paths. It also implies an ear-decomposition of criticals, as a generalization of the odd ear-decomposition of factor-critical graph