Packing and covering of crossing families of cuts

AbstractLet C be a crossing family of subsets of the finite set V (i.e., if T, U ∈ C and T ⋔ U ≠ ⊘, T ⌣ U ≠ V, then T ⋔ U ∈ C and T ⌣ U ∈ C). If D = (V, A) is a directed graph on V, then a cut induced by C is the set of arcs entering some set in C. A covering for C is a set of arcs entering each set in C, i.e., intersecting all cuts induced by C. It is shown that the following three conditions are equivalent for any given crossing family C: 1.(P1) For every directed graph D = (V, A), the minimum cardinality of a cut induced by C is equal to the maximum number of pairwise disjoint coverings for C.2.(P2) For every directed graph D = (V, A), and for every length function l: A → Z+, the minimum length of a covering for C is equal to the maximum number t of cuts C1,…, Ct induced by C (repetition allowed) such that no arc a is in more than l(a) of these cuts.3.(P3) ⊘ ∈ C, or V ∈ C, or there are no V1, V2, V3, V4, V5 in C such that V1 ⊆ V2 ⋔ V3, V2 ⌣ V3 = V, V3 ⌣ V4 ⊆ V5, V3 ⋔ V4 = ⊘.Directed graphs are allowed to have parallel arcs, so that (P1) is equivalent to its capacity version. (P1) and (P2) assert that certain hypergraphs, as well as their blockers, have the “Z+-max-flow min-cut property”. The equivalence of (P1), (P2), and (P3) implies Menger's theorem, the König-Egerváry theorem, the König-Gupta edge-colouring theorem for bipartite graphs, Fulkerson's optimum branching theorem, Edmonds' disjoint branching theorem, and theorems of Frank, Feofiloff and Younger, and the present author

Almost spanning subgraphs of random graphs after adversarial edge removal

Let Delta>1 be a fixed integer. We show that the random graph G(n,p) with p>>(log n/n)^{1/Delta} is robust with respect to the containment of almost spanning bipartite graphs H with maximum degree Delta and sublinear bandwidth in the following sense: asymptotically almost surely, if an adversary deletes arbitrary edges in G(n,p) such that each vertex loses less than half of its neighbours, then the resulting graph still contains a copy of all such H.Comment: 46 pages, 6 figure

Minimum Cuts in Near-Linear Time

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n^2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n^2 log^3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04)

Mutations and short geodesics in hyperbolic 3-manifolds

In this paper, we explicitly construct large classes of incommensurable hyperbolic knot complements with the same volume and the same initial (complex) length spectrum. Furthermore, we show that these knot complements are the only knot complements in their respective commensurabiltiy classes by analyzing their cusp shapes. The knot complements in each class differ by a topological cut-and-paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial (complex) length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold.Comment: This is the final (accepted) version of this pape

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