Global and Fixed-Terminal Cuts in Digraphs

The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut. 1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC. 2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC. 3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem. Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances

Cuts and connectivity in graphs and hypergraphs

In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following: - We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation. - We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2+epsilon)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier. - We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span. - We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-1/448 approximation algorithm for the global bicut problem

Every countable model of set theory embeds into its own constructible universe

The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random Q-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph, closely connected with the surreal numbers. The proof shows that $L^M$ contains a submodel that is a universal acyclic digraph of rank $Ord^M$. The method of proof also establishes that the countable models of set theory are linearly pre-ordered by embeddability: for any two countable models of set theory, one of them is isomorphic to a submodel of the other. Indeed, they are pre-well-ordered by embedability in order-type exactly $\omega_1+1$. Specifically, the countable well-founded models are ordered by embeddability in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory $M$ is universal for all countable well-founded binary relations of rank at most $Ord^M$; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the same proof method shows that if $M$ is any nonstandard model of PA, then every countable model of set theory---in particular, every model of ZFC---is isomorphic to a submodel of the hereditarily finite sets $HF^M$ of $M$. Indeed, $HF^M$ is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe. (v2 adds a reference and makes minor corrections) (v3 includes further changes, and removes the previous theorem 15, which was incorrect.

Single Source - All Sinks Max Flows in Planar Digraphs

Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

Parameterized Algorithms for Min-Max Multiway Cut and List Digraph Homomorphism

In this paper we design {\sf FPT}-algorithms for two parameterized problems. The first is \textsc{List Digraph Homomorphism}: given two digraphs $G$ and $H$ and a list of allowed vertices of $H$ for every vertex of $G$, the question is whether there exists a homomorphism from $G$ to $H$ respecting the list constraints. The second problem is a variant of \textsc{Multiway Cut}, namely \textsc{Min-Max Multiway Cut}: given a graph $G$, a non-negative integer $\ell$, and a set $T$ of $r$ terminals, the question is whether we can partition the vertices of $G$ into $r$ parts such that (a) each part contains one terminal and (b) there are at most $\ell$ edges with only one endpoint in this part. We parameterize \textsc{List Digraph Homomorphism} by the number $w$ of edges of $G$ that are mapped to non-loop edges of $H$ and we give a time $2^{O(\ell\cdot\log h+\ell^2\cdot \log \ell)}\cdot n^{4}\cdot \log n$ algorithm, where $h$ is the order of the host graph $H$. We also prove that \textsc{Min-Max Multiway Cut} can be solved in time $2^{O((\ell r)^2\log \ell r)}\cdot n^{4}\cdot \log n$. Our approach introduces a general problem, called {\sc List Allocation}, whose expressive power permits the design of parameterized reductions of both aforementioned problems to it. Then our results are based on an {\sf FPT}-algorithm for the {\sc List Allocation} problem that is designed using a suitable adaptation of the {\em randomized contractions} technique (introduced by [Chitnis, Cygan, Hajiaghayi, Pilipczuk, and Pilipczuk, FOCS 2012]).Comment: An extended abstract of this work will appear in the Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC), Patras, Greece, September 201

Packing tight Hamilton cycles in 3-uniform hypergraphs

Let H be a 3-uniform hypergraph with N vertices. A tight Hamilton cycle C \subset H is a collection of N edges for which there is an ordering of the vertices v_1, ..., v_N such that every triple of consecutive vertices {v_i, v_{i+1}, v_{i+2}} is an edge of C (indices are considered modulo N). We develop new techniques which enable us to prove that under certain natural pseudo-random conditions, almost all edges of H can be covered by edge-disjoint tight Hamilton cycles, for N divisible by 4. Consequently, we derive the corollary that random 3-uniform hypergraphs can be almost completely packed with tight Hamilton cycles w.h.p., for N divisible by 4 and P not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudo-random digraphs with even numbers of vertices.Comment: 31 pages, 1 figur

Finding Even Subgraphs Even Faster

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on $n$ vertices and a positive integer parameter $k$, find if there exist $k$ edges (arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the vertices have even degrees---where the resulting graph is \emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica, 2014}] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time $2^{O(k \log k)}n^{O(1)}$. They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. In this paper we answer their question in the affirmative: using the technique of computing \emph{representative families of co-graphic matroids} we design algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity

On width measures and topological problems on semi-complete digraphs

Under embargo until: 2021-02-01The topological theory for semi-complete digraphs, pioneered by Chudnovsky, Fradkin, Kim, Scott, and Seymour [10], [11], [12], [28], [43], [39], concentrates on the interplay between the most important width measures — cutwidth and pathwidth — and containment relations like topological/minor containment or immersion. We propose a new approach to this theory that is based on outdegree orderings and new families of obstacles for cutwidth and pathwidth. Using the new approach we are able to reprove the most important known results in a unified and simplified manner, as well as provide multiple improvements. Notably, we obtain a number of efficient approximation and fixed-parameter tractable algorithms for computing width measures of semi-complete digraphs, as well as fast fixed-parameter tractable algorithms for testing containment relations in the semi-complete setting. As a direct corollary of our work, we also derive explicit and essentially tight bounds on duality relations between width parameters and containment orderings in semi-complete digraphs.acceptedVersio

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application