Minimal multicut and maximal integer multiflow: A survey

We provide combinatorial algorithms for the unsplittable flow problem (UFP) that either match or improve the previously best results. In the UFP we are given a (possibly directed) capacitated graph with n vertices and m edges, and a set of terminal pairs each with its own demand and profit. The objective is to connect a subset of the terminal pairs each by a single flow path subject to the capacity constraints such that the total profit of the connected pairs is maximized.We consider three variants of the problem. First is the classical UFP in which the maximum demand is at most the minimum edge capacity. It was previously known to have an O(√m) approximation algorithm; the algorithm is based on the randomized rounding technique and its analysis makes use of the Chernoff bound and the FKG inequality.We provide a combinatorial algorithm that achieves the same approximation ratio and whose analysis is considerably simpler. Second is the extended UFP in which some demands might be higher than edge capacities. Our algorithm for this case improves the best known approximation ratio. We also give a lower bound that shows that the extended UFP is provably harder than the classical UFP. Finally, we consider the bounded UFP in which the maximum demand is at most 1/K times the minimum edge capacity for some K > 1. Here we provide combinatorial algorithms that match the currently best known algorithms. All of our algorithms are strongly polynomial and some can even be used in the online setting.

Finding Paths between graph colourings: PSPACE-completeness and superpolynomial distances

On The Number of Subgraphs of Prescribed Type of Planar Graphs With A Given Number of Vertices

For any positive integer k, the total k-domatic partition problem is to partition the vertices of a graph G into k pairwise disjoint total dominating sets. In this paper, we study the problem for planar graphs, chordal bipartite graphs, convex bipartite graphs, and bipartite permutation graphs. We show that the total 3-domatic partition problem on planar graphs is NP-complete. Moreover, we give an alternative algorithm to solve the total k-domatic partition problem for chordal bipartite graphs with weak elimination orderings, and adapt it to solve the problem in linear time for bipartite permutation graphs and convex bipartite graphs even if Gamma-free forms of the adjacency matrices of the considered graphs are not given.

In this paper, we present a constant-factor approximation algorithm for the unsplittable flow problem on a path. This improves on the previous best known approximation factor of O(log n). The approximation ratio of our algorithm is 7+e for any e>;0. In the unsplittable flow problem on a path, we are given a capacitated path P and n tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge e of P, the total demand of selected tasks that use e does not exceed the capacity of e. This is a well-studied problem that occurs naturally in various settings, and therefore it has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. Polynomial time constant factor approximation algorithms for the problem were previously known only under the no-bottleneck assumption (in which the maximum task demand must be no greater than the minimum edge capacity). We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either 1, 2, or 3.

Faster Shortest-Path Algorithms for Planar Graphs

Greenʼs theorem and isolation in planar graphs

I have always used concrete marupulatives, such as marshmallows and toothpicks, to create models for my geometry and discrete-mathematics courses. These models have come in handy when discussing volume, introducing the 4-cube, or illustrating isomorphic or bipartite graphs. However, after discovering what a dynamic geometry–software package could do for geometry teaching, which has been well documented by research (e.g., Battista and Clements [1995]), I realized that this type of technology also had much to offer for teaching graph theory in my discrete-mathematics course. Although this article discusses The Geometer's Sketchpad 3 (Jackiw 1995), any software that can draw, label, and drag figures can be substituted for Sketchpad.

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330–338) by proving that it is \(\textsc {NP}\)-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into k independent dominating sets is \(\textsc {NP}\)-complete for every \(k \ge 3\). We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451–6458) by showing that it is \(\textsc {NP}\)-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every \(k \ge 3\), we show that there is some constant t depending only on k such that deciding whether a k-regular graph can be partitioned into t independent dominating sets is \(\textsc {NP}\)-complete. We conclude by deriving moderately exponential-time algorithms for the problem.

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized online algorithms.

Given a graph with a source and a sink node, the NP-hard maximum k-splittable s,t-flow (MkSF) problem is to find a flow of maximum value from s to t with a flow decomposition using at most k paths. The multicommodity variant of this problem is a natural generalization of disjoint paths and unsplittable flow problems. Constructing a k-splittable flow requires two interdepending decisions. One has to decide on k paths (routing) and on the flow values for the paths (packing). We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step. Usually the routing is considered before the packing. Our main contributions are as follows: (i) We show that for constant k a polynomial number of packing alternatives containing at least one packing used by an optimal M kSF solution can be constructed in polynomial time. If k is part of the input, we obtain a slightly weaker result. In this case we can guarantee that, for any fixed e>0, the computed set of alternatives contains a packing used by a (1−e)-approximate solution. The latter result is based on the observation that (1−e)-approximate flows only require constantly many different flow values. We believe that this observation is of interest in its own right. (ii) Based on (i), we prove that, for constant k, the MkSF problem can be solved in polynomial time on graphs of bounded treewidth. If k is part of the input, this problem is still NP-hard and we present a polynomial time approximation scheme for it.

In this thesis, we study the Multidimensional Knapsack Problem (MKP) and two closely related special cases: the Unsplittable Flow Problem on Trees (UFPT), and the Unsplittable Flow Problem on Paths (UFPP). For these problems, when the natural integer programming formulation is relaxed to a linear program, the integrality gap is O(n). Previous work on UFPT and UFPP has established that the addition of rank constraints to the linear program can be effective in some cases at reducing the integrality gap. However, Friggstad and Gao proved that even with all rank constraints added, the integrality gap of UFPT is Ω(√(log n)). Faenza and Sanità showed that a formulation for these problems which approximates the integer hull arbitrarily well must either have exponentially many facets or be described as an extended formulation (i.e., described in a higher dimensional space). We are interested in polynomially sized formulations, so we focus our study on extended formulations. Our first contribution is a greedy algorithm which finds an optimal solution to the linear programming relaxation for the special case of UFPT where all requests share a common endpoint. We apply this to tighten the analysis of hard instances described in the literature. Following this, we describe two approximate extended formulations for MKP using disjunctive programming. The first is a formulation which improves upon the integrality gap of the linear relaxation using only a small number of extra variables and constraints. The second is a polyhedral (1 - ε)-approximate extended formulation for MKP for any 0 &lt; ε ≤ 1, which was originally given by Pritchard. We then introduce a new hierarchy of strengthened formulations for MKP, which we study in the context of UFPT. The strengthened formulations are NP-Hard to separate over, but they can be (1 - ε)-approximated using the aforementioned result. We conclude by evaluating the strength of this hierarchy when applied to the known hard instances from the literature. Our results suggest that this hierarchy may be most useful in conjunction with the rank constraints, but open questions remain.

The Disjoint Connecting Paths problem and its capacitated generalization, called Unsplittable Flow problem, play an important role in practical applications such as communication network design and routing. These tasks are NP-hard in general, but various polynomial-time approximations are known. Nevertheless, the approximations tend to be either too loose (allowing large deviation from the optimum), or too complicated, often rendering them impractical in large, complex networks. Therefore, our goal is to present a solution that provides a relatively simple, efficient algorithm for the unsplittable flow problem in large directed graphs, where the task is NP-hard, and is known to remain NP-hard even to approximate up to a large factor. The efficiency of our algorithm is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm for approximation. Rather than giving up the search for an exact solution, we restrict the solution space to a subset that is the most important for applications, and excludes only a small part that is marginal in some well-defined sense. Specifically, the sacrificed part only contains scenarios where some edges are very close to saturation. Since nearly saturated links are undesirable in practical applications, therefore, excluding near saturation is quite reasonable from the practical point of view. We refer the solutions that contain no nearly saturated edges as safe solutions, and call the approach safe approximation. We prove that this safe approximation can be carried out efficiently. That is, once we restrict ourselves to safe solutions, we can find the exact optimum by a randomized polynomial time algorithm.

Although Min-CDS in general graphs is hard to approximate, the restriction to certain special graph classes admits much better approximation results. Min-CDS in planar graphs remains NP-hard even for planar graphs that are regular of degree 4 [57]. The related problem, Min-DS in planar graphs, is also NP-hard even for planar graphs with maximum vertex degree 3 and planar graphs that are regular of degree 4 [57]. It is well known that Min-DS in planar graphs possesses a polynomial-time approximation scheme (PTAS) based on the shifting strategy [3]: For any constant e >0, there is a polynomial-time 1+e-approximation algorithm. Thus, it is immediate to conclude that Min-CDS in planar graphs can be approximated within a factor 3+e for any e>0 in polynomial time. However, the degree of the polynomial grows with 1∕e and hence, the approximation scheme is hardly practical.