Jakub Onufry Wojtaszczyk

Solving Connectivity Problems Parameterized by Treewidth in Single Exponential Time

For the vast majority of local problems on graphs of small tree width (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c^tw |V|^O(1) time algorithms, where tw is the tree width of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best -- known algorithms were naive dynamic programming schemes running in at least tw^tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c^tw |V|^O(1) time Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Steiner Tree, Feedback Vertex Set and Connected Dominating Set. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and H-minor-free graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like Cycle Packing.

On multiway cut parameterized above lower bounds

In this paper we consider two above lower bound parameterizations of the NodeMultiway Cut problem -- above the maximum separating cut and above a natural LP-relaxation -- and prove them to be fixed-parameter tractable. Our results imply O*(4k) algorithms for Vertex Coverabove Maximum Matching and Almost 2-SAT as well as an O*(2k) algorithm for NodeMultiway Cut with a standard parameterization by the solution size, improving previous bounds for these problems.

On Some Extensions of the FKN Theorem

Let S = a1r1+ a2r2+ + anrn be a weighted Rademacher sum. Friedgut, Kalai, and Naor have shown that if Var(jSj) is much smaller than Var(S), then the sum is largely determined by one of the summands. We provide a simple and elementary proof of this result, strengthen it, and extend it in various ways to a more general setting.

Sitting closer to friends than enemies, revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves [13] initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space ℝl in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into ℝ1 can be expressed as a purely combinatorial problem. In this paper we pursue a deeper study of this case, answering several questions posed by Kermarrec and Thraves. First, we refine the approach of Kermarrec and Thraves for the case of complete signed graphs by showing that the problem is closely related to the recognition of proper interval graphs. Second, we prove that the general case, whose polynomial-time tractability remained open, is in fact NP-complete. Finally, we provide lower and upper bounds for the time complexity of the general case: we prove that the existence of a subexponential time (in the number of vertices and edges of the input signed graph) algorithm would violate the Exponential Time Hypothesis, whereas a simple dynamic programming approach gives a running time single-exponential in the number of vertices.

Solving the 2-disjoint connected subgraphs problem faster than 2 n

The 2-Disjoint Connected Subgraphs problem, given a graph along with two disjoint sets of terminals Z1, Z2, asks whether it is possible to find disjoint sets A1, A2, such that Z1⊆A1, Z2⊆A2 and A1, A2 induce connected subgraphs. While the naive algorithm runs in O(2nnO(1)) time, solutions with complexity of form O((2−e)n) have been found only for special graph classes [15, 19]. In this paper we present an O(1.933n) algorithm for 2-Disjoint Connected Subgraphs in general case, thus breaking the 2n barrier. As a counterpoise of this result we show that if we parameterize the problem by the number of non-terminal vertices, it is hard both to speed up the brute-force approach and to find a polynomial kernel.

Scheduling partially ordered jobs faster than 2 n

In a scheduling problem, denoted by 1|prec|∑Ci in the Graham notation, we are given a set of n jobs, together with their processing times and precedence constraints. The task is to order the jobs so that their total completion time is minimized. 1|prec|∑Ci is a special case of the Traveling Repairman Problem with precedences. A natural dynamic programming algorithm solves both these problems in 2nnO(1) time, and whether there exists an algorithms solving 1|prec|∑Ci in O(cn) time for some constant c<2 was an open problem posted in 2004 by Woeginger. In this paper we answer this question positively.

Sitting Closer to Friends than Enemies, Revisited

Signed graphs, i.e., undirected graphs with edges labelled with a plus or minus sign, are commonly used to model relationships in social networks. Recently, Kermarrec and Thraves (2011) initiated the study of the problem of appropriately visualising the network: They asked whether any signed graph can be embedded into the metric space ℝl

Approximation schemes for capacitated geometric network design

{{\mathbb {R}}}^{l}

Dominating set is fixed parameter tractable in claw-free graphs

in such a manner that every vertex is closer to all its friends (neighbours via positive edges) than to all its enemies (neighbours via negative edges). Interestingly, embeddability into ℝ1

Solving the 2-Disjoint Connected Subgraphs Problem Faster than 2n

{{\mathbb {R}}}^{1}