Space Saving by Dynamic Algebraization

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof such that the dynamic programming algorithm runs in time $O^*(2^h)$, where $h$ is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.Comment: 14 pages, 1 figur

On the Equivalence among Problems of Bounded Width

In this paper, we introduce a methodology, called decomposition-based reductions, for showing the equivalence among various problems of bounded-width. First, we show that the following are equivalent for any $\alpha > 0$: * SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * 3-SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * Max 2-SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * Independent Set can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, and * Independent Set can be solved in $O^*(2^{\alpha \mathrm{cw}})$ time, where tw and cw are the tree-width and clique-width of the instance, respectively. Then, we introduce a new parameterized complexity class EPNL, which includes Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and Independent Set parameterized by path-width are EPNL-complete. This implies that if one of these EPNL-complete problems can be solved in $O^*(c^k)$ time, then any problem in EPNL can be solved in $O^*(c^k)$ time.Comment: accepted to ESA 201

Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220

The Polygenic and Monogenic Basis of Blood Traits and Diseases

Blood cells play essential roles in human health, underpinning physiological processes such as immunity, oxygen transport, and clotting, which when perturbed cause a significant global health burden. Here we integrate data from UK Biobank and a large-scale international collaborative effort, including data for 563,085 European ancestry participants, and discover 5,106 new genetic variants independently associated with 29 blood cell phenotypes covering a range of variation impacting hematopoiesis. We holistically characterize the genetic architecture of hematopoiesis, assess the relevance of the omnigenic model to blood cell phenotypes, delineate relevant hematopoietic cell states influenced by regulatory genetic variants and gene networks, identify novel splice-altering variants mediating the associations, and assess the polygenic prediction potential for blood traits and clinical disorders at the interface of complex and Mendelian genetics. These results show the power of large-scale blood cell trait GWAS to interrogate clinically meaningful variants across a wide allelic spectrum of human variation. Analysis of blood cell traits in the UK Biobank and other cohorts illuminates the full genetic architecture of hematopoietic phenotypes, with evidence supporting the omnigenic model for complex traits and linking polygenic burden with monogenic blood diseases

Graph Minors and Parameterized Algorithm Design

Abstract. The Graph Minors Theory, developed by Robertson and Sey-mour, has been one of the most influential mathematical theories in pa-rameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct meta-algorithmic consequences, we present the algorithmic appli-cations of core theorems such as the grid-exclusion theorem, and we give a brief description of the irrelevant vertex technique

Exact Exponential-Time Algorithms for Domination Problems in Graphs

This PhD thesis studies exact exponential-time algorithms for domination problems in graphs. Domination problems in graphs are a special kind of subset problems in graphs. A subset problem in a graph is a problem where one is given a graph G=(V,E), and one is asked whether there exist some subset S of a set of items U in the graph (mostly U is either the vertices V or the edges E) that satisfies certain properties. Domination problems in graphs are subset problems in which there is a domination criterion based on a neighbourhood relation in the graph that decides which elements of U are dominated by a given subset S, and where one of the properties that a solution subset S must satisfy is that S must dominate its complement U\S. The most well-known graph domination problem is the Dominating Set problem where the set U is the set of vertices V of G, a vertex subset S dominates all vertices in G that have a neighbour in S, and one is asked to compute the smallest vertex subset S of V that dominates all vertices in V\S. Other examples of domination problems in graphs are Independent Set, Edge Dominating Set, Total Dominating Set, Red-Blue Dominating Set, Partial Dominating Set, and #Dominating Set. We study exact exponential-time algorithms for these problems. These are algorithms that, when executed, use a number of operations that is exponential in a complexity parameter of the input in the worst case. That is, these algorithms use exponential time. Exact exponential-time algorithms then return an optimal solution to the problem. This in contrast to other fields of algorithm design where one sometimes trades running time for other properties of the algorithm or the returned solution. In this thesis, we also study parameterised algorithms for domination problems in graph on graph decompositions. These are algorithms whose worst-case running times are polynomial in the size of the graph and exponential in the graph-width parameter associated to the graph decomposition. Our study led to faster exact exponential-time algorithms for many well-known graph domination problems. This includes an O(1.4969^n)-time algorithm for Dominating Set, an O(1.2114^n)-time algorithm for Independent Set, an O(1.3226^n)-time algorithm for Edge Dominating Set, an O(1.4969^n)-time algorithm for Total Dominating Set, an O(1.2252^n)-time algorithm for Red-Blue Dominating Set, an O(1.5014^n)-time algorithm for Partial Dominating Set, and an O(1.5002^n)-time algorithm for #Dominating Set. We also obtained faster algorithms for these and many other graph domination problems on some prominent types of graph decompositions: tree decompositions, branch decompositions, and, for some problems, clique decompositions (also called k-expressions). A series of interesting new insights and techniques arose from this study. We mention the techniques of inclusion/exclusion-based branching and extended inclusion/exclusion-based branching. We also mention our generalisation of the fast subset convolution algorithm, which we translated to the setting of state-based dynamic programming algorithms on graph decompositions. This thesis also contains an accessible introduction to the field of exact exponential-time algorithms

On partitioning a graph into two connected subgraphs

Suppose a graph G is given with two vertex-disjoint sets of vertices Z1 and Z2. Can we partition the remaining vertices of G such that we obtain two connected vertex-disjoint subgraphs of G that contain Z1 and Z2, respectively? This problem is known as the 2-Disjoint Connected Subgraphs problem. It is already NP-complete for the class of n-vertex graphs G=(V,E) in which Z1 and Z2 each contain a connected set that dominates all vertices in V∖(Z1∪Z2). We present an O∗(1.2051n) time algorithm that solves it for this graph class. As a consequence, we can also solve this problem in O∗(1.2051n) time for the classes of n-vertex P6-free graphs and split graphs. This is an improvement upon a recent O∗(1.5790n) time algorithm for these two classes. Our approach translates the problem to a generalized version of hypergraph 2-coloring and combines inclusion/exclusion with measure and conquer