RMT 555 - PERSEKITARAN PERUNDANGAN APRIL-MAY 06.

Abstract. We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis (MCB) in weighted partial 2-trees, i.e., graphs of treewidth two. The implicit representation can be made explicit in a running time that is proportional to the size of the MCB. For planar graphs, Borradaile, Sankowski, and Wulff-Nilsen [Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time, FOCS 2010] showed how to compute an implicit O(n log n) space representation of an MCB in O(n log 5 n) time. For the special case of partial 2-trees, our algorithm improves this result to linear time and space. Such an improvement was achieved previously only for outerplanar graphs [Liu and Lu: Minimum Cycle Bases of Weighted Outerplanar Graphs, IPL 110:970–974, 2010]

SAT Modulo Monotonic Theories

We define the concept of a monotonic theory and show how to build efficient SMT (SAT Modulo Theory) solvers, including effective theory propagation and clause learning, for such theories. We present examples showing that monotonic theories arise from many common problems, e.g., graph properties such as reachability, shortest paths, connected components, minimum spanning tree, and max-flow/min-cut, and then demonstrate our framework by building SMT solvers for each of these theories. We apply these solvers to procedural content generation problems, demonstrating major speed-ups over state-of-the-art approaches based on SAT or Answer Set Programming, and easily solving several instances that were previously impractical to solve

All-Pairs Minimum Cuts in Near-Linear Time for Surface-Embedded Graphs

For an undirected $n$-vertex graph $G$ with non-negative edge-weights, we consider the following type of query: given two vertices $s$ and $t$ in $G$, what is the weight of a minimum $st$-cut in $G$? We solve this problem in preprocessing time $O(n\log^3 n)$ for graphs of bounded genus, giving the first sub-quadratic time algorithm for this class of graphs. Our result also improves by a logarithmic factor a previous algorithm by Borradaile, Sankowski and Wulff-Nilsen (FOCS 2010) that applied only to planar graphs. Our algorithm constructs a Gomory-Hu tree for the given graph, providing a data structure with space $O(n)$ that can answer minimum-cut queries in constant time. The dependence on the genus of the input graph in our preprocessing time is $2^{O(g^2)}$

On Minimum Average Stretch Spanning Trees in Polygonal 2-trees

A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.Comment: 17 pages, 12 figure

Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^\omega+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^\omega)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page

Parametric shortest-path algorithms via tropical geometry

We study parameterized versions of classical algorithms for computing shortest-path trees. This is most easily expressed in terms of tropical geometry. Applications include shortest paths in traffic networks with variable link travel times.Comment: 24 pages and 8 figure