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

Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a simple path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding $\Sigma_2$-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies. On the positive side, we give a polynomial-time algorithm for monomino clues, by reducing to hexagon clues on the boundary of the puzzle, even in the presence of broken edges, and solving "subset Hamiltonian path" for terminals on the boundary of an embedded planar graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of this paper appeared at the 9th International Conference on Fun with Algorithms (FUN 2018

Pure Nash Equilibria: Hard and Easy Games

We investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is SigmaP2-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each players payoff depends on moves of other players. We say that a game has small neighborhood if the utility function for each player depends only on (the actions of) a logarithmically small number of other players. The dependency structure of a game G can be expressed by a graph DG(G) or by a hypergraph H(G). By relating Nash equilibrium problems to constraint satisfaction problems (CSPs), we show that if G has small neighborhood and if H(G) has bounded hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class NC2 of highly parallelizable problems