A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N [v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N (v), for which vertex v is not a member of its neighborhood.For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k ∈ {1, 2, 3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general planar graph has a conflict-free coloring with at most eight colors.2. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color.For outerplanar graphs, this question can be decided in polynomial time.

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number χ CF (G) (the smallest k for which conflict-free k-colorings exist), with a focus on planar graphs.For general graphs, we prove the conflict-free variant of the famous Hadwiger Conjecture: If G does not contain K k+1 as a minor, then χ CF (G) ≤ k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. In addition, we give a complete characterization of the algorithmic/computational complexity of conflict-free coloring. It is NP-complete to decide whether a planar graph has a conflict-free coloring with one color, while for outerplanar graphs, this can be decided in polynomial time. Furthermore, it is NP-complete to decide whether a planar graph has a conflict-free coloring with two colors, while for outerplanar graphs, two colors always suffice. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs.

We consider problems in which a simple path of fixed length, in an undirected graph, is to be shifted from a start position to a goal position by moves that add an edge to either end of the path and remove an edge from the other end. We show that this problem may be solved in linear time in trees, and is fixed-parameter tractable when parameterized either by the cyclomatic number of the input graph or by the length of the path. However, it is PSPACE-complete for paths of unbounded length in graphs of bounded bandwidth.

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Abstract:We prove the computational intractability of rotating and placing n square tiles into a 1 × n array such that adjacent tiles are compatible-either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999702 (On the other hand, there is an easy 1 2 -approximation). This is the first (correct) proof of inapproximability for edge-matching and jigsaw puzzles. Along the way, we prove NP-hardness of distinguishing, for a directed graph on n nodes, between having a Hamiltonian path (length n − 1) and having at most 0.999999284(n − 1) edges that form a vertex-disjoint union of paths. We use this gap hardness and gap-preserving reductions to establish similar gap hardness for 1 × n jigsaw and edge-matching puzzles.

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When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special simple holes guarantee foldability.

Abstract:We consider variations on the classic video game Tetris where pieces are k-ominoes instead of the usual tetrominoes (k = 4), as popularized by the video games ntris and Pentris. We prove that it is NP-complete to survive or clear a given initial board with a given sequence of pieces for each k ≥ 5, complementing the previous NP-completeness result for k = 4. More surprisingly, we show that board clearing is NP-complete for k = 3; and if pieces may not be rotated, then clearing is NP-complete for k = 2 and survival is NP-complete for k = 3. All of these problems can be solved in polynomial time for k = 1.Keywords: NP-complete, complexity, puzzles, algorithms IntroductionTetris is one of the most famous puzzle computer games, originally created in 1984 and released in the west on the IBM PC in 1987 * 1 , and substantially popularized by being bundled with every Nintendo Game Boy (except in Japan) [7]. Today, Tetris and its many clones are available to play on almost every platform; the official Tetris mobile game sold over 425 million copies by 2014, eclipsing even the 35 million Game Boy copies [8].The popularity of Tetris has led to many variations, both official and unofficial, with various changes to the rules. Here we consider a theme introduced by Hunter Freyer's Pentris [3], where pieces are pentominoes instead of tetrominoes. Later, Shaunak Kishore's ntris [6] generalized to pieces that are k-ominoes (made of k unit squares joined edge to edge).Our results. In 2004, Breukelaar et al. [1] proved that Tetris is NP-complete for the original tetromino pieces; here we generalize this result to k-omino pieces. More precisely, Breukelaar et al. and we analyze an offline version of Tetris, where the board has a given configuration of occupied squares (resulting from past play, or a complex initial board) and the pieces come from a given sequence of n pieces, and the goals are to survive all the pieces (avoid stacking any piece too high), clear the entire board (remove all occupied squares), or maximize score (according to various measures). While the interesting score functions were clearer for tetrominoes, this aspect has many possible generalizations to andreali@mit.edu f) jaysonl@mit.edu g) ywy@mit.edu Table 1 Complexity results for Tetris with k-ominoes, with or without rotation. New results are marked with a section number ( §). k-ominoes, so we focus on the first two goals. Table 1 summarizes our (and past) results for Tetris with kominoes for the goals of clearing and survival. In addition to the standard rules where pieces can be rotated and translated left/right/down, we consider a variant that forbids rotations. This variation is particularly interesting for dominoes (k = 2), where we can show NP-completeness for clearing, while we conjecture polynomial time with rotation allowed. In total, our results categorize the computational complexity of most variants of Tetris with k-ominoes for most k. With RotationTetris with monominoes (k = 1) is easy (Section 3): we can always survive (by plac...