In this paper we present tight lower-bounds and new upper-bounds for hypergraph and database problems. We give tight lower-bounds for finding minimum hypercycles. We give tight lower-bounds for a substantial regime of unweighted hypercycle. We also give a new faster algorithm for longer unweighted hypercycles. We give a worst-case to average-case reduction from detecting a subgraph of a hypergraph in the worst-case to counting subgraphs of hypergraphs in the average-case. We demonstrate two applications of this worst-case to average-case reduction, which result in average-case lower bounds for counting hypercycles in random hypergraphs and queries in average-case databases. Our tight upper and lower bounds for hypercycle detection in the worst-case have immediate implications for the average-case via our worst-case to average-case reductions.

In this paper we focus on the intersection of tile assembling systems, edge-matching puzzles, combinatorial games, and knot construction and identity. As a basis, we utilize the game Celtic!, which is a 2-player board game where the goal of the game is to construct knots where one knot uses more of a player’s pieces than the other player over all knots. All pieces must build off an existing knot and a valid knot must be closed. We consider three variations: a 0-player self-assembly variation that deterministically places pieces to form a closed knot of some length, a 1-player puzzle variation where the goal is to form a closed knot of some length, and the original 2-player game with restricted pieces. We show these are P-complete, NP-complete (depending on the pieces), and PSPACE-complete (for a first-player win), respectively. We nearly fully characterize the hardness of the 1-player puzzle based on the pieces. We prove these results through standard hardness reductions and with constraint logic. Finally, we note some combinatorial game theory strategies to show certain configurations are a draw through strategy stealing.

Online voting is convenient and flexible, but amplifies the risks of voter coercion and vote buying. One promising mitigation strategy enables voters to give a coercer fake voting credentials, which silently cast votes that do not count. Current systems along these lines make problematic assumptions about credential issuance, however, such as strong trust in a registrar and/or in voter-controlled hardware, or expecting voters to interact with multiple registrars. Votegral is the first coercion-resistant voting architecture that leverages the physical security of in-person registration to address these credential-issuance challenges, amortizing the convenience costs of in-person registration by reusing credentials across successive elections. Votegral's registration component, TRIP, gives voters a kiosk in a privacy booth with which to print real and fake credentials on paper, eliminating dependence on trusted hardware in credential issuance. The voter learns and can verify in the privacy booth which credential is real, but real and fake credentials thereafter appear indistinguishable to others. Only voters actually under coercion, a hopefully-rare case, need to trust the kiosk. To achieve verifiability, each paper credential encodes an interactive zero-knowledge proof, which is sound in real credentials but unsound in fake credentials. Voters observe the difference in the order of printing steps, but need not understand the technical details. Experimental results with our prototype suggest that Votegral is practical and sufficiently scalable for real-world elections. User-visible latency of credential issuance in TRIP is at most 19.7 seconds even on resource-constrained kiosk hardware, making it suitable for registration at remote locations or on battery power. A companion usability study indicates that TRIP's usability is competitive with other e-voting systems including some lacking coercion resistance, and formal proofs support TRIP's combination of coercion-resistance and verifiability.

Advances in technology have given us the ability to create and manipulate robots for numerous applications at the molecular scale. At this size, fabrication tool limitations motivate the use of simple robots. The individual control of these simple objects can be infeasible. We investigate a model of robot motion planning, based on global external signals, known as the tilt model. Given a board and initial placement of polyominoes, the board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked. We propose a new hierarchy of shapes and design a single configuration that is strongly universal for any w × h bounded shape within this hierarchy (it can be reconfigured to construct any w × h bounded shape in the hierarchy). This class of shapes constitutes the most general set of buildable shapes in the literature, with most previous work consisting of just the first-level of our hierarchy. We accompany this result with a O(n4 log n)-time algorithm for deciding if a given hole-free shape is a member of the hierarchy. For our second result, we resolve a long-standing open problem within the field: We show that deciding if a given position may be covered by a tile for a given initial board configuration is PSPACEcomplete, even when all movable pieces are 1 × 1 tiles with no glues. We achieve this result by a reduction from Non-deterministic Constraint Logic for a one-player unbounded game.

We investigate the problem of assembling general shapes and patterns in a model in which particles move based on uniform external forces until they encounter an obstacle. In this model, corresponding particles may bond when adjacent with one another. Succinctly, this model considers a 2D grid of “open” and “blocked” spaces, along with a set of slidable polyominoes placed at open locations on the board. The board may be tilted in any of the 4 cardinal directions, causing all slidable polyominoes to move maximally in the specified direction until blocked. By successively applying a sequence of such tilts, along with allowing different polyominoes to stick when adjacent, tilt sequences provide a method to reconfigure an initial board configuration so as to assemble a collection of previous separate polyominoes into a larger shape. While previous work within this model of assembly has focused on designing a specific board configuration for the assembly of a specific given shape, we propose the problem of designing universal configurations that are capable of constructing a large class of shapes and patterns. For these constructions, we present the notions of weak and strong universality which indicate the presence of “excess” polyominoes after the shape is constructed. In particular, for given integers h, w, we show that there exists a weakly universal configuration with O(hw) 1 × 1 slidable particles that can be reconfigured to build any h × w patterned rectangle. We then expand this result to show that there exists a weakly universal configuration that can build any h × w-bounded size connected shape. Following these results, which require an admittedly relaxed assembly definition, we go on to show the existence of a strongly universal configuration (no excess particles) which can assemble any shape within a previously studied “drop” class, while using quadratically less space than previous results. Finally, we include a study of the complexity of deciding if a particle within a configuration may be relocated to another position, and deciding if a given configuration may be transformed into a second given configuration. We show both problems to be PSPACE-complete even when no particles stick to one another and movable particles are restricted to 1 × 1 tiles and a single 2 × 2 polyomino.