arXiv subjects

Adam Hesterberg

Publications and source records attributed to Adam Hesterberg.

24 records · Page 2Linked to original sources

arXiv2017

Upward Partitioned Book Embeddings

We analyze a directed variation of the book embedding problem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into $k$ pages, can we linearly order the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the problem is NP-complete for $k\ge 3$, and for $k\ge 4$ even in the special case when each page is a matching. By contrast, the problem can be solved in linear time for $k=2$ pages when pages are restricted to matchings. The problem comes from Jack Edmonds (1997), motivated as a generalization of the map folding problem from computational origami.

cs.CG↗

arXiv2016

Even $1 \times n$ Edge-Matching and Jigsaw Puzzles are Really Hard

We prove the computational intractability of rotating and placing $n$ square tiles into a $1 \times n$ 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 0.9999999851. (On the other hand, there is an easy $1 \over 2$-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 $1 \times n$ jigsaw and edge-matching puzzles.

cs.CC↗

arXiv2016

Single-Player and Two-Player Buttons & Scissors Games

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.

cs.CC↗

arXiv2016

Pachinko

Inspired by the Japanese game Pachinko, we study simple (perfectly "inelastic" collisions) dynamics of a unit ball falling amidst point obstacles (pins) in the plane. A classic example is that a checkerboard grid of pins produces the binomial distribution, but what probability distributions result from different pin placements? In the 50-50 model, where the pins form a subset of this grid, not all probability distributions are possible, but surprisingly the uniform distribution is possible for $\{1,2,4,8,16\}$ possible drop locations. Furthermore, every probability distribution can be approximated arbitrarily closely, and every dyadic probability distribution can be divided by a suitable power of $2$ and then constructed exactly (along with extra "junk" outputs). In a more general model, if a ball hits a pin off center, it falls left or right accordingly. Then we prove a universality result: any distribution of $n$ dyadic probabilities, each specified by $k$ bits, can be constructed using $O(n k^2)$ pins, which is close to the information-theoretic lower bound of $Ω(n k)$.

cs.CG↗

arXiv2015

Improved Connectivity Condition for Byzantine Fault Tolerance

Given a network in which some pairs of nodes can communicate freely, and some subsets of the nodes could be faulty and colluding to disrupt communication, when can messages reliably be sent from one given node to another? We give a new characterization of when the agreement problem can be solved and provide an agreement algorithm which can reach agreement when the number of Byzantine nodes along each minimal vertex cut is bounded. Our new bound holds for a strict superset of cases than the previously known bound. We show that the new bound is tight. Furthermore, we show that this algorithm does not require the processes to know the graph structure, as the previously known algorithm did. Finally, we explore some of the situations in which we can reach agreement if we assume that individual nodes or entire subgraphs are trustworthy.

cs.DC↗

arXiv2014

Folding a Paper Strip to Minimize Thickness

In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a "flat folding" is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where "thicker" creases consume more paper). We prove both problems strongly NP-complete even for 1D folding. On the other hand, we prove the first problem fixed-parameter tractable in 1D with respect to the number of layers.

cs.DS↗