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Wednesday, 4 November 2026: 3 new papers, 2 cross-lists and 11 replacements in Computational Geometry (cs.CG), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
We present new results on polyomino nets that fold into four or five distinct boxes. With the help of AI, we first found examples with areas 532 and 2128, respectively. We analyze these examples and reduce their areas to 178 and 358, respectively. We also generalize the construction of the five-box net and show that infinitely many polyomino nets fold into five distinct boxes.
We study the geometric bin packing problem for simple polygonal regions, i.e., non-degenerate, connected, rational regions, in the minimum number of unit-square bins, allowing only translations. We even consider packing general non-convex objects, possibly with holes, where we can then pack smaller objects. Previous schemes cover squares (MOR'06), circles and other $L_p$-balls (Algorithmica'16), sufficiently skewed rectangles (Algorithmica'23), and convex fat objects with a constant number of distinct shapes under resource augmentation (arXiv'24, WAOA'23). The result for circles and other $L_p$-balls needs bin augmentation to obtain rational packing realizations. For restricted classes of $L$-shaped polygons, Gerlach et al. (SoCG'26) give constant worst-case guarantees, while for general non-convex objects, bounding-box reductions give just approximation factors that depend on fatness. Against this background, we present a general framework that provides asymptotic polynomial-time approximation schemes (APTAS) for bin packing of any fixed class of simple polygonal regions satisfying just two assumptions. First, covering any object by the grid cells intersecting its interior in any sufficiently fine square grid, relative to the object's bounding box, increases its area by at most a small fraction. Second, the class can be partitioned into a constant number of sequences such that each object in a sequence can be translated to fit inside every preceding object. In particular, certain classes of star-shaped polygons and more general polygonal regions, including some with simple holes, satisfy both assumptions. In addition, our framework does not require a bound on the objects' boundary complexity. We emphasize that our results require no bin augmentation whatsoever and yield explicit, exact rational packings of polynomial encoding length for all input objects.
We study the problem of reconfiguring plane, straight-line spanning trees on point sets in the plane. A flip is a reconfiguration step in which one edge of a tree is replaced with another, yielding again a plane spanning tree. For a given point set $P$, the flip graph has a vertex for each plane spanning tree on $P$, and an edge between any two trees that differ by a single flip. The folklore happy edge conjecture suggests that any edge shared by both the initial and the target tree (a happy edge) is never flipped in a shortest flip sequence. This conjecture was computationally verified for up to 10 points in convex position. In this paper, we disprove this conjecture both on general point sets and on point sets in convex position by providing two counterexamples on $16$ and $30$ points, respectively. The counterexample for point sets in general position also holds for compatible flips, while the happy edge property holds for compatible flips in convex point sets as shown by Aichholzer, Dorfer, and Vogtenhuber [GD 2025].
Binary prefix coding assigns prefix-free binary words to symbols to minimize their frequency-weighted cost. Huffman solved the equal-letter-cost case in polynomial time in 1952 [Proc. IRE'52]. For unequal letter costs, Karp gave an integer-programming formulation in 1961 [IRE Trans. Inf. Theory'61]. Exact polynomial-time algorithms for fixed integer costs were given by Golin and Rote [IEEE Trans. Inf. Theory'98] and Bradford, Golin, Larmore, and Rytter [J. Algorithms'02]. For input-dependent costs, Golin, Mathieu, and Young gave a polynomial-time approximation scheme [SICOMP'12]. The exact complexity when letter costs are part of the input had remained unresolved. We resolve this $65$-year-old problem by proving that deciding whether there exists a binary prefix-free code of frequency-weighted cost at most a given threshold is NP-complete, even when the letter costs are consecutive positive integers $(M,M+1)$ supplied as part of the input. All frequencies, letter costs, and thresholds are encoded in binary. Consequently, finding an optimal code is NP-hard. Our proof uses two polynomial-time reductions, starting from Planar Independent Set. The intermediate problem prescribes the number of leaves at each depth of a full ordered binary tree and minimizes a signed linear score on their sorted right-edge counts. The first reduction represents an independent-set verification program by a tree growing level by level. A fixed-profile identity converts narrow coordinate restrictions into linear penalties that enforce the computation. The second reduction adds large dyadic terms to the symbol frequencies to force prescribed depths. These terms contribute a constant to the weighted number of ones, leaving smaller signed perturbations to reproduce the tree score.
We give an improved algorithm for weighted directed planar diameter that runs in $\widetilde{O}(n^{3/2})$ time. This improves upon the $\widetilde{O}(n^{5/3})$ time algorithm for weighted directed planar diameter of Gawrychowski, Kaplan, Mozes, Sharir, and Weimann [SIAM J. Comput. 2021] and the recent $\widetilde{O}(n^{8/5})$ time algorithm for unweighted undirected planar diameter of Fredslund-Hansen, Mozes, and Weimann [SOSA 2027]. The main new result is an improved data structure for computing the farthest vertex when given feasible planar distances to the vertices on the outer face. The critical insight is the fact that normalized distances to the vertices of a face behave like pseudolines, for which we can leverage range searching and upper envelope data structures. Using the farthest vertex data structure, we are able to compute all vertex eccentricities and diameter in $\widetilde{O}(n^{3/2})$.