We consider the problem of fairly allocating a set of indivisible goods among n agents with additive valuations, using the popular fairness notion of maximin share (MMS). Since MMS allocations do not always exist, a series of works provided existence and algorithms for approximate MMS allocations. The Garg-Taki algorithm gives the current best approximation factor of (3/4 + 1/12n). Most of these results are based on complicated analyses, especially those providing better than 2/3 factor. Moreover, since no tight example is known of the Garg-Taki algorithm, it is unclear if this is the best factor of this approach. In this paper, we significantly simplify the analysis of this algorithm and also improve the existence guarantee to a factor of (3/4 + min(1/36, 3/(16n-4))). For small n, this provides a noticeable improvement. Furthermore, we present a tight example of this algorithm, showing that this may be the best factor one can hope for with the current techniques.
We study a generalization of the knapsack problem with geometric and vector constraints. The input is a set of rectangular items, each with an associated profit and d nonnegative weights (d-dimensional vector), and a square knapsack. The goal is to find a non-overlapping axis-parallel packing of a subset of items into the given knapsack such that the vector constraints are not violated, i.e., the sum of weights of all the packed items in any of the d dimensions does not exceed one. We consider two variants of the problem: (i) the items are not allowed to be rotated, (ii) items can be rotated by 90 degrees.We give a (2 + ε)-approximation algorithm for this problem (both versions). In the process, we also study a variant of the maximum generalized assignment problem (Max-GAP), called Vector-Max-GAP, and design a PTAS for it.
In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is 1.406 by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let λ be the smallest constant such that for every set I of items, the number of bins in the optimal solution to G2BP for I is upper bounded by λ opt(I) + c, where opt(I) is the number of bins in the optimal solution to 2BP for I and c is a constant. It is known that 4/3 ≤ λ ≤ 1.692. Bansal and Khan [SODA'14] conjectured that λ = 4/3. The conjecture, if true, will imply a (4/3+ε)-approximation algorithm for 2BP. According to convention, for a given constant δ > 0, a rectangle is large if both its height and width are at least δ, and otherwise it is called skewed. We make progress towards the conjecture by showing λ = 4/3 for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on λ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS.
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