The paper examines a new problem in the irregular packing literature that has existed in industry for decades; two-dimensional irregular (convex) bin packing with guillotine constraints. Due to the cutting process of certain materials, cuts are restricted to extend from one edge of the stock-sheet to another, called guillotine cutting. This constraint is common place in glass cutting and is an ...

Let b ∈ Zd be an integer conic combination of a finite set of integer vectors X ⊂ Zd . In this note we provide upper bounds on the size of a smallest subset X̃ ⊆ X such that b is an integer conic combination of elements of X̃ . We apply our bounds to general integer programming and to the cutting stock problem and provide an NP certificate for the latter, whose existence has not been known so far.

The algorithmic aspects of the following problem are investigated: n (22) persons want to cut a cake into n shares so that every person will get at least l/n of the cake by his own measure and so that the number of cuts made on the cake is minimal. The cutting process is to be governed by a protocol (computer program). It is shown that no deterministic protocol exists which is fair (in a sense ...

Type–token ratio (TTR), or vocabulary size divided by text length (V/N), is a timehonoured but unsatisfactory measure of lexical diversity. The problem is that the TTR of a text sample is affected by its length. We present an algorithm for rapidly computing TTR through a moving window that is independent of text length, and we demonstrate that this measurement can detect changes within a text a...

This paper presents a way to construct the Sylvester A-resultant matrix for three bi-degree (m; n) polynomials whose exponent set is cut oo by rectangles at the corners. The paper also shows that the determinant of this matrix does give the resultant of the three polynomials.

The cake cutting problem models the fair division of a heterogeneous good between multiple agents. Previous work assumes that each agent derives value only from its own piece. However, agents may also care about the pieces assigned to other agents; such externalities naturally arise in fair division settings. We extend the classical model to capture externalities, and generalize the classical f...

We consider discrete protocols for the classical Steinhaus cake cutting problem. Under mild technical conditions, we show that any deterministic strategy-proof protocol for two agents in the standard Robertson-Webb query model is dictatorial, that is, there is a fixed agent to which the protocol allocates the entire cake. For n > 2 agents, a similar impossibility holds, namely there always exis...

We consider the problem of envy-free cake cutting, which is the distribution of a continuous heterogeneous resource among self interested players such that nobody prefers what somebody else receives to what they get. Existing work has focused on two distinct classes of solutions to this problem allocations which give each player a continuous piece of cake and allocations which give each player ...