An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w then Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound of ckw^{2} on the number of chains used by First-Fit for some constant c, while Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this paper we prove an upper bound of the form ckw. This is best possible up to the value of c.Comment: v3: referees' comments incorporate

Foundations of Online Structure Theory II: The Operator Approach

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. We give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms

Foundations of Online Structure Theory II: The Operator Approach

We introduce a framework for online structure theory. Our approach generalises notions arising independently in several areas of computability theory and complexity theory. We suggest a unifying approach using operators where we allow the input to be a countable object of an arbitrary complexity. We give a new framework which (i) ties online algorithms with computable analysis, (ii) shows how to use modifications of notions from computable analysis, such as Weihrauch reducibility, to analyse finite but uniform combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine structure of finite analogs of infinite combinatorial problems, and (iv) see how similar ideas can be amalgamated from areas such as EX-learning, computable analysis, distributed computing and the like. One of the key ideas is that online algorithms can be viewed as a sub-area of computable analysis. Conversely, we also get an enrichment of computable analysis from classical online algorithms