We analyze the competitive ratio and the advice complexity of the online unbounded knapsack problem. An instance is given as a sequence of n items with a size and a value each, and an algorithm has to decide whether or not and how often to pack each item into a knapsack of bounded capacity. The items are given online and the total size of the packed items must not exceed the knapsack’s capacity, while the objective is to maximize the total value of the packed items. While each item can only be packed once in the classical knapsack problem (also called the 0-1 knapsack problem), the unbounded version allows for items to be packed multiple times. We show that the simple unbounded knapsack problem, where the size of each item is equal to its value, allows for a competitive ratio of 2. We also analyze randomized algorithms and show that, in contrast to the 0-1 knapsack problem, one uniformly random bit cannot improve an algorithm’s performance. More randomness lowers the competitive ratio to less than 1.736, but it can never be below 1.693. In the advice complexity setting, we measure how many bits of information (so-called advice bits) the algorithm has to know to achieve some desired solution quality. For the simple unbounded knapsack problem, one advice bit lowers the competitive ratio to 3/2. While this cannot be improved with fewer than log2n advice bits for instances of length n, a competitive ratio of 1+ε can be achieved with O(ε-1·log(nε-1)) advice bits for any ε>0}\\varvec{0}$$\\end{document}]]>. We further show that no amount of advice bounded by a function f(n) allows an algorithm to be optimal. We also study the online general unbounded knapsack problem and show that it does not allow for any bounded competitive ratio for both deterministic and randomized algorithms, as well as for algorithms using fewer than log2n advice bits. We also provide a surprisingly simple algorithm that uses O(ε-1·log(nε-1)) advice bits to achieve a competitive ratio of 1+ε for any ε>0}\\varvec{0}$$\\end{document}]]>.

We show how to obtain, via a unified framework provided by logic and automata theory, many classical results of Brillhart and Morton on Rudin-Shapiro sums. The techniques also facilitate easy proofs for new results.

We study a variant of the offline Dial-a-Ride problem, where each request has a source and destination and the goal is to maximize the number of requests served within a specified time limit. We investigate this problem for the uniform metric space and show that the problem is NP-hard. We then present a 2/3 approximation algorithm called TWOCHAIN, which simply looks for pairs of requests that are “chained” together and serves those before serving requests that are not connected to any others. We also show that a natural generalization of this algorithm, k-chain, has an approximation ratio at most 7/9. We also analyze the longest-chain-first algorithm for the problem, characterizing graphs on which it is optimal, and showing that it has an approximation ratio no better than 5/6. Our experiments on all of these algorithms show that TWOCHAIN is a promising algorithm, performing nearly as well as more computationally intensive variants. We dedicate this article to the memory of Gerhard Woeginger, whose life and work greatly influenced our professional lives, as expanded upon in the Acknowledgments. Woeginger’s prolific research in scheduling, matching, bin-packing, TSP, and online algorithms in general, all served as important parts of the foundation on which our own scholarly pursuits were shaped and formed over the years. Woeginger also studied Dial-a-Ride (DARP) Problems, as DARP is a generalization both of scheduling problems and of TSP, which were two of his most active areas of research.

Motivated by the success of the serial dictatorship mechanism in social choice settings, we explore its usefulness in tackling various combinatorial optimization problems. We do so by considering an abstract model, in which a set of agents are asked to act in a particular ordering, called the action sequence. Each agent acts in a way that gives her the maximum possible value, given the actions of the agents who preceded her in the action sequence. Our goal is to compute action sequences that yield approximately optimal total value to the agents (a.k.a., social welfare). We assume query access to the value vi(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v_i(S)$$\\end{document} that the agent i gets when she acts after the agents in the ordered set S. We establish tight bounds on the social welfare that can be achieved using polynomially many queries. Even though these bounds show a marginally sublinear approximation of optimal social welfare in general, excellent approximations can be obtained when the valuations stem from an underlying combinatorial domain. Indicatively, when the valuations are defined using bipartite matchings, arborescences in directed graphs, and satisfiability of Boolean expressions, simple query-efficient algorithms yield 2-approximations. We discuss issues related to truthfulness and show how some of our algorithms can be implemented truthfully using VCG-like payments. Finally, we introduce and study the price of serial dictatorship, a notion that provides an optimistic measure of the quality of combinatorial optimization solutions generated by action sequences.

This collection includes seven papers which are based on work presented at ICALP 2020, the 47th International Colloquium on Automata, Languages and Programming. ICALP is the annual conference of the European Association for Theoretical Computer Science. The 2020 edition of the conference was held in conjunction with the annual ACM/IEEE conference on Logic in Computer Science (LICS). The ICALP 2020 conference was held entirely online. The organization of the conference was severely affected in the early days of the COVID-19 pandemic. The conference had been scheduled to be held in Beijing. When travel to China from other parts of the world was curtailed in early 2020, a decision was made to relocate the conference to Saarbrücken. In the end, with the spread of the epidemic, a physical gathering there also proved impossible and the conference was held entirely online. It was experimental, but we discovered just how successful an online conference could be. The papers at ICALP were presented in two tracks: Track A (Algorithms, Complexity and Games) and Track B (Automata, Logic, Semantics and Theory of Programming). The present issue contains seven papers that are based on work presented in these two tracks. The authors of these selected papers were invited to submit extended versions, incorporating new material and proofs that had been omitted from the conference proceedings. The papers were selected from among the 138 papers presented at ICALP (102 in Track A and 36 in Track B). These papers had themselves been selected by the ICALP programme committee from a total of 470 submissions that were received. All manuscripts invited to this collection were rigorously refereed to the usual standards of ToCS. This collection includes two prize winning papers. The paper by Gawrychowski, Moses and Weimann was the winner of the award for best paper in Track (A) It pres-

We give a randomized algorithm that finds a minimum cut in an undirected weighted m-edge n-vertex graph G with high probability in O(mlog2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(m \\log ^2 n)$$\\end{document} time. This is the first improvement to Karger’s celebrated O(mlog3n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(m \\log ^3 n)$$\\end{document} time algorithm from 1996. Our main technical contribution is a deterministic O(mlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(m \\log n)$$\\end{document} time algorithm that, given a spanning tree T of G, finds a minimum cut of G that 2-respects (cuts two edges of) T.