Yair Dombb

The efficiency of fair division with connected pieces

We consider the issue of fair division of goods, using the cake cutting abstraction, and aim to bound the possible degradation in social welfare due to the fairness requirements. Previous work has considered this problem for the setting where the division may allocate each player any number of unconnected pieces. Here, we consider the setting where each player must receive a single connected piece. For this setting, we provide tight bounds on the maximum possible degradation to both utilitarian and egalitarian welfare due to three fairness criteria -- proportionality, envy-freeness and equitability.

Throw one's cake: and eat it too

We consider the problem of fairly dividing a heterogeneous cake between a number of players with different tastes. In this setting, it is known that fairness requirements may result in a suboptimal division from the social welfare standpoint. Here, we show that in some cases, discarding some of the cake and fairly dividing only the remainder may be socially preferable to any fair division of the entire cake. We study this phenomenon, providing asymptotically-tight bounds on the social improvement achievable by such discarding.

The Efficiency of Fair Division with Connected Pieces

The cake-cutting setting, in which a single heterogeneous good must be divided between multiple parties with different tastes, is a classic model for studying questions regarding fairness in resource allocation. In this work, we turn our attention to (economic) efficiency considerations in cake cutting, examining the possible trade-offs between meeting the fairness criteria, on the one hand, and maximizing social welfare, on the other. We focus on divisions that give each agent a single (contiguous) piece of the cake and provide tight bounds (or, in some cases, nearly tight) on the possible degradation in utilitarian and egalitarian welfare resulting from meeting the fairness requirements.

The approximate swap and mismatch edit distance

There is no known algorithm that solves the general case of the approximate edit distance problem, where the edit operations are insertion, deletion, mismatch, and swap, in time o(nm), where n is the length of the text and m is the length of the pattern. In the effort to study this problem, the edit operations have been analyzed independently. Karloff [10] showed an algorithm that approximates the edit distance problem with only the mismatch operation in time O(1@e^2nlog^3m). Amir et al. [4] showed that if the only edit operations allowed are swap and mismatch, then the exact edit distance problem can be solved in time O(nmlogm). In this paper, we discuss the problem of approximate edit distance with swap and mismatch. We show a randomized O(1@e^3nlognlog^3m) time algorithm for the problem. The algorithm guarantees an approximation factor of (1+@e) with probability of at least 1-1n.

Fixed structure complexity

We consider a non-standard parametrization, where, forproblems consisting of a combinatorial structure and a number, we parameterizeby the combinatorial structure, rather than by the number.For example, in the Short-Nondeterministic-Halt problem, which is todetermine if a nondeterministic machine M accepts the empty string int steps, we parameterize by |M|, rather than t. We call such parametrizationfixed structure parametrization. Fixed structure parametrization notonly provides a new set of parameterized problems, but also results inproblems that do not seem to fall within the classical parameterizedcomplexity classes. In this paper we take the first steps in understandingthese problems. We define fixed structure analogues of various classicalproblems, including graph problems, and provide complexity, hardnessand equivalence results.

Approximate swap and mismatch edit distance

There is no known algorithm that solves the general case of the Approximate Edit Distance problem, where the edit operations are: insertion, deletion, mismatch, and swap, in time o(nm), where n is the length of the text and m is the length of the pattern. In the effort to study this problem, the edit operations were analyzed independently. Karloff [10] showed an algorithm that approximates the edit distance problem with only the mismatch operation in time O(1/Ɛ2n log3 m). Amir et. al. [3] showed that if the only edit operations allowed are swap and mismatch, then the exact edit distance problem can be solved in time O(n√m log m). In this paper, we discuss the problem of approximate edit distance with swap and mismatch. We show a randomized O(1/Ɛ3n log n log3 m) time algorithm for the problem. The algorithm guarantees an approximation factor of (1 + Ɛ) with probability of at least 1 - 1/n.

Toss one’s cake, and eat it too: partial divisions can improve social welfare in cake cutting

We consider the problem of fairly dividing a heterogeneous good (a.k.a. “cake”) between a number of players with different tastes. In this setting, it is known that fairness requirements may result in a suboptimal division from the social welfare standpoint. Here we show that, in some cases, leaving some of the cake unallocated, and fairly dividing only the remainder of the cake may be socially preferable to any fair division of the entire cake. We study this phenomenon, providing asymptotically-tight bounds on the social improvement achievable by such partial divisions.

Auctioning Time: Truthful Auctions of Heterogeneous Divisible Goods

We consider the problem of auctioning time - a one-dimensional continuously-divisible heterogeneous good - among multiple agents. Applications include auctioning time for using a shared device, auctioning TV commercial slots, and more. Different agents may have different valuations for the different possible intervals; the goal is to maximize the aggregate utility. Agents are self-interested and may misrepresent their true valuation functions if this benefits them. Thus, we seek auctions that are truthful. Considering the case that each agent may obtain a single interval, the challenge is twofold, as we need to determine both where to slice the interval, and who gets what slice. We consider two settings: discrete and continuous. In the discrete setting, we are given a sequence of m indivisible elements (e1, …, em), and the auction must allocate each agent a consecutive subsequence of the elements. In the continuous setting, we are given a continuous, infinitely divisible interval, and the auction must allocate each agent a subinterval. The agents’ valuations are nonatomic measures on the interval. We show that, for both settings, the associated computational problem is NP-complete even under very restrictive assumptions. Hence, we provide approximation algorithms. For the discrete case, we provide a truthful auctioning mechanism that approximates the optimal welfare to within a log m factor. The mechanism works for arbitrary monotone valuations. For the continuous setting, we provide a truthful auctioning mechanism that approximates the optimal welfare to within an O(log n) factor (where n is the number of agents). Additionally, we provide a truthful 2-approximation mechanism for the case that all pieces must be of some fixed size.