In voting with ranked ballots each agent submits a strict ranking of the form a > b > c > d over the alternatives, and the voting rule decides on the winner based on these rankings. Although this ballot format has desirable characteristics, there is a question of whether it is expressive enough for the agents. Kahng et. al. address this issue by adding intensities to the rankings. They introduce ranking with intensities ballot format, where agents can use both >> and > in their rankings to express intensive and normal preferences between consecutive alternatives in their rankings. While Kahng et. al. focus on analyzing this ballot format in the utilitarian distortion framework, in this work, we look at the potentials of using this ballot format from the metric distortion view point. We design a class of voting rules coined Positional Scoring Rules, which can be used for different problems in the metric setting, and show that by solving a zero-sum game we can find the optimal member of this class for our problem. This rule takes intensities into account and achieves a lower distortion. In addition, by proving a bound on the price of ignoring intensities, we show that we might lose a great deal in terms of distortion by not taking the intensities into account.

The maximin share (\(\textsf{MMS}\)) is the most prominent share-based fairness notion in the fair allocation of indivisible goods. Recent years have seen significant efforts to improve the approximation guarantees for \(\textsf{MMS}\) for different valuation classes, particularly for additive valuations. For the additive setting, it has been shown that for some instances, no allocation can guarantee a factor better than \(1 - \frac{1}{n^4}\) of maximin share value to all agents. However, the best currently known algorithm achieves an approximation guarantee of \(\frac{3}{4} + \frac{3}{3836}\) for \(\textsf{MMS}\). In this work, we narrow this gap and improve the best-known approximation guarantee for \(\textsf{MMS}\) to \(\frac{10}{13}\).

We study the fully dynamic pattern matching problem where the pattern may contain up to k wildcard symbols, each matching any symbol of the alphabet. Both the text and the pattern are subject to updates (insert, delete, change). We design an algorithm with 𝒪(n log² n) preprocessing and update/query time 𝒪̃(kn^{k/{k+1}} + k² log n). The bound is truly sublinear for a constant k, and sublinear when k = o(log n). We further complement our results with a conditional lower bound: assuming subquadratic preprocessing time, achieving truly sublinear update time for the case k = Ω(log n) would contradict the Strong Exponential Time Hypothesis (SETH). Finally, we develop sublinear algorithms for two special cases: - If the pattern contains w non-wildcard symbols, we give an algorithm with preprocessing time 𝒪(nw) and update time 𝒪(w + log n), which is truly sublinear whenever w is truly sublinear. - Using FFT technique combined with block decomposition, we design a deterministic truly sublinear algorithm with preprocessing time 𝒪(n^{1.8}) and update time 𝒪(n^{0.8} log n) for the case that there are at most two non-wildcards.

We study metric distortion in distributed voting, where $n$ voters are partitioned into $k$ groups, each selecting a local representative, and a final winner is chosen from these representatives (or from the entire set of candidates). This setting models systems like U.S. presidential elections, where state-level decisions determine the national outcome. We focus on four cost objectives from Anshelevich \et~\cite{anshelevich2022distortion}: $\avgavg$, $\avgmax$, $\maxavg$, and $\maxmax$. We present improved distortion bounds for both deterministic and randomized mechanisms, offering a near-complete characterization of distortion in this model. For deterministic mechanisms, we reduce the upper bound for $\avgmax$ from $11$ to $7$, establish a tight lower bound of $5$ for $\maxavg$ (improving on $2+\sqrt{5}$), and tighten the upper bound for $\maxmax$ from $5$ to $3$. For randomized mechanisms, we consider two settings: (i) only the second stage is randomized, and (ii) both stages may be randomized. In case (i), we prove tight bounds: $5\!-\!2/k$ for $\avgavg$, $3$ for $\avgmax$ and $\maxmax$, and $5$ for $\maxavg$. In case (ii), we show tight bounds of $3$ for $\maxavg$ and $\maxmax$, and nearly tight bounds for $\avgavg$ and $\avgmax$ within $[3\!-\!2/n,\ 3\!-\!2/(kn^*)]$ and $[3\!-\!2/n,\ 3]$, respectively, where $n^*$ denotes the largest group size.

We study the problem of fair allocation of indivisible goods for subadditive agents. While constant-MMS bounds have been given for additive and fractionally subadditive agents, the best existential bound for the case of subadditive agents is 1/O(log n log log n). In this work, we improve this bound to a 1/O((log log n)2)-MMS guarantee. To this end, we introduce new matching techniques and rounding methods for subadditive valuations that we believe are of independent interest and will find their applications in future work.

Recently, some studies on the fair allocation of indivisible goods notice a connection between a purely combinatorial problem called the Rainbow Cycle problem and a fairness notion known as EFX: assuming that the rainbow cycle number for parameter d (i.e. R(d)) is O(d^β .log(d)^γ), we can find a (1 − ϵ)-EFX allocation with O_ϵ(n^(β/β+1) .log(n)^(γ/β+1)) number of discarded goods. The best upper bound on R(d) is improved in a series of works to O(d^4), O(d^(2+o(1))), and finally to O(d^2). Also, via a simple observation, we have R(d) ∈ Ω(d). In this paper, we introduce another problem in extremal combinatorics. For a parameter l, we define the rainbow path degree and denote it by H(l). We show that any lower bound on H(l) yields an upper bound on R(d). Next, we prove that H(l) ∈ Ω(l^2 / log(l)) which yields an almost tight upper bound of R(d) ∈ Ω(d.log(d)). This, in turn, proves the existence of (1−ϵ)-EFX allocation with O_ϵ(√n .log(n)) number of discarded goods. In addition, for the special case of the Rainbow Cycle problem that the edges in each part form a permutation, we improve the upper bound to R(d) ≤ 2d−4. We leverage H(l) to achieve this bound. Our conjecture is that the exact value of H(l) is ⌊l^2/2⌋ − 1. We provide some experiments that support this conjecture. Assuming this conjecture is correct, we have R(d) ∈ θ(d).

We consider the problem of guaranteeing maximin-share ($\MMS$) when allocating a set of indivisible items to a set of agents with fractionally subadditive ($\XOS$) valuations. For $\XOS$ valuations, it has been previously shown that for some instances no allocation can guarantee a fraction better than $1/2$ of maximin-share to all the agents. Also, a deterministic allocation exists that guarantees $0.219225$ of the maximin-share of each agent. Our results involve both deterministic and randomized allocations. On the deterministic side, we improve the best approximation guarantee for fractionally subadditive valuations to $3/13 = 0.230769$. We develop new ideas on allocating large items in our allocation algorithm which might be of independent interest. Furthermore, we investigate randomized algorithms and the Best-of-both-worlds fairness guarantees. We propose a randomized allocation that is $1/4$-$\MMS$ ex-ante and $1/8$-$\MMS$ ex-post for $\XOS$ valuations. Moreover, we prove an upper bound of $3/4$ on the ex-ante guarantee for this class of valuations.

In this work, we study the maximin share fairness notion for allocation of indivisible goods in the subadditive and fractionally subadditive settings. While previous work refutes the possibility of obtaining an allocation which is better than 1/2-MMS, the only positive result for the subadditive setting states that when the number of items is equal to m, there always exists an Ω(1/log m)-\MMS allocation. Since the number of items may be larger than the number of agents (n), such a bound can only imply a weak bound of Ω(1/(n log n))-MMS allocation in general. In this work, we improve this gap exponentially to an Ω(1/(log n log log n))-MMS guarantee. In addition to this, we prove that when the valuation functions are fractionally subadditive, a 1/4.6-MMS allocation is guaranteed to exist. This also improves upon the previous bound of 1/5-MMS guarantee for the fractionally subadditive setting.

We study the problem of fairly allocating a set of indivisible goods to a set of n agents. Envy-freeness up to any good (EFX) criterion (which requires that no agent prefers the bundle of another agent after the removal of any single good) is known to be a remarkable analogue of envy-freeness when the resource is a set of indivisible goods. In this paper, we investigate EFX for restricted additive valuations, that is, every good has a non-negative value, and every agent is interested in only some of the goods. We introduce a natural relaxation of EFX called EFkX which requires that no agent envies another agent after the removal of any k goods. Our main contribution is an algorithm that finds a complete (i.e., no good is discarded) EF2X allocation for restricted additive valuations. In our algorithm we devise new concepts, namely configuration and envy-elimination that might be of independent interest. We also use our new tools to find an EFX allocation for restricted additive valuations that discards at most n/2 -1 goods.

Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is (φ − 1) ≃ 0.61 by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time 0.73-approximation allocation algorithm. For our algorithm we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness.

In Spatial Voting Theory, distortion is a measure of how good the winner is. It has been proved that no deterministic voting mechanism can guarantee a distortion better than 3, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election? We consider an election with two candidates and suggest an abstention model, which is a general form of the abstention model proposed by Kirchgässner. Our results characterize the distortion ¨ value and provide a rather complete picture of the model.

Edit distance and longest common subsequence are among the most fundamental problems in combinatorial optimization. Recent developments have proven strong lower bounds against subquadratic time solutions for both problems. Moreover, the best approximation factors for subquadratic time solutions have been limited to 3 for edit distance and super constant for longest common subsequence. Improved approximation algorithms for these problems1 are some of the biggest open questions in combinatorial optimization. In this work, we present improved algorithms for both edit distance and longest common subsequence. The running times are truly subquadratic, though we obtain 1 + o(1) approximate solutions for both problems if the input satisfies a mild condition. In this setting, first, an adversary chooses one of the input strings. Next, this string is perturbed by a random procedure, and then the adversary chooses the second string after observing the perturbed one.

In Spatial Voting Theory, distortion is a measure of how good the winner is. It is proved that no deterministic voting mechanism can guarantee a distortion better than 3, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election?We consider an election with two candidates and suggest an abstention model, which is a more general form of the abstention model proposed by Kirchgässner (2003). We define the¨ concepts of the expected winner and the expected distortion to evaluate the distortion of an election in our model. Our results fully characterize the distortion value and provide a rather complete picture of the model.

Longest common subsequence (LCS) is one of the most fundamental problems in combinatorial optimization. Apart from theoretical importance, LCS has enormous applications in bioinformatics, revision control systems, and data comparison programs1. Although a simple dynamic program computes LCS in quadratic time, it has been recently proven that the problem admits a conditional lower bound and may not be solved in truly subquadratic time [2]. In addition to this, LCS is notoriously hard with respect to approximation algorithms. Apart from a trivial sampling technique that obtains a nx approximation solution in time O(n2–2x) nothing else is known for LCS. This is in sharp contrast to its dual problem edit distance for which several linear time solutions are obtained in the past two decades [4, 5, 9, 10, 16]. In this work, we present the first nontrivial algorithm for approximating LCS in linear time. Our main result is a linear time algorithm for the longest common subsequence which has an approximation factor of O(n0.497956). This beats the barrier for approximating LCS in linear time.

One of the important yet insufficiently studied subjects in fair allocation is the externality effect among agents. For a resource allocation problem, externalities imply that the share allocated to an agent may affect the utilities of other agents.

We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang (2014) wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with n agents, no allocation can guarantee better than 1/n approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a 1/n approximation guarantee. Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem without loss of generality, we show it enables us to find a 1/2 approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

We study the problem of fair allocation for indivisible goods. We use the maxmin share paradigm introduced by Budish~\citeBudish:first as a measure for fairness. \procacciafirst ~\citeProcaccia:first were the first to investigate this fundamental problem in the additive setting. They show that a maxmin guarantee (1-$\MMS$ allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a $1/2$-$\MMS$ allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. \sprocacciafirst ~\citeProcaccia:first provide a proof for the existence of a $2/3$-$\MMS$ allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of \sprocacciafirst~to a $3/4$ factor in the additive setting. The main idea for our $3/4$-$\MMS$ allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time. The reader can find a summary of our results in Table \refresultstable.

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players in a way that meets the following criteria: (I) every player(weakly) prefers his allocated cake to any other player’s share (such notion is known as envy-freeness), (II) the mechanism is strategy-proof (truthful), and (III) the number of cuts made on the cake is minimal. We provide methods, namely expansion process and expansion process with unlocking, for dividing the cake under different assumptions on the valuation functions of the players.