arXiv subjects
Publications and source records attributed to Vignesh Viswanathan.
We present an $(e^{1/e} - c)$-approximation algorithm for maximizing Nash social welfare under additive valuations, for some constant $c > 0$. This result improves upon the previous best-known approximation factor of $e^{1/e}$ [Barman, Krishnamurthy and Vaish, EC 2018].
We consider the problem of dividing a set of indivisible goods among agents with additive valuations. This problem has been studied under various objectives in both the computer science and the operations research literature. Our main contribution is a novel dictator test using this problem, which can separate a dictator from any function sufficiently far from a dictator. We use this test to prove the following hardness results (assuming the unique games conjecture is true): (1) We show that it is NP-hard to approximate the max Nash welfare by a factor better than $\sqrt[3]{\frac{81}{65}} - \varepsilon \approx 1.0761$. This improves on the previous best known inapproximability factor of $\sqrt{\frac87} - \varepsilon \approx 1.069$. (2) We show that it is NP-hard to approximate the maximum budgeted allocation by a factor better than $\frac{243}{227} - \varepsilon \approx 1.07$. This improves on the previous best known inapproximability factor of $\frac{16}{15} - \varepsilon \approx 1.067$. (3) We show that it is NP-hard to approximate the max generalized assignment problem (GAP) by a factor better than $\frac{145}{129} - \varepsilon \approx 1.124$. This improves on the previous best known inapproximability factor of $\frac{11}{10} - \varepsilon \approx 1.10$.
We study a generalization of the classical Hajnal-Szemer\'edi theorem to vertex-weighted graphs. Given a graph with nonnegative vertex weights, a coloring is called $\alpha$-approximately equitable up to one vertex ($\alpha$-EQ1) if, for each color class, the total weight remaining after removing its maximum-weight vertex is at most $\alpha \geq 1$ times the weight of any other color class. For vertex-weighted graphs with maximum degree $\Delta$, we show that there exist instances for which no $k$-coloring is $\alpha$-EQ1 for any $k < \frac{3\Delta}{2}$ and $\alpha < \sqrt{2}$. In light of this impossibility, we relax these parameters and establish the following results for any vertex-weighted graph $G$ with maximum degree $\Delta$: (1) for any $\varepsilon \in (0,1)$ and all $k \geq (\frac{c}{\varepsilon^2}\ln{\frac{1}{\varepsilon}}) \Delta$, there exists a $(1 + \varepsilon)$-EQ1 $k$-coloring of $G$, where $c$ is a fixed constant; and (2) for all $k \ge \Delta + 1$, there exists a $2$-EQ1 $k$-coloring of $G$. Furthermore, such equitable colorings can be computed in polynomial time. En route to our results on equitability under vertex weights, we establish sufficient conditions for the existence of $k$-colorings that are equitable with respect to any given partition of the vertex set. Our coloring results correspond to fairness guarantees in a constrained fair division setting and lead to concentration inequalities for partly dependent random variables.
We consider the problem of partitioning an undirected graph (representing a social network) over $n$ nodes and max degree $\Delta$ into $k$ equally sized parts. Each node in the graph, representing an agent, derives utility proportional to the number of their neighbors in their assigned part. Our goal is to find a balanced partitioning that is fair. The two notions of fairness we consider are the core and envy-freeness. A partition is envy-free if no node gains utility from moving to a different part, and a partition is in the core if no set of $n/k$ nodes can deviate to form a new part with all nodes gaining in utility. We show that there exists a balanced partition which is both $O(\max\{\sqrt{\Delta}, k^2\} \ln n)$-approximately envy-free and in the $(k + o(k))$-approximate core. Taken separately, these two guarantees are comparable to (and in some cases, better than) the best known envy-freeness and core guarantees for this problem. Moreover, we show that these desirable partitions can be computed efficiently if we slightly relax the balancedness constraint. In addition, when $k = 2$, we show that a $(1.618 + o(1))$-core exists, and a $(2 + \varepsilon)$-core can be computed in polynomial time. The last two results make progress on two open questions from Li et al. [AAAI, 2023].
Background: Clinical linear accelerators are an accessible platform for preclinical research on the biological effects of ultra rapid electron irradiation (FLASH). However, they are not inherently designed for the accurate pulse control required for experiments using a small number of relatively high-dose pulses, and available methods for beam control such as respiratory gating can be error prone owing to system latency. Here we experimentally characterize the temporal latency of the respiratory gating system for controlling beam-on and beam-off at the individual linac pulse level. Methods and Materials: We used programmable controller boards and a relay circuit to monitor and control delivery of specific numbers of pulses through the built-in monitor chamber and respiratory gating system of a Varian Trilogy linac. We implemented two methods an adaptive method using only the delivered pulse signal, and a synchronization method additionally using the linac internal pulse-timing signal and characterized their performance for standard and customized pulse sequences. Results: Characterizing the latency parameters permitted choosing optimal timing parameters that maximized the rate of successfully delivering the desired number of pulses using both adaptive and synchronization methods. Conclusions: We demonstrated that accounting for latency and/or using the ability to read the prior information on expected pulse timing can provide high accuracy in delivering specified numbers of pulses. This reliability is critical for accurate dose delivery in preclinical FLASH research of single fraction and especially fractionated dosing regimens. The ability to generate custom pulse sequences enables more detailed exploration of the temporal dependence of biological FLASH effects.
Fair allocation of indivisible goods is a fundamental problem at the interface of economics and computer science. Traditional approaches focus either on randomized allocations that are fair in expectation or deterministic allocations that are approximately fair. Recent work reconciles both these approaches via best-of-both-worlds guarantees, wherein one seeks randomized allocations that are fair in expectation (ex-ante fair) while being supported on approximately fair allocations (ex-post fair). Prior work has shown that under additive valuations, there always exists a randomized allocation that is ex-ante stochastic-dominance envy-free (sd-EF) and ex-post envy-free up to one good (EF1). Our work is motivated by the goal of achieving stronger ex-post fairness guarantees such as envy-freeness up to any good (EFX) along with meaningful ex-ante guarantees. We make the following contributions: 1) We first consider lexicographic preferences, a subdomain of additive valuations where ex-post EFX allocations always exist and can be computed efficiently. On the negative side, we show that ex-ante sd-EF is fundamentally incompatible with ex-post EFX, prompting a relaxation of the ex-ante benchmark. We then present a poly. time algorithm that achieves ex-post EFX and PO together with ex-ante 9/10-EF. Our algorithm uses dependent rounding and leverages structural properties of EFX and PO allocations. 2)For monotone valuations, we study EFX-with-charity: a relaxation of EFX where some goods remain unallocated, with no agent envying the unallocated pool. We show that ex-post EFX-with-charity can be achieved alongside ex-ante 0.5-EF. 3)Finally, for subadditive valuations, we strengthen our previous ex-post guarantee to EFX-with-bounded-charity, where at most n-1 goods (n= no. of agents) remain unallocated, at the price of weakening the ex-ante guarantee to 0.5-proportionality.
We study a two-sided matching model where one side of the market (hospitals) has combinatorial preferences over the other side (doctors). Specifically, we consider the setting where hospitals have matroid rank valuations over the doctors, and doctors have either ordinal or cardinal unit-demand valuations over the hospitals. While this setting has been extensively studied in the context of one-sided markets, it remains unexplored in the context of two-sided markets. When doctors have ordinal preferences over hospitals, we present simple sequential allocation algorithms that guarantee stability, strategyproofness for doctors, and approximate strategyproofness for hospitals. When doctors have cardinal utilities over hospitals, we present an algorithm that finds a stable allocation maximizing doctor welfare; subject to that, we show how one can maximize either the hospital utilitarian or hospital Nash welfare. Moreover, we show that it is NP-hard to compute stable allocations that approximately maximize hospital Nash welfare.
Large Language Models (LLMs) have become an indispensable part of natural language processing tasks. However, autoregressive sampling has become an efficiency bottleneck. Multi-Draft Speculative Decoding (MDSD) is a recent approach where, when generating each token, a small draft model generates multiple drafts, and the target LLM verifies them in parallel, ensuring that the final output conforms to the target model distribution. The two main design choices in MDSD are the draft sampling method and the verification algorithm. For a fixed draft sampling method, the optimal acceptance rate is a solution to an optimal transport problem, but the complexity of this problem makes it difficult to solve for the optimal acceptance rate and measure the gap between existing verification algorithms and the theoretical upper bound. This paper discusses the dual of the optimal transport problem, providing a way to efficiently compute the optimal acceptance rate. For the first time, we measure the theoretical upper bound of MDSD efficiency for vocabulary sizes in the thousands and quantify the gap between existing verification algorithms and this bound. We also compare different draft sampling methods based on their optimal acceptance rates. Our results show that the draft sampling method strongly influences the optimal acceptance rate, with sampling without replacement outperforming sampling with replacement. Additionally, existing verification algorithms do not reach the theoretical upper bound for both without replacement and with replacement sampling. Our findings suggest that carefully designed draft sampling methods can potentially improve the optimal acceptance rate and enable the development of verification algorithms that closely match the theoretical upper bound.
We study the problem of maximizing Nash social welfare, which is the geometric mean of agents' utilities, in two well-known models. The first model involves one-sided preferences, where a set of indivisible items is allocated among a group of agents (commonly studied in fair division). The second model deals with two-sided preferences, where a set of workers and firms, each having numerical valuations for the other side, are matched with each other (commonly studied in matching-under-preferences literature). We study these models under capacity constraints, which restrict the number of items (respectively, workers) that an agent (respectively, a firm) can receive. We develop constant-factor approximation algorithms for both problems under a broad class of valuations. Specifically, our main results are the following: (a) For any $\epsilon > 0$, a $(6+\epsilon)$-approximation algorithm for the one-sided problem when agents have submodular valuations, and (b) a $1.33$-approximation algorithm for the two-sided problem when the firms have subadditive valuations. The former result provides the first constant-factor approximation algorithm for Nash welfare in the one-sided problem with submodular valuations and capacities, while the latter result improves upon an existing $\sqrt{OPT}$-approximation algorithm for additive valuations. Our result for the two-sided setting also establishes a computational separation between the Nash and utilitarian welfare objectives. We also complement our algorithms with hardness-of-approximation results. Additionally, for the case of additive valuations, we modify the configuration LP of Feng and Li [ICALP 2024] to obtain an $(e^{1/e}+\epsilon)-$ approximation algorithm for weighted two-sided Nash social welfare under capacity constraints.
We study the problem of fair allocation of indivisible items when agents have ternary additive valuations -- each agent values each item at some fixed integer values $a$, $b$, or $c$ that are common to all agents. The notions of fairness we consider are max Nash welfare (MNW), when $a$, $b$, and $c$ are non-negative, and max egalitarian welfare (MEW). We show that for any distinct non-negative $a$, $b$, and $c$, maximizing Nash welfare is APX-hard -- i.e., the problem does not admit a PTAS unless P = NP. We also show that for any distinct $a$, $b$, and $c$, maximizing egalitarian welfare is APX-hard except for a few cases when $b = 0$ that admit efficient algorithms. These results make significant progress towards completely characterizing the complexity of computing exact MNW allocations and MEW allocations. En route, we resolve open questions left by prior work regarding the complexity of computing MNW allocations under bivalued valuations, and MEW allocations under ternary mixed manna.
We characterized the dosimetric properties of a clinical linear accelerator configured to deliver ultra-high dose rate (UHDR) irradiation to mice and cell-culture FLASH radiobiology experiments. UHDR electron beams were controlled by a microcontroller and relay interfaced with the respiratory gating system. We produced beam collimators with indexed stereotactic mouse positioning devices to provide anatomically specific preclinical treatments. Treatment delivery was monitored directly with an ionization chamber, and charge measurements were correlated with radiochromic film at the entry surface of the mice. The setup for conventional (CONV) dose rate irradiation was similar but the source-to-surface distance was longer. Monte Carlo simulations and film dosimetry were used to characterize beam properties and dose distributions. The mean electron beam energies before the flattening filter were 18.8 MeV (UHDR) and 17.7 MeV (CONV), with corresponding values at the mouse surface of 17.2 MeV and 16.2 MeV. The charges measured with an external ion chamber were linearly correlated with the mouse entrance dose. Use of relay gating for pulse control initially led to a delivery failure rate of 20% ($+/-$ 1 pulse); adjustments to account for the linac latency improved this rate to <1/20. Beam field sizes for two anatomically specific mouse collimators (4x4 $cm^2$ for whole-abdomen and 1.5x1.5 $cm^2$ for unilateral lung irradiation) were accurate within <5% and had low radiation leakage (<4%). Normalizing the dose at the center of the mouse (~0.75 cm depth) produced UHDR and CONV doses to the irradiated volumes with >95% agreement. We successfully configured a clinical linear accelerator for increased output and developed a robust preclinical platform for anatomically specific irradiation, with highly accurate and precise temporal and spatial dose delivery, for both CONV and UHDR applications.
The recent criticisms of the robustness of post hoc model approximation explanation methods (like LIME and SHAP) have led to the rise of model-precise abductive explanations. For each data point, abductive explanations provide a minimal subset of features that are sufficient to generate the outcome. While theoretically sound and rigorous, abductive explanations suffer from a major issue -- there can be several valid abductive explanations for the same data point. In such cases, providing a single abductive explanation can be insufficient; on the other hand, providing all valid abductive explanations can be incomprehensible due to their size. In this work, we solve this issue by aggregating the many possible abductive explanations into feature importance scores. We propose three aggregation methods: two based on power indices from cooperative game theory and a third based on a well-known measure of causal strength. We characterize these three methods axiomatically, showing that each of them uniquely satisfies a set of desirable properties. We also evaluate them on multiple datasets and show that these explanations are robust to the attacks that fool SHAP and LIME.
The Graphical House Allocation problem asks: how can $n$ houses (each with a fixed non-negative value) be assigned to the vertices of an undirected graph $G$, so as to minimize the "aggregate local envy", i.e., the sum of absolute differences along the edges of $G$? This problem generalizes the classical Minimum Linear Arrangement problem, as well as the well-known House Allocation Problem from Economics, the latter of which has notable practical applications in organ exchanges. Recent work has studied the computational aspects of Graphical House Allocation and observed that the problem is NP-hard and inapproximable even on particularly simple classes of graphs, such as vertex disjoint unions of paths. However, the dependence of any approximations on the structural properties of the underlying graph had not been studied. In this work, we give a complete characterization of the approximability of the Graphical House Allocation problem. We present algorithms to approximate the optimal envy on general graphs, trees, planar graphs, bounded-degree graphs, bounded-degree planar graphs, and bounded-degree trees. For each of these graph classes, we then prove matching lower bounds, showing that in each case, no significant improvement can be attained unless P = NP. We also present general approximation ratios as a function of structural parameters of the underlying graph, such as treewidth; these match the aforementioned tight upper bounds in general, and are significantly better approximations for many natural subclasses of graphs. Finally, we present constant factor approximation schemes for the special classes of complete binary trees and random graphs.
We study the problem of fairly allocating indivisible goods (positively valued items) and chores (negatively valued items) among agents with decreasing marginal utilities over items. Our focus is on instances where all the agents have simple preferences; specifically, we assume the marginal value of an item can be either $-1$, $0$ or some positive integer $c$. Under this assumption, we present an efficient algorithm to compute leximin allocations for a broad class of valuation functions we call order-neutral submodular valuations. Order-neutral submodular valuations strictly contain the well-studied class of additive valuations but are a strict subset of the class of submodular valuations. We show that these leximin allocations are Lorenz dominating and approximately proportional. We also show that, under further restriction to additive valuations, these leximin allocations are approximately envy-free and guarantee each agent their maxmin share. We complement this algorithmic result with a lower bound showing that the problem of computing leximin allocations is NP-hard when $c$ is a rational number.
Algorithmic recourse is a process that leverages counterfactual explanations, going beyond understanding why a system produced a given classification, to providing a user with actions they can take to change their predicted outcome. Existing approaches to compute such interventions -- known as recourse -- identify a set of points that satisfy some desiderata -- e.g. an intervention in the underlying causal graph, minimizing a cost function, etc. Satisfying these criteria, however, requires extensive knowledge of the underlying model structure, an often unrealistic amount of information in several domains. We propose a data-driven and model-agnostic framework to compute counterfactual explanations. We introduce StEP, a computationally efficient method that offers incremental steps along the data manifold that directs users towards their desired outcome. We show that StEP uniquely satisfies a desirable set of axioms. Furthermore, via a thorough empirical and theoretical investigation, we show that StEP offers provable robustness and privacy guarantees while outperforming popular methods along important metrics.
We study the problem of allocating indivisible chores among agents with binary supermodular cost functions. In other words, each chore has a marginal cost of $0$ or $1$ and chores exhibit increasing marginal costs (or decreasing marginal utilities). In this note, we combine the techniques of Viswanathan and Zick (2022) and Barman et al. (2023) to present a general framework for fair allocation with this class of valuation functions. Our framework allows us to generalize the results of Barman et al. (2023) and efficiently compute allocations which satisfy weighted notions of fairness like weighted leximin or min weighted $p$-mean malfare for any $p \ge 1$.
We study the problem of fairly allocating a set of indivisible goods among agents with {\em bivalued submodular valuations} -- each good provides a marginal gain of either $a$ or $b$ ($a < b$) and goods have decreasing marginal gains. This is a natural generalization of two well-studied valuation classes -- bivalued additive valuations and binary submodular valuations. We present a simple sequential algorithmic framework, based on the recently introduced Yankee Swap mechanism, that can be adapted to compute a variety of solution concepts, including max Nash welfare (MNW), leximin and $p$-mean welfare maximizing allocations when $a$ divides $b$. This result is complemented by an existing result on the computational intractability of MNW and leximin allocations when $a$ does not divide $b$. We show that MNW and leximin allocations guarantee each agent at least $\frac25$ and $\frac{a}{b+2a}$ of their maximin share, respectively, when $a$ divides $b$. We also show that neither the leximin nor the MNW allocation is guaranteed to be envy free up to one good (EF1). This is surprising since for the simpler classes of bivalued additive valuations and binary submodular valuations, MNW allocations are known to be envy free up to any good (EFX).
Moving Target Defense (MTD) has emerged as a key technique in various security applications as it takes away the attacker's ability to perform reconnaissance for exploiting a system's vulnerabilities. However, most of the existing research in the field assumes unrealistic access to information about the attacker's motivations and/or actions when developing MTD strategies. Many of the existing approaches also assume complete knowledge regarding the vulnerabilities of a system and how each of these vulnerabilities can be exploited by an attacker. In this work, we aim to create algorithms that generate effective Moving Target Defense strategies that do not rely on prior knowledge about the attackers. Our work assumes that the only way the defender receives information about its own reward is via interaction with the attacker in a repeated game setting. Depending on the amount of information that can be obtained from the interactions, we devise two different algorithms using multi-armed bandit formulation to identify efficient strategies. We then evaluate our algorithms using data mined from the National Vulnerability Database to showcase that they match the performance of the state-of-the-art techniques, despite using a lot less amount of information.