Palash Dey

Ashlesha Hota, Shashwat Kumar, Daman Deep Singh et al.

The concentration of digital payment transactions in just two UPI apps like PhonePe and Google Pay has raised concerns of duopoly in India s digital financial ecosystem. To address this, the National Payments Corporation of India (NPCI) has mandated that no single UPI app should exceed 30 percent of total transaction volume. Enforcing this cap, however, poses a significant computational challenge: how to redistribute user transactions across apps without causing widespread user inconvenience while maintaining capacity limits? In this paper, we formalize this problem as the Minimum Edge Activation Flow (MEAF) problem on a bipartite network of users and apps, where activating an edge corresponds to a new app installation. The objective is to ensure a feasible flow respecting app capacities while minimizing additional activations. We further prove that Minimum Edge Activation Flow is NP-Complete. To address the computational challenge, we propose scalable heuristics, named Decoupled Two-Stage Allocation Strategy (DTAS), that exploit flow structure and capacity reuse. Experiments on large semi-synthetic transaction network data show that DTAS finds solutions close to the optimal ILP within seconds, offering a fast and practical way to enforce transaction caps fairly and efficiently.

Ashlesha Hota, Susobhan Bandopadhyay, Palash Dey

In shift bribery, a briber seeks to promote his preferred candidate by paying voters to raise their ranking. Classical models of shift bribery assume voters act independently, overlooking the role of social influence. However, in reality, individuals are social beings and are often represented as part of a social network, where bribed voters may influence their neighbors, thereby amplifying the effect of persuasion. We study Shift bribery over Networks, where voters are modeled as nodes in a directed weighted graph, and arcs represent social influence between them. In this setting, bribery is not confined to directly targeted voters its effects can propagate through the network, influencing neighbors and amplifying persuasion. Given a budget and individual cost functions for shifting each voter's preference toward a designated candidate, the goal is to determine whether a shift strategy exists within budget that ensures the preferred candidate wins after both direct and network-propagated influence takes effect. We show that the problem is NP-Complete even with two candidates and unit costs, and W[2]-hard when parameterized by budget or maximum degree. On the positive side, we design polynomial-time algorithms for complete graphs under plurality and majority rules and path graphs for uniform edge weights, linear-time algorithms for transitive tournaments for two candidates, linear cost functions and uniform arc weights, and pseudo-polynomial algorithms for cluster graphs. We further prove the existence of fixed-parameter tractable algorithms with treewidth as parameter for two candidates, linear cost functions and uniform arc weights and pseudo-FPT with cluster vertex deletion number for two candidates and uniform arc weights. Together, these results give a detailed complexity landscape for shift bribery in social networks.

Palash Dey, Anubhav Dhar, Ashlesha Hota et al.

In this paper, we study the Maximum Common Vertex Subgraph problem: Given two input graphs $G_1,G_2$ and a non-negative integer $h$, is there a common subgraph $H$ on at least $h$ vertices such that there is no isolated vertex in $H$. In other words, each connected component of $H$ has at least $2$ vertices. This problem naturally arises in graph theory along with other variants of the well-studied Maximum Common Subgraph problem and also has applications in computational social choice. We show that this problem is NP-hard and provide an FPT algorithm when parameterized by $h$. Next, we conduct a study of the problem on common structural parameters like vertex cover number, maximum degree, treedepth, pathwidth and treewidth of one or both input graphs. We derive a complete dichotomy of parameterized results for our problem with respect to individual parameterizations as well as combinations of parameterizations from the above structural parameters. This provides us with a deep insight into the complexity theoretic and parameterized landscape of this problem.

Palash Dey, Sudeshna Kolay, Sipra Singh

We study the knapsack problem with graph theoretic constraints. That is, we assume that there exists a graph structure on the set of items of knapsack and the solution also needs to satisfy certain graph theoretic properties on top of knapsack constraints. In particular, we need to compute in the connected knapsack problem a connected subset of items which has maximum value subject to the size of knapsack constraint. We show that this problem is strongly NP-complete even for graphs of maximum degree four and NP-complete even for star graphs. On the other hand, we develop an algorithm running in time $O\left(2^{tw\log tw}\cdot\text{poly}(\min\{s^2,d^2\})\right)$ where $tw,s,d$ are respectively treewidth of the graph, size, and target value of the knapsack. We further exhibit a $(1-ε)$ factor approximation algorithm running in time $O\left(2^{tw\log tw}\cdot\text{poly}(n,1/ε)\right)$ for every $ε>0$. We show similar results for several other graph theoretic properties, namely path and shortest-path under the problem names path-knapsack and shortestpath-knapsack. Our results seems to indicate that connected-knapsack is computationally hardest followed by path-knapsack and shortestpath-knapsack.

Koustav De, Harshil Mittal, Palash Dey et al.

The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is NP-hard. Consequently, the computational task of finding a Kemeny ranking has been studied under the lens of parameterized complexity with respect to many parameters. We first present a comprehensive relationship, both theoretical and empirical, among these parameters. Further, we study the problem of computing all distinct Kemeny rankings under the lens of parameterized complexity. We consider the target Kemeny score, number of candidates, average distance of input rankings, maximum range of any candidate, and unanimity width as our parameters. For all these parameters, we already have FPT algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present FPT approximation algorithms for Kemeny rank aggregation with respect to these parameters.

Aritra Banik, Sujoy Bhore, Palash Dey et al.

The kidney exchange mechanism allows many patient-donor pairs who are otherwise incompatible with each other to come together and exchange kidneys along a cycle. However, due to infrastructure and legal constraints, kidney exchange can only be performed in small cycles in practice. In reality, there are also some altruistic donors who do not have any paired patients. This allows us to also perform kidney exchange along paths that start from some altruistic donor. Unfortunately, the computational task is NP-complete. To overcome this computational barrier, an important line of research focuses on designing faster algorithms, both exact and using the framework of parameterized complexity. The standard parameter for the kidney exchange problem is the number $t$ of patients that receive a healthy kidney. The current fastest known deterministic FPT algorithm for this problem, parameterized by $t$, is $O^\star\left(14^t\right)$. In this work, we improve this by presenting a deterministic FPT algorithm that runs in time $O^\star\left((4e)^t\right)\approx O^\star\left(10.88^t\right)$. This problem is also known to be W[1]-hard parameterized by the treewidth of the underlying undirected graph. A natural question here is whether the kidney exchange problem admits an FPT algorithm parameterized by the pathwidth of the underlying undirected graph. We answer this negatively in this paper by proving that this problem is W[1]-hard parameterized by the pathwidth of the underlying undirected graph. We also present some parameterized intractability results improving the current understanding of the problem under the framework of parameterized complexity.

Dhruv Sarkar, Aprameyo Chakrabartty, Subhamon Supantha et al.

We study a generalization of the Online Convex Optimization (OCO) framework with time-varying adversarial constraints. In this problem, after selecting a feasible action from the convex decision set $X,$ a convex constraint function is revealed alongside the cost function in each round. Our goal is to design a computationally efficient learning policy that achieves a small regret with respect to the cost functions and a small cumulative constraint violation (CCV) with respect to the constraint functions over a horizon of length $T$. It is well-known that the projection step constitutes the major computational bottleneck of the standard OCO algorithms. However, for many structured decision sets, linear functions can be efficiently optimized over the decision set. We propose a *projection-free* online policy which makes a single call to a Linear Program (LP) solver per round. Our method outperforms state-of-the-art projection-free online algorithms with adversarial constraints, achieving improved bounds of $\tilde{O}(T^{\frac{3}{4}})$ for both regret and CCV. The proposed algorithm is conceptually simple - it first constructs a surrogate cost function as a non-negative linear combination of the cost and constraint functions. Then, it passes the surrogate costs to a new, adaptive version of the online conditional gradient subroutine, which we propose in this paper.

Palash Dey, Debajyoti Kar, Swagato Sanyal

In a district-based election, we apply a voting rule $r$ to decide the winners in each district, and a candidate who wins in a maximum number of districts is the winner of the election. We present efficient sampling-based algorithms to predict the winner of such district-based election systems in this paper. When $r$ is plurality and the margin of victory is known to be at least $\varepsilon$ fraction of the total population, we present an algorithm to predict the winner. The sample complexity of our algorithm is $\mathcal{O}\left(\frac{1}{\varepsilon^4}\log \frac{1}{\varepsilon}\log\frac{1}δ\right)$. We complement this result by proving that any algorithm, from a natural class of algorithms, for predicting the winner in a district-based election when $r$ is plurality, must sample at least $Ω\left(\frac{1}{\varepsilon^4}\log\frac{1}δ\right)$ votes. We then extend this result to any voting rule $r$. Loosely speaking, we show that we can predict the winner of a district-based election with an extra overhead of $\mathcal{O}\left(\frac{1}{\varepsilon^2}\log\frac{1}δ\right)$ over the sample complexity of predicting the single-district winner under $r$. We further extend our algorithm for the case when the margin of victory is unknown, but we have only two candidates. We then consider the median voting rule when the set of preferences in each district is single-peaked. We show that the winner of a district-based election can be predicted with $\mathcal{O}\left(\frac{1}{\varepsilon^4}\log\frac{1}{\varepsilon}\log\frac{1}δ\right)$ samples even when the harmonious order in different districts can be different and even unknown. Finally, we also show some results for estimating the margin of victory of a district-based election within both additive and multiplicative error bounds.

Debajyoti Kar, Mert Kosan, Debmalya Mandal et al.

Ensuring fairness in machine learning algorithms is a challenging and essential task. We consider the problem of clustering a set of points while satisfying fairness constraints. While there have been several attempts to capture group fairness in the $k$-clustering problem, fairness at an individual level is relatively less explored. We introduce a new notion of individual fairness in $k$-clustering based on features not necessarily used for clustering. We show that this problem is NP-hard and does not admit a constant factor approximation. Therefore, we design a randomized algorithm that guarantees approximation both in terms of minimizing the clustering distance objective and individual fairness under natural restrictions on the distance metric and fairness constraints. Finally, our experimental results against six competing baselines validate that our algorithm produces individually fairer clusters than the fairest baseline by 12.5% on average while also being less costly in terms of the clustering objective than the best baseline by 34.5% on average.

Aditya Anand, Palash Dey

We introduce the notion of {\em Distance Restricted Manipulation}, where colluding manipulator(s) need to compute if there exist votes which make their preferred alternative win the election when their knowledge about the others' votes is a little inaccurate. We use the Kendall-Tau distance to model the manipulators' confidence in the non-manipulators' votes. To this end, we study this problem in two settings - one where the manipulators need to compute a manipulating vote that succeeds irrespective of perturbations in others' votes ({\em Distance Restricted Strong Manipulation}), and the second where the manipulators need to compute a manipulating vote that succeeds for at least one possible vote profile of the others ({\em Distance Restricted Weak Manipulation}). We show that {\em Distance Restricted Strong Manipulation} admits polynomial-time algorithms for every scoring rule, maximin, Bucklin, and simplified Bucklin voting rules for a single manipulator, and for the $k$-approval rule for any number of manipulators, but becomes intractable for the Copeland$^α$ voting rule for every $α\in[0,1]$ even for a single manipulator. In contrast, {\em Distance Restricted Weak Manipulation} is intractable for almost all the common voting rules, with the exception of the plurality rule. For a constant number of alternatives, we show that both the problems are polynomial-time solvable for every anonymous and efficient voting rule.

Palash Dey, Neeldhara Misra, Swaprava Nath et al.

We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers $k_a$ and $k_d$ corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most $k_d$ voter groups such that, no matter which $k_a$ voter groups the attacker attacks, the outcome of the election does not change. In the optimal attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking $k_a$ voter groups such that, no matter which $k_d$ voter groups the defender defends, the outcome of the election is always different from the original (without any attack) one.

Palash Dey, Swaprava Nath, Garima Shakya

A preferential domain is a collection of sets of preferences which are linear orders over a set of alternatives. These domains have been studied extensively in social choice theory due to both its practical importance and theoretical elegance. Examples of some extensively studied preferential domains include single peaked, single crossing, Euclidean, etc. In this paper, we study the sample complexity of testing whether a given preference profile is close to some specific domain. We consider two notions of closeness: (a) closeness via preferences, and (b) closeness via alternatives. We further explore the effect of assuming that the {\em outlier} preferences/alternatives to be random (instead of arbitrary) on the sample complexity of the testing problem. In most cases, we show that the above testing problem can be solved with high probability for all commonly used domains by observing only a small number of samples (independent of the number of preferences, $n$, and often the number of alternatives, $m$). In the remaining few cases, we prove either impossibility results or $Ω(n)$ lower bound on the sample complexity. We complement our theoretical findings with extensive simulations to figure out the actual constant factors of our asymptotic sample complexity bounds.

Palash Dey

Studying complexity of various bribery problems has been one of the main research focus in computational social choice. In all the models of bribery studied so far, the briber has to pay every voter some amount of money depending on what the briber wants the voter to report and the briber has some budget at her disposal. Although these models successfully capture many real world applications, in many other scenarios, the voters may be unwilling to deviate too much from their true preferences. In this paper, we study the computational complexity of the problem of finding a preference profile which is as close to the true preference profile as possible and still achieves the briber's goal subject to budget constraints. We call this problem Optimal Bribery. We consider three important measures of distances, namely, swap distance, footrule distance, and maximum displacement distance, and resolve the complexity of the optimal bribery problem for many common voting rules. We show that the problem is polynomial time solvable for the plurality and veto voting rules for all the three measures of distance. On the other hand, we prove that the problem is NP-complete for a class of scoring rules which includes the Borda voting rule, maximin, Copeland$^α$ for any $α\in[0,1]$, and Bucklin voting rules for all the three measures of distance even when the distance allowed per voter is $1$ for the swap and maximum displacement distances and $2$ for the footrule distance even without the budget constraints (which corresponds to having an infinite budget). For the $k$-approval voting rule for any constant $k>1$ and the simplified Bucklin voting rule, we show that the problem is NP-complete for the swap distance even when the distance allowed is $2$ and for the footrule distance even when the distance allowed is $4$ even without the budget constraints.

Palash Dey

This thesis is in the area called computational social choice which is an intersection area of algorithms and social choice theory.

Palash Dey, Nimrod Talmon, Otniel van Handel

We consider elections where the voters come one at a time, in a streaming fashion, and devise space-efficient algorithms which identify an approximate winning committee with respect to common multiwinner proportional representation voting rules; specifically, we consider the Approval-based and the Borda-based variants of both the Chamberlin-- ourant rule and the Monroe rule. We complement our algorithms with lower bounds. Somewhat surprisingly, our results imply that, using space which does not depend on the number of voters it is possible to efficiently identify an approximate representative committee of fixed size over vote streams with huge number of voters.

Arnab Maiti, Palash Dey

A directed graph where there is exactly one edge between every pair of vertices is called a {\em tournament}. Finding the "best" set of vertices of a tournament is a well studied problem in social choice theory. A {\em tournament solution} takes a tournament as input and outputs a subset of vertices of the input tournament. However, in many applications, for example, choosing the best set of drugs from a given set of drugs, the edges of the tournament are given only implicitly and knowing the orientation of an edge is costly. In such scenarios, we would like to know the best set of vertices (according to some tournament solution) by "querying" as few edges as possible. We, in this paper, precisely study this problem for commonly used tournament solutions: given an oracle access to the edges of a tournament T, find $f(T)$ by querying as few edges as possible, for a tournament solution f. We first show that the set of Condorcet non-losers in a tournament can be found by querying $2n-\lfloor \log n \rfloor -2$ edges only and this is tight in the sense that every algorithm for finding the set of Condorcet non-losers needs to query at least $2n-\lfloor \log n \rfloor -2$ edges in the worst case, where $n$ is the number of vertices in the input tournament. We then move on to study other popular tournament solutions and show that any algorithm for finding the Copeland set, the Slater set, the Markov set, the bipartisan set, the uncovered set, the Banks set, and the top cycle must query $Ω(n^2)$ edges in the worst case. On the positive side, we are able to circumvent our strong query complexity lower bound results by proving that, if the size of the top cycle of the input tournament is at most $k$, then we can find all the tournament solutions mentioned above by querying $O(nk + \frac{n\log n}{\log(1-\frac{1}{k})})$ edges only.

Palash Dey

We introduce and study the weakly single-crossing domain on trees which is a generalization of the well-studied single-crossing domain in social choice theory. We design a polynomial-time algorithm for recognizing preference profiles which belong to this domain. We then develop an efficient elicitation algorithm for this domain which works even if the preferences can be accessed only sequentially and the underlying single-crossing tree structure is not known beforehand. We also prove matching lower bound on the query complexity of our elicitation algorithm when the number of voters is large compared to the number of candidates. We also prove a lower bound of $Ω(m^2\log n)$ on the number of queries that any algorithm needs to ask to elicit single crossing profile when random queries are allowed. This resolves an open question in an earlier paper and proves optimality of their preference elicitation algorithm when random queries are allowed.

Palash Dey, Neeldhara Misra

Eliciting the preferences of a set of agents over a set of alternatives is a problem of fundamental importance in social choice theory. Prior work on this problem has studied the query complexity of preference elicitation for the unrestricted domain and for the domain of single peaked preferences. In this paper, we consider the domain of single crossing preference profiles and study the query complexity of preference elicitation under various settings. We consider two distinct situations: when an ordering of the voters with respect to which the profile is single crossing is known versus when it is unknown. We also consider different access models: when the votes can be accessed at random, as opposed to when they are coming in a pre-defined sequence. In the sequential access model, we distinguish two cases when the ordering is known: the first is that sequence in which the votes appear is also a single-crossing order, versus when it is not. The main contribution of our work is to provide polynomial time algorithms with low query complexity for preference elicitation in all the above six cases. Further, we show that the query complexities of our algorithms are optimal up to constant factors for all but one of the above six cases. We then present preference elicitation algorithms for profiles which are close to being single crossing under various notions of closeness, for example, single crossing width, minimum number of candidates | voters whose deletion makes a profile single crossing.

Palash Dey, Neeldhara Misra

In multiagent systems, we often have a set of agents each of which have a preference ordering over a set of items and one would like to know these preference orderings for various tasks, for example, data analysis, preference aggregation, voting etc. However, we often have a large number of items which makes it impractical to ask the agents for their complete preference ordering. In such scenarios, we usually elicit these agents' preferences by asking (a hopefully small number of) comparison queries --- asking an agent to compare two items. Prior works on preference elicitation focus on unrestricted domain and the domain of single peaked preferences and show that the preferences in single peaked domain can be elicited by much less number of queries compared to unrestricted domain. We extend this line of research and study preference elicitation for single peaked preferences on trees which is a strict superset of the domain of single peaked preferences. We show that the query complexity crucially depends on the number of leaves, the path cover number, and the distance from path of the underlying single peaked tree, whereas the other natural parameters like maximum degree, diameter, pathwidth do not play any direct role in determining query complexity. We then investigate the query complexity for finding a weak Condorcet winner for preferences single peaked on a tree and show that this task has much less query complexity than preference elicitation. Here again we observe that the number of leaves in the underlying single peaked tree and the path cover number of the tree influence the query complexity of the problem.

Palash Dey, Neeldhara Misra, Y. Narahari

The Coalitional Manipulation problem has been studied extensively in the literature for many voting rules. However, most studies have focused on the complete information setting, wherein the manipulators know the votes of the non-manipulators. While this assumption is reasonable for purposes of showing intractability, it is unrealistic for algorithmic considerations. In most real-world scenarios, it is impractical for the manipulators to have accurate knowledge of all the other votes. In this paper, we investigate manipulation with incomplete information. In our framework, the manipulators know a partial order for each voter that is consistent with the true preference of that voter. In this setting, we formulate three natural computational notions of manipulation, namely weak, opportunistic, and strong manipulation. We say that an extension of a partial order is if there exists a manipulative vote for that extension. 1. Weak Manipulation (WM): the manipulators seek to vote in a way that makes their preferred candidate win in at least one extension of the partial votes of the non-manipulators. 2. Opportunistic Manipulation (OM): the manipulators seek to vote in a way that makes their preferred candidate win in every viable extension of the partial votes of the non-manipulators. 3. Strong Manipulation (SM): the manipulators seek to vote in a way that makes their preferred candidate win in every extension of the partial votes of the non-manipulators. We consider several scenarios for which the traditional manipulation problems are easy (for instance, Borda with a single manipulator). For many of them, the corresponding manipulative questions that we propose turn out to be computationally intractable. Our hardness results often hold even when very little information is missing, or in other words, even when the instances are quite close to the complete information setting.

Palash Dey, Neeldhara Misra, Y. Narahari

We study the computational complexity of committee selection problem for several approval-based voting rules in the presence of outliers. Our first result shows that outlier consideration makes committee selection problem intractable for approval, net approval, and minisum approval voting rules. We then study parameterized complexity of this problem with five natural parameters, namely the target score, the size of the committee (and its dual parameter, the number of candidates outside the committee), the number of outliers (and its dual parameter, the number of non-outliers). For net approval and minisum approval voting rules, we provide a dichotomous result, resolving the parameterized complexity of this problem for all subsets of five natural parameters considered (by showing either FPT or W[1]-hardness for all subsets of parameters). For the approval voting rule, we resolve the parameterized complexity of this problem for all subsets of parameters except one. We also study approximation algorithms for this problem. We show that there does not exist any alpha(.) factor approximation algorithm for approval and net approval voting rules, for any computable function alpha(.), unless P=NP. For the minisum voting rule, we provide a pseudopolynomial (1+eps) factor approximation algorithm.

Arnab Bhattacharyya, Palash Dey

We investigate the problem of winner determination from computational social choice theory in the data stream model. Specifically, we consider the task of summarizing an arbitrarily ordered stream of $n$ votes on $m$ candidates into a small space data structure so as to be able to obtain the winner determined by popular voting rules. As we show, finding the exact winner requires storing essentially all the votes. So, we focus on the problem of finding an {\em $\eps$-winner}, a candidate who could win by a change of at most $\eps$ fraction of the votes. We show non-trivial upper and lower bounds on the space complexity of $\eps$-winner determination for several voting rules, including $k$-approval, $k$-veto, scoring rules, approval, maximin, Bucklin, Copeland, and plurality with run off.

Palash Dey, Y. Narahari

The margin of victory of an election is a useful measure to capture the robustness of an election outcome. It also plays a crucial role in determining the sample size of various algorithms in post election audit, polling etc. In this work, we present efficient sampling based algorithms for estimating the margin of victory of elections. More formally, we introduce the \textsc{$(c, ε, δ)$--Margin of Victory} problem, where given an election $\mathcal{E}$ on $n$ voters, the goal is to estimate the margin of victory $M(\mathcal{E})$ of $\mathcal{E}$ within an additive factor of $c MoV(\mathcal{E})+εn$. We study the \textsc{$(c, ε, δ)$--Margin of Victory} problem for many commonly used voting rules including scoring rules, approval, Bucklin, maximin, and Copeland$^α.$ We observe that even for the voting rules for which computing the margin of victory is NP-Hard, there may exist efficient sampling based algorithms, as observed in the cases of maximin and Copeland$^α$ voting rules.

Palash Dey, Neeldhara Misra, Y. Narahari

The Coalitional Manipulation (CM) problem has been studied extensively in the literature for many voting rules. The CM problem, however, has been studied only in the complete information setting, that is, when the manipulators know the votes of the non-manipulators. A more realistic scenario is an incomplete information setting where the manipulators do not know the exact votes of the non- manipulators but may have some partial knowledge of the votes. In this paper, we study a setting where the manipulators know a partial order for each voter that is consistent with the vote of that voter. In this setting, we introduce and study two natural computational problems - (1) Weak Manipulation (WM) problem where the manipulators wish to vote in a way that makes their preferred candidate win in at least one extension of the partial votes of the non-manipulators; (2) Strong Manipulation (SM) problem where the manipulators wish to vote in a way that makes their preferred candidate win in all possible extensions of the partial votes of the non-manipulators. We study the computational complexity of the WM and the SM problems for commonly used voting rules such as plurality, veto, k-approval, k-veto, maximin, Copeland, and Bucklin. Our key finding is that, barring a few exceptions, manipulation becomes a significantly harder problem in the setting of incomplete votes.

Palash Dey, Neeldhara Misra, Y. Narahari

Bribery in elections is an important problem in computational social choice theory. However, bribery with money is often illegal in elections. Motivated by this, we introduce the notion of frugal bribery and formulate two new pertinent computational problems which we call Frugal-bribery and Frugal- $bribery to capture bribery without money in elections. In the proposed model, the briber is frugal in nature and this is captured by her inability to bribe votes of a certain kind, namely, non-vulnerable votes. In the Frugal-bribery problem, the goal is to make a certain candidate win the election by changing only vulnerable votes. In the Frugal-{dollar}bribery problem, the vulnerable votes have prices and the goal is to make a certain candidate win the election by changing only vulnerable votes, subject to a budget constraint of the briber. We further formulate two natural variants of the Frugal-{dollar}bribery problem namely Uniform-frugal-{dollar}bribery and Nonuniform-frugal-{dollar}bribery where the prices of the vulnerable votes are, respectively, all the same or different. We study the computational complexity of the above problems for unweighted and weighted elections for several commonly used voting rules. We observe that, even if we have only a small number of candidates, the problems are intractable for all voting rules studied here for weighted elections, with the sole exception of the Frugal-bribery problem for the plurality voting rule. In contrast, we have polynomial time algorithms for the Frugal-bribery problem for plurality, veto, k-approval, k-veto, and plurality with runoff voting rules for unweighted elections. However, the Frugal-{dollar}bribery problem is intractable for all the voting rules studied here barring the plurality and the veto voting rules for unweighted elections.