arXiv subjects

Pranjal Pandey

Publications and source records attributed to Pranjal Pandey.

5 recordsLinked to original sources

arXiv2026

Reaching Fairness by Reallocating Goods

Fair allocation of indivisible goods has largely been studied under the assumption that no prior allocation exists. Motivated by practical settings with pre-existing (and possibly unfair) allocations, we study how to achieve fairness through limited reallocations. Building on recent work on reformability/reallocations, we consider three fairness notions---envy-freeness (EF), envy-freeness up to one good (EF1), and envy-freeness up to any good (EFX)---and optimize the number of goods reallocated. We analyze both the classical and parameterized complexity of these problems, providing a comprehensive analysis across multiple fairness notions.

cs.GT↗

arXiv2026

Minimum Envy Graphical House Allocation Beyond Identical Valuations

House allocation is an extremely well-studied problem in the field of fair allocation, where the goal is to assign $n$ houses to $n$ agents while satisfying certain fairness criterion, e.g., envy-freeness. To model social interactions, the Graphical House Allocation framework introduces a social graph $G$, in which each vertex corresponds to an agent, and an edge $(u, v)$ corresponds to the potential of agent $u$ to envy the agent $v$, based on their allocations and valuations. In undirected social graphs, the potential for envy is in both the directions. In the Minimum Envy Graphical House Allocation (ME-GHA) problem, given a set of $n$ agents, $n$ houses, a social graph, and agent's valuation functions, the goal is to find an allocation that minimizes the total envy summed up over all the edges of $G$. Recent work, [Hosseini et al., AAMAS 2023, AAMAS 2024] studied ME-GHA in the regime of polynomial-time algorithms, and designed exact and approximation algorithms, for certain graph classes under identical agent valuations. We initiate the study of \gha with non-identical valuations, a setting that has so far remained unexplored. We investigate the multivariate (parameterized) complexity of \gha by identifying structural restrictions on the social graph and valuation functions that yield tractability. We also design moderately exponential-time algorithms for several graph classes, and a polynomial-time algorithm for {binary valuations that returns an allocation with envy at most one when the social graph has maximum degree at most one.

cs.GT↗

arXiv2019

Modified celestial amplitude in Einstein gravity

In this paper we evaluate the modified celestial amplitude for gravitons and gluons, as defined in arXiv:1801.10171[hep-th]. We find that the modified (tree) amplitude is finite for gravitons in Einstein gravity. The modified amplitude behaves like correlation function of operators inserted at various points of null-infinity in the Minkowski space-time. Therefore, unlike the standard celestial amplitudes, these are three dimensional objects. We also show that this amplitude admits conformal soft factorization recently studied in the literature.

hep-th↗

arXiv2019

Conformal properties of soft operators -- 2 : Use of null-states

Representations of the (Lorentz) conformal group with the soft operators as highest weight vectors have two universal properties, which we clearly state in this paper. Given a soft operator with a certain dimension and spin, the first property is about the existence of "(large) gauge transformation" that acts on the soft operator. The second property is the decoupling of (large) gauge-invariant null-states of the soft operators from the $S$-matrix elements. In each case, the decoupling equation has the form of zero field-strength condition with the soft operator as the (gauge) potential. Null-state decoupling effectively reduces the number of polarisation states of the soft particle and is crucial in deriving soft-theorems from the Ward identities of asymptotic symmetries. To the best of our understanding, these properties are not directly related to the Lorentz invariance of the $S$-matrix or the existence of asymptotic symmetries. We also verify that the results obtained from the decoupling of null-states are consistent with the leading and subleading soft-theorems with finite energy massive and massless particles in the external legs.

hep-th↗

arXiv2019

Conformal properties of soft-operators - 1 : Use of null-states

Soft-operators, loosely speaking, are operators which create or annihilate zero energy massless particles on the celestial sphere in Minkowski space. The Lorentz group acts on the celestial sphere by conformal transformation and the soft-operators transform as conformal primary operators of various dimension and spin. Working in space-time dimensions $D=4$ and $6$, we study some properties of the conformal representations with the (leading) soft photon and graviton as the highest weight vectors. Typically these representations contain null-vectors. We argue, from the $S$-matrix point of view, that infinite dimensional asymptotic symmetries and conformal invariance require us to set some of these null-vectors to zero. As a result, the corresponding soft-operator satisfies linear PDE on the celestial sphere. Curiously, these PDEs are equations of motion of Euclidean gauge theories on the celestial sphere with scalar gauge-invariance, i.e, the gauge parameter is a scalar field on the sphere. These are probably related to large $U(1)$ and supertranslation transformations at infinity. Now, the PDE satisfied by the soft-operator can be converted into PDE for the $S$-matrix elements with the insertion of the soft-operator. These equations can then be solved subject to appropriate boundary conditions on the celestial sphere, provided by conformal invariance. The solutions determine the soft $S$-matrix elements, for different helicities of the soft-particle, in terms of a single scalar function. This makes the Ward-identity for the asymptotic symmetry almost integrable. The result of the integration, which we are not able to perform completely, should of course be Weinberg's soft-theorem. Finally, we comment on the similarity between the roles played by null-states in the context of asymptotic symmetry and in string theory in relation to space-time gauge symmetry.

hep-th↗