arXiv subjects

Raghuvansh Saxena

Publications and source records attributed to Raghuvansh Saxena.

4 recordsLinked to original sources

arXiv2026

Fair Allocation under Conflict Constraints

We study the fair allocation of indivisible items subject to conflict constraints. In this framework, the items are represented as the vertices of a graph, with edges corresponding to conflicts between pairs of items. Each agent is assigned an independent set of items from the graph. Our goal is to achieve a fair and efficient allocation of these items. Fairness pertains to satisfying envy-freeness up to one item (EF1), while efficiency is defined by maximality, meaning that no unallocated item can be feasibly assigned to any agent. First, we explore the case of two agents. For monotone valuations, we show that a maximal EF1 allocation always exists on any graph. Our existence proof relies on a color-switching technique, which locally modifies a maximal allocation while preserving feasibility and restoring EF1. We further show that such allocations can be computed in pseudopolynomial time in general, and in polynomial time for additive valuations on arbitrary graphs, as well as for monotone valuations on interval and bipartite graphs. By contrast, once monotonicity is dropped, maximal EF1 allocations need not exist even for identical additive valuations, and deciding existence becomes NP-hard. Next, we consider the case with a general number of agents. Again, we arrive at a negative result: An EF1 and maximal allocation fails to exist even for three agents under identical monotone valuations, and determining the existence of such an allocation is NP-hard. On the positive side, we show that under identical non-monotone additive valuations on a path graph, an EF[1,1] and maximal allocation always exists. This result involves a novel application of the "cycle plus triangles" theorem.

cs.GT↗

arXiv2026

Fair Interval Scheduling of Indivisible Chores

We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with a start and finish time, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i.e., no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomial-time algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents with identical additive valuations, we show the existence of an EF1 and maximal schedule when the constraints constitute a path graph. This result uses a reduction to the ``cycle-plus-triangles'' theorem. Using different techniques, we provide an efficient algorithm for finding such a schedule when there are four or more agents and the valuations are further assumed to be dichotomous. We also show that stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) along with maximality and EF1 along with Pareto optimality, cannot be achieved.

cs.GT↗

arXiv2021

Near-Optimal Two-Pass Streaming Algorithm for Sampling Random Walks over Directed Graphs

For a directed graph $G$ with $n$ vertices and a start vertex $u_{\sf start}$, we wish to (approximately) sample an $L$-step random walk over $G$ starting from $u_{\sf start}$ with minimum space using an algorithm that only makes few passes over the edges of the graph. This problem found many applications, for instance, in approximating the PageRank of a webpage. If only a single pass is allowed, the space complexity of this problem was shown to be $\tildeΘ(n \cdot L)$. Prior to our work, a better space complexity was only known with $\tilde{O}(\sqrt{L})$ passes. We settle the space complexity of this random walk simulation problem for two-pass streaming algorithms, showing that it is $\tildeΘ(n \cdot \sqrt{L})$, by giving almost matching upper and lower bounds. Our lower bound argument extends to every constant number of passes $p$, and shows that any $p$-pass algorithm for this problem uses $\tildeΩ(n \cdot L^{1/p})$ space. In addition, we show a similar $\tildeΘ(n \cdot \sqrt{L})$ bound on the space complexity of any algorithm (with any number of passes) for the related problem of sampling an $L$-step random walk from every vertex in the graph.

cs.DS↗

arXiv2020

Optimal Mechanism Design for Single-Minded Agents

We consider revenue-optimal mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and one is a 'type' that captures some auxiliary information. One setting is the FedEx Problem, for which FGKK [2016] characterize the optimal mechanism for a single agent. We ask: how far can such characterizations go? In particular, we consider single-minded agents. A seller has heterogenous items. A buyer has a value v for a specific subset of items S, and obtains value v iff he gets (at least) all the items in S. We show: 1. Deterministic mechanisms are optimal for distributions that satisfy the "declining marginal revenue" (DMR) property; we give an explicit construction of the optimal mechanism. 2. Without DMR, the result depends on the structure of the directed acyclic graph (DAG) representing the partial order among types. When the DAG has out-degree at most 1, we characterize the optimal mechanism a la FedEx. 3. Without DMR, when the DAG has some node with out-degree at least 2, we show that in this case the menu complexity is unbounded: for any M, there exist distributions over (v,S) pairs such that the menu complexity of the optimal mechanism is at least M. 4. For the case of 3 types, we show that for all distributions there exists an optimal mechanism of finite menu complexity. This is in contrast to 2 additive heterogenous items or which the menu complexity could be uncountable [MV07; DDT15]. In addition, we prove that optimal mechanisms for Multi-Unit Pricing (without DMR) can have unbounded menu complexity. We also propose an extension where the menu complexity of optimal mechanisms can be countable but not uncountable. Together these results establish that optimal mechanisms in interdimensional settings are both much richer than single-dimensional settings, yet also vastly more structured than multi-dimensional settings.

cs.GT↗