7,910 research outputs found
Robust Combinatorial Optimization with Locally Budgeted Uncertainty
Budgeted uncertainty sets have been established as a major influence on
uncertainty modeling for robust optimization problems. A drawback of such sets
is that the budget constraint only restricts the global amount of cost increase
that can be distributed by an adversary. Local restrictions, while being
important for many applications, cannot be modeled this way.
We introduce new variant of budgeted uncertainty sets, called locally
budgeted uncertainty. In this setting, the uncertain parameters become
partitioned, such that a classic budgeted uncertainty set applies to each
partition, called region.
In a theoretical analysis, we show that the robust counterpart of such
problems for a constant number of regions remains solvable in polynomial time,
if the underlying nominal problem can be solved in polynomial time as well. If
the number of regions is unbounded, we show that the robust selection problem
remains solvable in polynomial time, while also providing hardness results for
other combinatorial problems.
In computational experiments using both random and real-world data, we show
that using locally budgeted uncertainty sets can have considerable advantages
over classic budgeted uncertainty sets
Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty
We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min-max-min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Γ of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min-max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min-max-min problem is N P-hard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times
A Variant of the Maximum Weight Independent Set Problem
We study a natural extension of the Maximum Weight Independent Set Problem
(MWIS), one of the most studied optimization problems in Graph algorithms. We
are given a graph , a weight function ,
a budget function , and a positive integer .
The weight (resp. budget) of a subset of vertices is the sum of weights (resp.
budgets) of the vertices in the subset. A -budgeted independent set in
is a subset of vertices, such that no pair of vertices in that subset are
adjacent, and the budget of the subset is at most . The goal is to find a
-budgeted independent set in such that its weight is maximum among all
the -budgeted independent sets in . We refer to this problem as MWBIS.
Being a generalization of MWIS, MWBIS also has several applications in
Scheduling, Wireless networks and so on. Due to the hardness results implied
from MWIS, we study the MWBIS problem in several special classes of graphs. We
design exact algorithms for trees, forests, cycle graphs, and interval graphs.
In unweighted case we design an approximation algorithm for -claw free
graphs whose approximation ratio () is competitive with the approximation
ratio () of MWIS (unweighted). Furthermore, we extend Baker's
technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page
Opinion dynamics with varying susceptibility to persuasion
A long line of work in social psychology has studied variations in people's susceptibility to persuasion -- the extent to which they are willing to modify their opinions on a topic. This body of literature suggests an interesting perspective on theoretical models of opinion formation by interacting parties in a network: in addition to considering interventions that directly modify people's intrinsic opinions, it is also natural to consider interventions that modify people's susceptibility to persuasion. In this work, we adopt a popular model for social opinion dynamics, and we formalize the opinion maximization and minimization problems where interventions happen at the level of susceptibility. We show that modeling interventions at the level of susceptibility lead to an interesting family of new questions in network opinion dynamics. We find that the questions are quite different depending on whether there is an overall budget constraining the number of agents we can target or not. We give a polynomial-time algorithm for finding the optimal target-set to optimize the sum of opinions when there are no budget constraints on the size of the target-set. We show that this problem is NP-hard when there is a budget, and that the objective function is neither submodular nor supermodular. Finally, we propose a heuristic for the budgeted opinion optimization and show its efficacy at finding target-sets that optimize the sum of opinions compared on real world networks, including a Twitter network with real opinion estimates
Coverage, Matching, and Beyond: New Results on Budgeted Mechanism Design
We study a type of reverse (procurement) auction problems in the presence of
budget constraints. The general algorithmic problem is to purchase a set of
resources, which come at a cost, so as not to exceed a given budget and at the
same time maximize a given valuation function. This framework captures the
budgeted version of several well known optimization problems, and when the
resources are owned by strategic agents the goal is to design truthful and
budget feasible mechanisms, i.e. elicit the true cost of the resources and
ensure the payments of the mechanism do not exceed the budget. Budget
feasibility introduces more challenges in mechanism design, and we study
instantiations of this problem for certain classes of submodular and XOS
valuation functions. We first obtain mechanisms with an improved approximation
ratio for weighted coverage valuations, a special class of submodular functions
that has already attracted attention in previous works. We then provide a
general scheme for designing randomized and deterministic polynomial time
mechanisms for a class of XOS problems. This class contains problems whose
feasible set forms an independence system (a more general structure than
matroids), and some representative problems include, among others, finding
maximum weighted matchings, maximum weighted matroid members, and maximum
weighted 3D-matchings. For most of these problems, only randomized mechanisms
with very high approximation ratios were known prior to our results
Curriculum learning for multilevel budgeted combinatorial problems
Learning heuristics for combinatorial optimization problems through graph
neural networks have recently shown promising results on some classic NP-hard
problems. These are single-level optimization problems with only one player.
Multilevel combinatorial optimization problems are their generalization,
encompassing situations with multiple players taking decisions sequentially. By
framing them in a multi-agent reinforcement learning setting, we devise a
value-based method to learn to solve multilevel budgeted combinatorial problems
involving two players in a zero-sum game over a graph. Our framework is based
on a simple curriculum: if an agent knows how to estimate the value of
instances with budgets up to , then solving instances with budget can
be done in polynomial time regardless of the direction of the optimization by
checking the value of every possible afterstate. Thus, in a bottom-up approach,
we generate datasets of heuristically solved instances with increasingly larger
budgets to train our agent. We report results close to optimality on graphs up
to nodes and a speedup on average compared to the quickest
exact solver known for the Multilevel Critical Node problem, a max-min-max
trilevel problem that has been shown to be at least -hard
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