Complexity of Manipulating and Controlling Approval-Based Multiwinner Voting

**Read paper on the following link:** https://ifaamas.org/Proceedings/aamas2022/pdfs/p263.pdf **Abstract:** We simulate voting rules for multiwinner elections in a model of judgment aggregation that distinguishes between rationality and feasibility constraints. These constraints restrict the structure of the individual judgments and of the collective outcome computed by the rule, respectively. We extend known results regarding the simulation of single-winner voting rules to the multiwinner setting, both for elections with ordinal preferences and for elections with approval-based preferences. This not only provides us with a new tool to analyse multiwinner elections, but it also suggests the definition of new judgment aggregation rules, by generalising some of the principles at the core of well-known multiwinner voting rules to this richer setting. We explore this opportunity with regards to the principle of proportionality. Finally, in view of the computational difficulty associated with many judgment aggregation rules, we investigate the computational complexity of our embeddings and of the new judgment aggregation rules we put forward.

Robustness among multiwinner voting rules

Suppose you want to design a voting rule that can be used to elect a committee or parliament by asking each voter to approve of a subset of the candidates standing. There are several properties you may want that rule to satisfy. First, voters should enjoy some form of proportional representation. Second, voters should not have an incentive to misrepresent their preferences. Third, outcomes should be Pareto efficient. We show that it is impossible to design a voting rule that satisfies all three properties. We also explore what possibilities there are when we weaken our requirements. Of special interest is the methodology we use, as a significant part of the proof is outsourced to a SAT solver. While prior work has considered similar questions for the special case of resolute voting rules, which do not allow for ties between outcomes, we focus on the fact that, in practice, most voting rules allow for the possibility of such ties.

Method of Moment Restrictions in Robust Control and Filtering

The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines.While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far—only its generalization to three or more dimensions was known to be W[1]-hard. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k, which in particular means that there is no hope for fixed-parameter tractability with respect to the parameter k. Our reductions show also the W[1]-completeness of the more general problem Set Cover on instances that “almost have the consecutive-ones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row.For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]-hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k ·n O(1) time, we show that Rectangle Stabbing is fixed-parameter tractable in the still NP-hard case where both these restrictions apply.

This paper considers the “Frobenius problem”: Givenn natural numbersa1,a2,...an such that their greatest common divisor is 1, find the largest natural number that is not expressible as a nonnegative integer combination of them. This problem can be seen to be NP-hard. For the casesn=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixedn. This is done by first proving an exact relation between the Frobenius problem and a geometric concept called the “covering radius”. Then a polynomial time algorithm is developed for finding the covering radius of any polytope in a fixed number of dimensions. The last algorithm relies on a structural theorem proved here that describes for any polytopeK, the setK+ℤh={x∶x∈ℝn;x=y+z;y∈K;z∈ℤn} which is the portion of space covered by all lattice translates ofK. The proof of the structural theorem relies on some recent developments in the Geometry of Numbers. In particular, it uses a theorem of Kannan and Lovasz [11], bounding the width of lattice-point-free convex bodies and the techniques of Kannan, Lovasz and Scarf [12] to study the shapes of a polyhedron obtained by translating each facet parallel, to itself. The concepts involved are defined from first principles. In a companion paper [10], I extend the structural result and use that to solve a general problem of which the Frobenius problem is a special case.

Optimal, constant I/O similarity scaling for full-information and state-feedback control problems

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

We introduce natural strategic games on graphs, which capture the idea of coordination in a local setting. We study the existence of equilibria that are resilient to coalitional deviations of unbounded and bounded size (i.e., strong equilibria and k-equilibria respectively). We show that pure Nash equilibria and 2-equilibria exist, and give an example in which no 3-equilibrium exists. Moreover, we prove that strong equilibria exist for various special cases. We also study the price of anarchy (PoA) and price of stability (PoS) for these solution concepts. We show that the PoS for strong equilibria is 1 in almost all of the special cases for which we have proven strong equilibria to exist. The PoA for pure Nash equilbria turns out to be unbounded, even when we fix the graph on which the coordination game is to be played. For the PoA for k-equilibria, we show that the price of anarchy is between (Formula presented.) and (Formula presented.). The latter upper bound is tight for (Formula presented.) (i.e., strong equilibria). Finally, we consider the problems of computing strong equilibria and of determining whether a joint strategy is a k-equilibrium or strong equilibrium. We prove that, given a coordination game, a joint strategy s, and a number k as input, it is co-NP complete to determine whether s is a k-equilibrium. On the positive side, we give polynomial time algorithms to compute strong equilibria for various special cases.

We consider the problem of rescheduling a facility modeled as a single machine in the face of newly arrived jobs with part-type dependent setup times. The facility contains a number of jobs that have been assigned due dates and scheduled so as to meet them. We wish to insert the new jobs into the existing schedule in a manner that will minimize the disruption of the jobs in the system and minimize the total weighted completion time or the maximum completion time of the new jobs. We provide a polynomial-time algorithm for the maximum completion time problem, prove that the total weighted completion time problem is NP-hard in the strong sense and study several of its special cases. In particular, we show that the case with reverse-agreeable weights (of which the unit weight problem is a special case) can be solved in polynomial time when the number of part types is fixed. We also present two heuristics for the problem with arbitrary weights and develop data-dependent worst-case error bounds. Extensive computational experiments show that the heuristics consistently obtain near-optimal solutions in very reasonable CPU times.

This paper formulates a nonlinear time series model which encompasses several standard nonlinear models for time series as special cases. It also offers two methods for estimating missing observations, one using prediction and fixed point smoothing algorithms and the other using optimal estimating equation theory. Recursive estimation of missing observations in an autoregressive conditionally heteroscedastic (ARCH) model and the estimation of missing observations in a linear time series model are shown to be special cases. Construction of optimal estimates of missing observations using estimating equation theory is discussed and applied to some nonlinear models.

Optimal schedule for works running on machines, in a general case, is a hard problem and there is no complete optimal deterministic algorithm in polynomial time. Optimal and approximated solutions were issued for some specific cases with constraints. One can find the solutions for the cases 1 and 2 machines [4,16] as the initial algorithms. In [16], author solved late works problem using algorithm with setup times included. [4] solve the due works problem on two machines, and can be extended for the case of 3 machines with some conditions on works. Other authors looked at schedule problems in specific cases like [5,8,17]. In this paper, we apply duration automata [1,2] to solve the schedule problem dynamically for the works with uncertain processing time in a cluster computer, which is a system consisting of many computers (computing nodes) co-working together. We solve the schedule problem in a cluster with $m$ machines for two cases: all machines are the same and different in configurations (machine's resources are formally considered as a configuration information, represented by the time need to finish the works). Because of uncertainly processing time, tranditional algorithms can not be used effectively. By using DA model with DA's order criterion, we issue schedule algorithms and practically prove to be better in time consuming compare to FIFO (natural order), the fastest first (greedy) and the longest first (safety) methods without synchronized points.

Polynomial time algorithms for two special classes of the proportionate multiprocessor open shop

Generalizing many well-known and natural scheduling problems, scheduling with job-specific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4-approximation algorithm. This is true even for a special case that is equivalent to the covering version of the well-studied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasi-polynomial time (1+varepsilon )-approximation algorithm that yields an (e+varepsilon )-approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with optimal cost at 1+varepsilon speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most log n many classes. This algorithm allows the jobs even to have up to log n many distinct release dates. All proposed quasi-polynomial time algorithms require the input data to be quasi-polynomially bounded.

In this thesis, we study the theoretical approach on energy-efficient scheduling problems arising in demand response management in the modern electrical smart grid. Consumers send in power requests with flexible feasible timeslots during which their requests can be served. The grid controller, upon receiving power requests, schedules each request within the specified interval. The electricity cost is measured by a convex function of the load in each timeslot. The objective is to schedule all requests with the minimum total electricity cost. We study the smart grid scheduling problem in different models. For the offline model, we prove the problem is NP-hard for the general case. We propose a polynomial time algorithm for special input where jobs have unit power request and unit time duration. By adapting the polynomial time algorithm for unit-size jobs, we propose an approximation algorithm for more general input. On the other hand, we also present an exact algorithm to find the optimal schedule for the problem with general input. For the online model, we propose an online algorithm for jobs with jobs with arbitrary power request, arbitrary time duration, and arbitrary contiguous feasible intervals. We also show a lower bound of the competitive ratio for the smart grid scheduling problem with unit height and arbitrary width. For special cases, we design different online algorithms with better competitive ratios. Finally, we look at other optimization problems and show how to solve them by adapting our techniques. We prove that our online algorithm can solve the machine minimization problem with an asymptotically optimal competitive ratio. We also show that our exact algorithm can be adapted to solve other demand response management problems.