scholarly journals The Computational Complexity of Understanding Binary Classifier Decisions

2021 ◽  
Vol 70 ◽  
Author(s):  
Stephan Waeldchen ◽  
Jan Macdonald ◽  
Sascha Hauch ◽  
Gitta Kutyniok
Keyword(s):  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Boolean Circuits ◽  
Binary Classifier ◽  
Minimal Set ◽  
Limited Size ◽  
Approximation Factor ◽  
Special Cases ◽  
Relevant Variables ◽  
Binary Classifiers

For a d-ary Boolean function Φ: {0, 1}d → {0, 1} and an assignment to its variables x = (x1, x2, . . . , xd) we consider the problem of finding those subsets of the variables that are sufficient to determine the function value with a given probability δ. This is motivated by the task of interpreting predictions of binary classifiers described as Boolean circuits, which can be seen as special cases of neural networks. We show that the problem of deciding whether such subsets of relevant variables of limited size k ≤ d exist is complete for the complexity class NPPP and thus, generally, unfeasible to solve. We then introduce a variant, in which it suffices to check whether a subset determines the function value with probability at least δ or at most δ − γ for 0 < γ < δ. This promise of a probability gap reduces the complexity to the class NPBPP. Finally, we show that finding the minimal set of relevant variables cannot be reasonably approximated, i.e. with an approximation factor d1−α for α > 0, by a polynomial time algorithm unless P = NP. This holds even with the promise of a probability gap.

scholarly journals Makespan minimization with OR-precedence constraints

2021 ◽  
Author(s):  
Felix Happach
Keyword(s):  
Scheduling Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Precedence Constraints ◽  
Np Hard ◽  
Approximation Factor ◽  
Processing Times ◽  
Special Cases ◽  
Approximation Guarantee ◽  
Unit Processing

AbstractWe consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In this paper, we consider a different type of precedence relation that has not been discussed as extensively and is called OR-precedence. In order for a job to start, we require that at least one of its predecessors is completed—in contrast to all its predecessors. Additionally, we assume that each job has a release date before which it must not start. We prove that a simple List Scheduling algorithm due to Graham (Bell Syst Tech J 45(9):1563–1581, 1966) has an approximation guarantee of 2 and show that obtaining an approximation factor of $$4/3 - \varepsilon $$ 4 / 3 - ε is NP-hard. Further, we present a polynomial-time algorithm that solves the problem to optimality if preemptions are allowed. The latter result is in contrast to classical precedence constraints where the preemptive variant is already NP-hard. Our algorithm generalizes previous results for unit processing time jobs subject to OR-precedence constraints, but without release dates. The running time of our algorithm is $$O(n^2)$$ O ( n 2 ) for arbitrary processing times and it can be reduced to O(n) for unit processing times, where n is the number of jobs. The performance guarantees presented here match the best-known ones for special cases where classical precedence constraints and OR-precedence constraints coincide.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

SCHEDULING TWO-MACHINE FLOW SHOPS WITH EXACT DELAYS

2007 ◽  
Vol 18 (02) ◽  
pp. 341-359 ◽  
Author(s):  
JOSEPH Y.-T. LEUNG ◽  
HAIBING LI ◽  
HAIRONG ZHAO
Keyword(s):  
Polynomial Time ◽  
Completion Time ◽  
Flow Shop ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Completion Time ◽  
Time Interval ◽  
Start Time ◽  
Special Cases ◽  
Efficient Approximation

We consider two-machine flow shop problems with exact delays. In this model, there are two machines, the upstream machine and the downstream machine. Each job j has two operations: the first operation has to be processed on the upstream machine and the second operation has to be processed on the downstream machine, subject to the constraint that the time interval between the completion time of the first operation and the start time of the second operation is exactly [Formula: see text]. We concentrate on the objectives of makespan and total completion time. For the makespan objective, we first show that the problem is strongly NP-hard even if there are only two possible delay values. We then show that some special cases of the problem are solvable in polynomial time. Finally, we design efficient approximation algorithms for the general case and some special cases. For the total completion time objective, we give optimal polynomial-time algorithm for a special case and an efficient approximation algorithm for another one.

scholarly journals Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses

2021 ◽  
Author(s):  
Yiannis Giannakopoulos ◽  
Georgy Noarov ◽  
Andreas S. Schulz
Keyword(s):  
Nash Equilibria ◽  
Price Of Anarchy ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Deterministic Algorithm ◽  
Congestion Games ◽  
Lambert W Function ◽  
Efficient Computation ◽  
Approximation Factor ◽  
Best Responses

We present a deterministic polynomial-time algorithm for computing [Formula: see text]-approximate (pure) Nash equilibria in (proportional sharing) weighted congestion games with polynomial cost functions of degree at most [Formula: see text]. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of the algorithm is that it only uses best-improvement steps in the actual game, as opposed to the previously best algorithms, that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis at al. [Caragiannis I, Fanelli A, Gravin N, Skopalik A (2011) Efficient computation of approximate pure Nash equilibria in congestion games. Ostrovsky R, ed. Proc. 52nd Annual Symp. Foundations Comput. Sci. (FOCS) (IEEE Computer Society, Los Alamitos, CA), 532–541; Caragiannis I, Fanelli A, Gravin N, Skopalik A (2015) Approximate pure Nash equilibria in weighted congestion games: Existence, efficient computation, and structure. ACM Trans. Econom. Comput. 3(1):2:1–2:32.], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of [Formula: see text] for the price of anarchy (PoA) of [Formula: see text]-approximate equilibria in weighted congestion games, where [Formula: see text] is the Lambert-W function. More specifically, we show that this PoA is exactly equal to [Formula: see text], where [Formula: see text] is the unique positive solution of the equation [Formula: see text]. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, whereas our lower bound is simple enough to apply even to singleton congestion games.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

A Polynomial-Time Algorithm for the Knapsack Problem with Two Variables

1976 ◽  
Vol 23 (1) ◽  
pp. 147-154 ◽  
Author(s):  
D. S. Hirschberg ◽  
C. K. Wong
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm
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