scholarly journals Bypassing Combinatorial Protections: Polynomial-Time Algorithms for Single-Peaked Electorates

2015 ◽  
Vol 53 ◽  
pp. 439-496 ◽  
Author(s):  
Felix Brandt ◽  
Markus Brill ◽  
Edith Hemaspaandra ◽  
Lane A. Hemaspaandra
Keyword(s):  
Political Science ◽  
Polynomial Time ◽  
Np Hard ◽  
Polynomial Time Algorithms ◽  
Election Systems ◽  
First Time

For many election systems, bribery (and related) attacks have been shown NP-hard using constructions on combinatorially rich structures such as partitions and covers. This paper shows that for voters who follow the most central political-science model of electorates---single-peaked preferences---those hardness protections vanish. By using single-peaked preferences to simplify combinatorial covering challenges, we for the first time show that NP-hard bribery problems---including those for Kemeny and Llull elections---fall to polynomial time for single-peaked electorates. By using single-peaked preferences to simplify combinatorial partition challenges, we for the first time show that NP-hard partition-of-voters problems fall to polynomial time for single-peaked electorates. We show that for single-peaked electorates, the winner problems for Dodgson and Kemeny elections, though Theta-two-complete in the general case, fall to polynomial time. And we completely classify the complexity of weighted coalition manipulation for scoring protocols in single-peaked electorates.

scholarly journals Equitable Coloring of Graphs. Recent Theoretical Results and New Practical Algorithms

2016 ◽  
Vol 26 (3) ◽  
pp. 281-295 ◽  
Author(s):  
Hanna Furmańczyk ◽  
Andrzej Jastrzębski ◽  
Marek Kubale
Keyword(s):  
Polynomial Time ◽  
Np Hard ◽  
Coloring Problem ◽  
Equitable Coloring ◽  
Practical Algorithms ◽  
Polynomial Time Algorithms ◽  
Sequencing And Scheduling ◽  
General Graphs ◽  
Theoretical Results

AbstractIn many applications in sequencing and scheduling it is desirable to have an underlaying graph as equitably colored as possible. In this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.

scholarly journals On the Computation of Fully Proportional Representation

2013 ◽  
Vol 47 ◽  
pp. 475-519 ◽  
Author(s):  
N. Betzler ◽  
A. Slinko ◽  
J. Uhlmann
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Proportional Representation ◽  
Fixed Parameter Tractability ◽  
Np Hard ◽  
Winner Determination ◽  
Negative Results ◽  
Polynomial Time Algorithms ◽  
Fixed Parameter

We investigate two systems of fully proportional representation suggested by Chamberlin Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

scholarly journals Llull and Copeland Voting Computationally Resist Bribery and Constructive Control

2009 ◽  
Vol 35 ◽  
pp. 275-341 ◽  
Author(s):  
P. Faliszewski ◽  
E. Hemaspaandra ◽  
L. A. Hemaspaandra ◽  
J. Rothe
Keyword(s):  
Polynomial Time ◽  
Network Flow ◽  
Np Hard ◽  
Voter Preferences ◽  
External Agent ◽  
Fixed Parameter ◽  
Election System ◽  
Election Systems ◽  
The Given ◽  
Flexible Models

Control and bribery are settings in which an external agent seeks to influence the outcome of an election. Constructive control of elections refers to attempts by an agent to, via such actions as addition/deletion/partition of candidates or voters, ensure that a given candidate wins. Destructive control refers to attempts by an agent to, via the same actions, preclude a given candidate's victory. An election system in which an agent can sometimes affect the result and it can be determined in polynomial time on which inputs the agent can succeed is said to be vulnerable to the given type of control. An election system in which an agent can sometimes affect the result, yet in which it is NP-hard to recognize the inputs on which the agent can succeed, is said to be resistant to the given type of control. Aside from election systems with an NP-hard winner problem, the only systems previously known to be resistant to all the standard control types were highly artificial election systems created by hybridization. This paper studies a parameterized version of Copeland voting, denoted by Copeland^\alpha, where the parameter \alpha is a rational number between 0 and 1 that specifies how ties are valued in the pairwise comparisons of candidates. In every previously studied constructive or destructive control scenario, we determine which of resistance or vulnerability holds for Copeland^\alpha for each rational \alpha, 0 \leq \alpha \leq 1. In particular, we prove that Copeland^{0.5}, the system commonly referred to as ``Copeland voting,'' provides full resistance to constructive control, and we prove the same for Copeland^\alpha, for all rational \alpha, 0 < \alpha < 1. Among systems with a polynomial-time winner problem, Copeland voting is the first natural election system proven to have full resistance to constructive control. In addition, we prove that both Copeland^0 and Copeland^1 (interestingly, Copeland^1 is an election system developed by the thirteenth-century mystic Llull) are resistant to all standard types of constructive control other than one variant of addition of candidates. Moreover, we show that for each rational \alpha, 0 \leq \alpha \leq 1, Copeland^\alpha voting is fully resistant to bribery attacks, and we establish fixed-parameter tractability of bounded-case control for Copeland^\alpha. We also study Copeland^\alpha elections under more flexible models such as microbribery and extended control, we integrate the potential irrationality of voter preferences into many of our results, and we prove our results in both the unique-winner model and the nonunique-winner model. Our vulnerability results for microbribery are proven via a novel technique involving min-cost network flow.

scholarly journals Hamiltonian Cycle in K1,r-Free Split Graphs — A Dichotomy

2021 ◽  
pp. 1-32
Author(s):  
P. Renjith ◽  
N. Sadagopan
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Optimization Problem ◽  
Hamiltonian Cycle ◽  
Hamiltonian Path ◽  
Np Hard ◽  
Polynomial Time Algorithms ◽  
Split Graphs ◽  
Np Complete ◽  
Cycle Path

For an optimization problem known to be NP-Hard, the dichotomy study investigates the reduction instances to determine the line separating polynomial-time solvable vs NP-Hard instances (easy vs hard instances). In this paper, we investigate the well-studied Hamiltonian cycle problem (HCYCLE), and present an interesting dichotomy result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete on planar bipartite graphs with maximum degree [Formula: see text]. We use this result to show that HCYCLE is NP-complete for [Formula: see text]-free split graphs. Further, we present polynomial-time algorithms for Hamiltonian cycle in [Formula: see text]-free and [Formula: see text]-free split graphs. We believe that the structural results presented in this paper can be used to show similar dichotomy result for Hamiltonian path problem and other variants of Hamiltonian cycle (path) problems.

scholarly journals k-Approximate Quasiperiodicity Under Hamming and Edit Distance

Algorithmica ◽  
2021 ◽  
Author(s):  
Aleksander Kędzierski ◽  
Jakub Radoszewski
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Edit Distance ◽  
Hamming Distance ◽  
Efficient Algorithms ◽  
Np Hard ◽  
Total Cost ◽  
Polynomial Time Algorithms ◽  
Np Hardness

AbstractQuasiperiodicity in strings was introduced almost 30 years ago as an extension of string periodicity. The basic notions of quasiperiodicity are cover and seed. A cover of a text T is a string whose occurrences in T cover all positions of T. A seed of text T is a cover of a superstring of T. In various applications exact quasiperiodicity is still not sufficient due to the presence of errors. We consider approximate notions of quasiperiodicity, for which we allow approximate occurrences in T with a small Hamming, Levenshtein or weighted edit distance. In previous work Sim et al. (J Korea Inf Sci Soc 29(1):16–21, 2002) and Christodoulakis et al. (J Autom Lang Comb 10(5/6), 609–626, 2005) showed that computing approximate covers and seeds, respectively, under weighted edit distance is NP-hard. They, therefore, considered restricted approximate covers and seeds which need to be factors of the original string T and presented polynomial-time algorithms for computing them. Further algorithms, considering approximate occurrences with Hamming distance bounded by k, were given in several contributions by Guth et al. They also studied relaxed approximate quasiperiods. We present more efficient algorithms for computing restricted approximate covers and seeds. In particular, we improve upon the complexities of many of the aforementioned algorithms, also for relaxed quasiperiods. Our solutions are especially efficient if the number (or total cost) of allowed errors is small. We also show conditional lower bounds for computing restricted approximate covers and prove NP-hardness of computing non-restricted approximate covers and seeds under the Hamming distance.

scholarly journals A Heuristic for a Mixed Integer Program using the Characteristic Equation Approach

2017 ◽  
Vol 2 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Philimon Nyamugure ◽  
Elias Munapo ◽  
‘Maseka Lesaoana ◽  
Santosh Kumar
Keyword(s):  
Characteristic Equation ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Integer Program ◽  
Mixed Integer ◽  
Mixed Integer Program ◽  
Np Hard ◽  
Equation Approach ◽  
Mip Model ◽  
Polynomial Time Algorithms

While most linear programming (LP) problems can be solved in polynomial time, pure and mixed integer problems are NP-hard and there are no known polynomial time algorithms to solve these problems. A characteristic equation (CE) was developed to solve a pure integer program (PIP). This paper presents a heuristic that generates a feasible solution along with the bounds for the NP-hard mixed integer program (MIP) model by solving the LP relaxation and the PIP, using the CE.

scholarly journals An algorithmic analysis of Flood-It and Free-Flood-It on graph powers

2014 ◽  
Vol Vol. 16 no. 3 (Analysis of Algorithms) ◽  
Author(s):  
Uéverton dos Santos Souza ◽  
Fábio Protti ◽  
Maise Silva
Keyword(s):  
Polynomial Time ◽  
Analysis Of Algorithms ◽  
Np Hard ◽  
Colored Graph ◽  
Combinatorial Game ◽  
Polynomial Time Algorithms ◽  
Minimum Number ◽  
International Audience ◽  
Algorithmic Analysis

Analysis of Algorithms International audience Flood-it is a combinatorial game played on a colored graph G whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a fixed pivot. Free-Flood-it is a variant where the pivot can be freely chosen for each move of the game. The standard versions of Flood-it and Free-Flood-it are played on m ×n grids. In this paper we analyze the behavior of these games when played on other classes of graphs, such as d-boards, powers of cycles and circular grids. We describe polynomial time algorithms to play Flood-it on C2n (the second power of a cycle on n vertices), 2 ×n circular grids, and some types of d-boards (grids with a monochromatic column). We also show that Free-Flood-it is NP-hard on C2n and 2 ×n circular grids.

GATE RESIZING TO REDUCE POWER CONSUMPTION

2002 ◽  
Vol 13 (03) ◽  
pp. 405-429
Author(s):  
EDWARD Y. C. CHENG ◽  
SARTAJ SAHNI
Keyword(s):  
Power Consumption ◽  
Polynomial Time ◽  
Timing Constraints ◽  
Np Hard ◽  
Reduce Power Consumption ◽  
Polynomial Time Algorithms ◽  
Gate Resizing ◽  
General Circuits

We study the problem of resizing gates so as to reduce overall power consumption while satisfying a circuit's timing constraints. Polynomial time algorithms for series-parallel and tree circuits are obtained. Gate resizing with multigate modules is shown to be NP-hard. Algorithms that improve upon those presented by Chen and Sarrafzadeh4 for general circuits are also developed.

scholarly journals Preferences Single-Peaked on a Tree: Multiwinner Elections and Structural Results

2022 ◽  
Vol 73 ◽  
pp. 231-276
Author(s):  
Dominik Peters ◽  
Lan Yu ◽  
Hau Chan ◽  
Edith Elkind
Keyword(s):  
Polynomial Time ◽  
Scoring Function ◽  
Large Family ◽  
Np Hard ◽  
Scoring Functions ◽  
Winner Determination ◽  
Polynomial Time Algorithms ◽  
Minimum Number ◽  
Number Of Leaves ◽  
Positive Results

A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989b) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin–Courant rule for preferences single-peaked on trees. We show that in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard for the Borda scoring function; indeed, this hardness results extends to a large family of scoring functions. However, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.

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.

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