By Warren D. Smith February 2010, updated July 2011 & Nov. 2015.
In 1995, B.L.Monroe invented a new multiwinner voting system he called "full proportional representation." Unfortunately, this system was formulated as an "integer program" (and Monroe left it at that) and general integer programming problems are "NP-complete."
Solving "low order polynomial-time" problems is generally considered to be practical. High order polynomial-time problems are worse. NP-complete problems are still worse and are generally believed to be insoluble by any polynomial-time algorithm (Garey & Johnson 1978).
To my knowledge no non-silly natural NP-complete problem has ever been solved at the size scales relevant to large elections, which might at the high end mean 10^{9} voters and 100 candidates. So unless a miracle occurred, we would expect Monroe's integer programs to be beyond the capabilities of forseeable computers and his voting system to be of little to no practical interest.
Indeed, Potthoff & Brams 1998 re-examined Monroe and formulated it as an integer program with VC+C integer variables in a V-voter, C-candidate election, which for any V and C with VC+C>1000 was just absolutely absurd for any practical purpose.
My purpose here is:
These realizations may be important for political scientists and democracies, and also might be important to computer scientists as a real-world source of important large degree-constrained subgraph problems.
Let there be V voters and C candidates, from whom we are to elect W winners, 0<W<C. Each voter, as her "vote," provides a real number "score" to each candidate, e.g. let S_{ij} be the score awarded the jth candidate by the ith voter.
There could be various rules about what scores are permitted, for example each score might be required to lie in the real interval [0, 1000]. One could also consider imposing additional restrictions on "allowed scores," but we shall not.
We then can regard there as being a complete bipartite graph G with V red vertices and C blue vertices (and hence VC edges) with the edge from voter i to candidate j having real "weight" S_{ij}. We shall assume for simplicity that V is an exact multiple of W.
If not, we may add a number <W of additional artificial "voters" each of whom is completely apathetic about who wins the election, to force the new augmented V to be divisible by W. It also is possible to stay with V that is not an exact multiple of W, with some slight additional algorithmic pain.
We then find the subgraph H of G which
The W election-winners are then the candidates with nonzero valency in H. Each voter is said to be "represented" by the unique candidate she is joined to by an H-edge.
How difficult is the task of finding such a subgraph H?
THEOREM (NP-completeness): Even in the "approval^{3}" case where each voter scores each candidates exactly 1 or exactly 0 and each candidate is approved by exactly three voters, the question "does there exist a set of V/3 winners with each voter approving at least one winner?" is NP-complete.
PROOF: Consult Garey & Johnson page 221 problem SP2 "exact cover by 3-sets." Q.E.D.
REMARK: Garey & Johnson's problems SP3 (set packing) and SP5 (minimum set cover) also are useful for proving results of this ilk, see also Dyer & Frieze 1986 for another useful NP-complete problem. Also Skowron et al 2015 go further by showing and/or citing various "inapproximability" results for "minimizing misrepresentation."
THEOREM (Polynomial time if C bounded or winners pre-specified): Given the V×C matrix S as input, there is an algorithm which runs in binomial(C,W)·O((V+W)VWlog(V+W)) steps to find a Monroe-optimal W-element winner-subset of the C candidates. That is because there is an algorithm to determine the Monroe quality-measure of any specified W-winner set in O((V+W)VWlog(V+W)) steps, and there are binomial(C,W) possible W-element subsets of C candidates.
PROOF: If the W winners are pre-specified, then restricting attention to the V voters and W winners we have a complete bipartite graph with V and W red and blue vertices and a V×W submatrix of S is all that matters. Then the subgraph H has valency V/W at each blue vertex and valency 1 at each red vertex. Among all such subgraphs we are to find the one with maximum possible sum of edge-weights. This is a "degree-constrained subgraph" problem known to be soluble in low-order polynomial time, e.g. see Szabo 2009, Lovasz & Plummer 1986, etc. Known techniques solve this problem in the time bound stated. Indeed this time bound probably is not best possible, and the techniques of Gabow & Tarjan 1991 may be better if the edge costs are integers none of which are absurdly large. [Also there are efficient techniques to generate all the binomial(C,W) subsets of binomial(C,W) steps, see Nijenhuis & Wilf.] Q.E.D.
Given a graph G the DCS existence problem is to find a subgraph H satisfying certain "constraints" on the allowed valencies of H at each G-vertex v (or prove no such H exists). If the valence of v in H is demanded to be in some specified set of integers (a possibly-different set being specified for each G-vertex v) then this existence problem is well known to be NP-complete.
But if each of the sets happens to be a set of consecutive integers then this problem is known to be soluble in low-order polynomial time. Of particular importance is the special case where every H-valence is demanded to be exactly equal to 1; then this is called the "matching problem." Matching problems are a lot easier to deal with if G is a bipartite, rather than a general graph.
These problems are also formulable as optimization rather than mere yes/no existence questions. The optimization problem is to find the H with maximum possible sum of edge weights (or minimum possible). These problems also are known to be soluble in low-order polynomial time. In particular, there is a known algorithm by Dinic to solve the maximum-cardinality bipartite matching problem in a V-vertex E-edge graph, in O(V^{1/2}E) operations. (The "cardinality" version involves edge weights which are 0 or 1 only.) The fastest algorithms for the other problem-variants all ultimately are based on Dinic's algorithm.
There also are slower and simpler algorithms. A quite simple algorithm similar to the "Hungarian method" for solving the "assignment problem" will work to solve the Monrovian optimized weighted-DCS problem instead in O(V^{3}) steps. Versions of this which incorporate "sparse graph" and "fibonacci heap priority queue" data structures are known which run in O(V^{2}logV + EV) steps.
If we are given the solution to a maximum-weight matching or DCS problem and all we are asked to do is to verify its optimality, even faster speeds are possible; essentially, we need only to run the final step of the augmenting paths algorithm to verify there are none. This can be done in O(V^{2}) steps or indeed O(E+VlogV) steps.
In the single-winner special case W=1, Monroe's voting method becomes range voting.
Dinic's algorithm solves a simpler problem (unweighted bipartite matching). Its runtime bound is O(V^{1/2}E) operations. The Gabow-Tarjan "scaling" approach can be used to achieve this time bound (times slow-growing log, inverse-Ackermann, and log-cost factors) for all the more general problem variants provided the integer edge-costs are not absurdly large. This is tremendously faster than all known algorithms for general integer programming problems (they take time exponential in the input size).
Assume for the purposes of argument that it is feasible to compute the W winners, nobody cares how complicated and slow the software and hardware are, and everybody trusts the computation. It then also is interesting to enquire: How good a voting method is this?
Monroe was trying to make his voting method be "optimal." However, there are two reasons to attack that:
I believe the correct thing to optimize is "Bayesian Regret" and have a multiwinner generalization. Although BR does appear to be somewhat related to Monroe's measure, it clearly is different.
Generalization of Monroe to involve parameter K: Let K be an integer with 1≤K≤W. Associate with each voter-candidate edge, that voter's score for that candidate. Assume each voter has valence K, that is, is joined by graph edges to exactly K winners. Further, assume each candidate is joined to exactly VK/W voters. Choose, among all possible such subgraphs, the one with the greatest sum-of-edge-scores.
This idea, or something very similar, also was discussed by Potthoff & Brams 1998. Monroe's system is the special case K=1 of this. The case K=W is "naive multiwinner range voting" (i.e, the top W finishers in ordinary range voting become the W winners) which I do not recommend for most purposes because it can be highly disproportional: E.g. with a 51% Democratic electorate, you could get 100% Democratic winners.
Andrew Jennings' greedy method: Let T=V/W. The candidate whose Tth highest score is the greatest, wins. We now eliminate him and the T voters who scored him highest from the election, and continue on to elect the next winner (until W have been elected).
Jennings notes that this can be regarded as a PR voting method. Each candidate is matched with the voters he "represents." (This matching, however, becomes less justifiable for later winners.) Jennings's same argument would also argue that Monroe's is a PR voting method too. One could try to argue the generalized-Monroe method with K=2 is PR also, but as we increase K any such attempts become shakier and shakier, until when K=W we clearly do not have PR anymore.
Skowron et al's approximation methods: Skowron et al 2015 invented and tested methods of approximately optimizing Monroe's quality measure, finding for problem sizes equivalent to Poland's parliament that it was feasible to obtain 99% or 96% of optimality. I am somewhat dubious that any complicated algorithm containing many arbitrary choices, which does not find the optimum, but only finds a parliament "guaranteed to be within 70% of optimum, and in practice often 96% or 99%" could be politically acceptable. Another problem is Skowron et al regard the ballots as rank-orderings, not ratings ballots, and use a scoring notion like Borda voting. That was just stupid because it instantly makes their voting system massively vulnerable to cloning.
Toby Pereira in 2016 pointed out that in this 2-winner, 2N-voter election:
N voters: A, B, C each scored 1, all other candidates scored 0 N voters: A, B, D each scored 1, all other candidates scored 0
Monroe would regard any of the following 2-winner sets: AB, AC, AD, BC, BD, CD (i.e. every possible 2-candidate subset) as equally good. However, in Pereira's view, AB clearly should have been regarded as the unique best. Pereira also points out that Monroe would elect CD in the following 2-winner, 2(x+1)N-voter election:
#voters approved candidates xN A, B, C xN A, B, D N C N D
regardless of the value of x>0. Pereira regards that as clearly silly in the limit x→∞. In view of these examples, Monroe's method seems clearly not to be the best possible multiwinner voting method, even if we did not need to worry about computational effort because of assuming the existence of arbitrarily fast computers.
I thank Andrew Jennings for bringing Monroe 1995 to my attention and thus stimulating me to produce this report. There is nothing really original here in the sense that our result arises instantly by combining Monroe's idea with work on the "degree-constrained subgraph problem" done by a large community of graph-theorists and algorithmicists during 1960-2010. My sole contribution was bridging the gap between these people and the political scientists (represented by Monroe) – neither set had been aware of the other.
Ravindra K. Ahuja, Thomas L. Magnanti, James B. Orlin: Network flows: theory, algorithms, and applications, Prentice Hall 1993.
Vasek Chvatal: Edmonds polyhedra and a hierarchy of combinatorial problems, Discrete Mathematics 4 (1973) 305-337.
Vasek Chvatal: Cutting planes in combinatorics, Europ. J. Combinatorics 6 (1985) 217-226.
V.Chvatal, W.Cook, M.Hartmann: On cutting-plane proofs in combinatorial optimization, Linear Algebra and its Applications 114/115 (1989) 455-499.
Gerard Cornuejols: General factors of graphs, J. Combinatorial Theory B 45,2 (Oct. 1988) 185-198.
Gerard Cornuejols & David Hartvigsen: An extension of matching theory, J. Combinatorial Theory B 40,3 (June 1986) 285-296.
Yefim Dinitz: Dinitz's Algorithm: The Original Version and Even's Version, pp. 218-240 in Oded Goldreich, Arnold L. Rosenberg, and Alan L. Selman. Theoretical Computer Science: Essays in Memory of Shimon Even. Springer 2006.
M.E.Dyer & A.M.Frieze: Planar 3-dimensional matching is NP-complete, J.Algorithms 7 (1986) 176-184.
Jack Edmonds: Paths, trees, and flowers, Canad. J. Maths. 17 (1965) 449-467.
Jack Edmonds & Ellis L. Johnson: Matchings, A Well-Solved Class of Integer Linear Programs, pp.89-92 in Combinatorial Structures and Their Applications (Gordon & Breach NY 1970); also in Springer LNCS #2570 pp.27-30. Results stated with no proofs.
Harold N. Gabow & Robert E. Tarjan: Faster scaling algorithms for general graph-matching problems, J. Assoc. Comput. Machinery 38,4 (1991) 815-853.
M.R. Garey & D.S. Johnson: Computers and Intractability: A Guide to the Theory of NP-completeness, W.H. Freeman 1979.
Andrew V. Goldberg & Satish Rao: Beyond the flow decomposition barrier, J. Assoc. Computing Machinery 45,5 (1998) 783-797.
L. Lovasz & M.D. Plummer: Matching theory, Elsevier Science Pub. Co. 1986.
L.Lovasz: The factorization of graphs II, Acta Math. Acad. Sci. Hungarica 23 (1972) 223-246.
Burt L. Monroe: Fully Proportional Representation, The American Political Science Review 89,4 (Dec. 1995) 925-940.
Albert Nijenhuis & Herbert S. Wilf: Combinatorial Algorithms, Academic Press 1978.
Richard F. Potthoff & Steven J. Brams: Proportional Representation: Broadening the Options, Journal of Theoretical Politics 10,2 (April 1998) 147-178.
Yossi Shiloach: Another look at the degree constrained subgraph problem, Information Processing Letters 12,2 (1981) 89-92.
Piotr Skowron, Piotr Faliszewski, Arkzadii Slinko: Achieving fully proportional representation: Approximability results, Artificial Intelligence 222 (May 2015) 67-103.
Arkadii Slinko & Shaun White: Proportional Representation and Strategic Voters, Journal of Theoretical Politics 22,3 (July 2010) 301-332.
Jacint Szabo: Good characterizations for some degree constrained subgraphs, J. Combinatorial Theory B 99,2 (2009) 436-446.
W.T.Tutte: A short proof of the factor theorem for finite graphs, Canadian J. Maths. 6 (1954) 347-352; The factors of graphs, Canadian J. Maths. 4 (1952) 314-328; The factorization of linear graphs, J.London Maths Soc. 22,2 (April 1947) 107-111.