Subelection Clones in Ordinal Elections

• Subelection clones are a relaxation of perfect clones in ordinal elections, where a candidate subset is ranked consecutively by a fixed group of voters.
• They enable efficient detection by scanning contiguous blocks within voter rankings, and the associated decision problems are solvable in polynomial time.
• Analyzing subelection clones reveals a practical contrast between easily detectable single blocks and the NP-complete challenge of partitioning the entire candidate set.

Subelection clones are relaxations of perfect clones in ordinal elections. In the classical formulation, a clone set is a subset of candidates that every voter ranks as a contiguous block; in the subelection setting, the same contiguity requirement is imposed only on a selected subelection or on a fixed subgroup of at least $w$ voters. This notion appears in work on hidden consistent subelections and on imperfect clones, where it serves both as a descriptive model of partial agreement and as an algorithmic object whose complexity depends sharply on whether one seeks a single clone block or a partition of the entire candidate set (Elkind et al., 2011, Janeczko et al., 2024, Faliszewski et al., 14 Sep 2025).

1. Formal definitions

Let $E=(C,V)$ be an election with candidate set $C$ and voter multiset $V$, where each voter $v\in V$ casts a strict linear order $\succ_v$ over $C$. The background notion is the perfect clone set: for a profile $R=(\succ_1,\dots,\succ_n)$, a nonempty $X\subseteq C$ is a clone set if, for every voter $i$, every two members $E=(C,V)$0, and every outside candidate $E=(C,V)$1,

$E=(C,V)$2

Equivalently, no voter ever splits $E=(C,V)$3 by inserting an outsider between two members of $E=(C,V)$4, so the members of $E=(C,V)$5 form a consecutive block in every ranking (Elkind et al., 2011).

A clone subelection weakens the universal quantifier over voters. In the formulation of "Discovering Consistent Subelections," a subelection is a pair $E=(C,V)$6 with $E=(C,V)$7 and $E=(C,V)$8, and $E=(C,V)$9 is a clone subelection if for every voter $C$0, the set $C$1 appears as a contiguous block in $C$2’s full ranking. Equivalently, for each $C$3 and each triple $C$4, $C$5,

$C$6

When $C$7 and $C$8, the subelection has size $C$9 (Janeczko et al., 2024).

A more parameterized version is the $V$0-subelection clone. A nonempty set $V$1 is a $V$2-subelection clone if $V$3 and there exists a subset $V$4 of at least $V$5 voters such that for every $V$6, the members of $V$7 are ranked consecutively in $V$8. In this terminology, $V$9 is recognized as a clone by the subelection $v\in V$0 (Faliszewski et al., 14 Sep 2025).

2. Relation to perfect, independent, and approximate clones

The imperfect-clone literature distinguishes three notions. Perfect clones require contiguity in all votes. Independent or subelection clones are sets of candidates that only some of the voters recognize as a perfect clone. Approximate clones require that all voters rank the candidates close together, but not necessarily consecutively (Faliszewski et al., 14 Sep 2025).

For approximate clones, the formal condition is different. A set $v\in V$1 of size at most $v\in V$2 is a $v\in V$3-approximate clone if for every voter $v\in V$4,

$v\in V$5

Equivalently, in each vote $v\in V$6, at most $v\in V$7 outsiders appear interspersed between members of $v\in V$8. The contrast is explicit: in a subelection clone each member-ranking is perfectly consecutive, but only for the same fixed subcollection of $v\in V$9 voters, whereas in approximate clones all voters almost rank $\succ_v$0 consecutively (Faliszewski et al., 14 Sep 2025).

A representative example separates the notions. For $\succ_v$1 and three voters

$\succ_v$2

the set $\succ_v$3 is a perfect clone of size $\succ_v$4; $\succ_v$5 is a $\succ_v$6-subelection clone but not a perfect clone; and $\succ_v$7 is a $\succ_v$8-approximate clone with $\succ_v$9 but fails to be a subelection clone for $C$0 (Faliszewski et al., 14 Sep 2025).

This distinction corrects a common simplification. Subelection clones are not merely “noisy” perfect clones: they preserve exact contiguity, but only on a fixed witness set of voters. Approximate clones relax contiguity itself.

3. Detection and optimization for a single clone block

For hidden subelections, the central optimization problem is

$C$1

The associated decision problem, $C$2, asks whether there exist $C$3 and $C$4 with $C$5 and $C$6 such that $C$7 is a clone subelection (Janeczko et al., 2024).

Both problems are solvable in polynomial time. The key observation is that any clone set $C$8 of size $C$9 must appear as a contiguous segment of length $R=(\succ_1,\dots,\succ_n)$0 in each of the voters who witness it. Hence one can enumerate all segments of length $R=(\succ_1,\dots,\succ_n)$1 in all votes. For each voter $R=(\succ_1,\dots,\succ_n)$2 and each segment of length $R=(\succ_1,\dots,\succ_n)$3 in $R=(\succ_1,\dots,\succ_n)$4’s ranking, extract that contiguous block $R=(\succ_1,\dots,\succ_n)$5; there are at most $R=(\succ_1,\dots,\succ_n)$6 such blocks. For each candidate-set $R=(\succ_1,\dots,\succ_n)$7, count how many voters rank the members of $R=(\succ_1,\dots,\succ_n)$8 contiguously, and accept if any block qualifies for at least $R=(\succ_1,\dots,\succ_n)$9 voters (Janeczko et al., 2024).

The runtime stated for the decision version is

$X\subseteq C$0

For $X\subseteq C$1, one replaces the decision threshold $X\subseteq C$2 by taking the maximum frequency over all blocks $X\subseteq C$3. No approximation or heuristics are needed: the exact solution is found in polynomial time by direct enumeration. The underlying reason is that contiguity is a strong, local condition detectable by scanning each vote (Janeczko et al., 2024).

4. Partition problems and parameterized complexity

The complexity landscape changes when the objective is not to find one clone block but to partition the entire candidate set into blocks recognized by the same subelection. The relevant problem is $X\subseteq C$4-Subelection-Clone Partition: given $X\subseteq C$5, integers $X\subseteq C$6, $X\subseteq C$7, and $X\subseteq C$8, determine whether there exists a subelection $X\subseteq C$9 with $i$0 and a partition $i$1 of $i$2 into subsets each of size at most $i$3, such that each $i$4 is a perfect clone for the same set of voters $i$5 (Faliszewski et al., 14 Sep 2025).

Problem	Formulation	Complexity status
$i$6	Find $i$7 with $i$8, $i$9 and $E=(C,V)$00 a clone subelection	Polynomial time
$E=(C,V)$01	Maximize the number of voters witnessing a clone block of size $E=(C,V)$02	Polynomial time
$E=(C,V)$03-Subelection-Clone Identification	Given $E=(C,V)$04, ask whether $E=(C,V)$05 is recognized by at least $E=(C,V)$06 voters and $E=(C,V)$07	In $E=(C,V)$08
$E=(C,V)$09-Subelection-Clone Partition	Partition $E=(C,V)$10 into $E=(C,V)$11 parts of size at most $E=(C,V)$12, all perfect clones for the same $E=(C,V)$13 with $E=(C,V)$14	NP-complete already for $E=(C,V)$15; FPT in $E=(C,V)$16 and $E=(C,V)$17; XP in $E=(C,V)$18; W[1]-hard for $E=(C,V)$19

The identification problem is described as obviously in $E=(C,V)$20: one merely scans all subsets $E=(C,V)$21 of size $E=(C,V)$22, or better, checks each vote and each interval of length $E=(C,V)$23, to see whether they rank $E=(C,V)$24 consecutively. By contrast, Subelection-Clone Partition is NP-complete already for $E=(C,V)$25. The proof sketch uses a reduction from Restricted Exact Cover by 3-Sets, where candidates represent set-indices and dummy plusses/minuses, voters enforce either grouping $E=(C,V)$26 or $E=(C,V)$27, and picking $E=(C,V)$28 voters corresponds to selecting an exact cover (Faliszewski et al., 14 Sep 2025).

Parameterized complexity gives a more granular picture. There is an FPT algorithm in parameter $E=(C,V)$29: guess the $E=(C,V)$30 voters, then solve a perfect-clone partition in polynomial time. There is also an FPT algorithm in parameter $E=(C,V)$31, since $E=(C,V)$32 is FPT in $E=(C,V)$33, and an XP algorithm in parameter $E=(C,V)$34: guess on one vote a length-$E=(C,V)$35 perfect-clone partition, then verify it extends to $E=(C,V)$36 voters. On the negative side, Subelection-Clone Partition is W[1]-hard for parameter $E=(C,V)$37 (Faliszewski et al., 14 Sep 2025).

These results correct another common simplification. Clone detection is easy; clone partition is not. The easy case concerns one hidden block. The hard case concerns a coordinated decomposition of the whole candidate set witnessed by the same subelection.

5. Empirical findings and applications

Empirical work on consistent subelections studies both synthetic and real-world profiles. One benchmark dataset consists of 344 elections of size $E=(C,V)$38 candidates and $E=(C,V)$39 voters, sampled from statistical cultures including Impartial Culture, urn models, Mallows, single-peaked, single-crossing, and spatial models. For each election $E=(C,V)$40 and each fixed $E=(C,V)$41, the experiments compute $E=(C,V)$42 and plot the results on a two-dimensional map of elections based on a Fruchterman–Reingold embedding of the isomorphic-swap-distance graph (Janeczko et al., 2024).

The reported pattern is that Impartial-Culture elections exhibit relatively low $E=(C,V)$43 on average out of $E=(C,V)$44 voters, whereas elections with latent structure, such as correlated Mallows with small dispersion and single-peaked elections, show substantially larger clone subelections: for correlated Mallows, one can find $E=(C,V)$45-candidate clone blocks witnessed by $E=(C,V)$46 voters. At the same time, no strong monotonic trend appears in the map for clones, since the swap-distance to perfect-clone elections is not tightly linked to large-scale contiguity patterns (Janeczko et al., 2024).

Two real-world examples are especially concrete. In the sushi dataset with $E=(C,V)$47 voters and $E=(C,V)$48 sushi types, $E=(C,V)$49 of the electorate agree that Tamago and Kappa-maki form a clone block, and $E=(C,V)$50 for the triple $E=(C,V)$51. In French presidential polling from 2014, with about $E=(C,V)$52 respondents and $E=(C,V)$53 candidates, $E=(C,V)$54 for $E=(C,V)$55, and $E=(C,V)$56 for $E=(C,V)$57. The clone-detection algorithm runs in seconds even on thousands of voters and ten candidates (Janeczko et al., 2024).

The applications discussed in that work are market segmentation, political profiling, and elicitation acceleration. Clone blocks may reveal categories of options that a subpopulation cannot distinguish, such as vegetarian sushi items or a narrow ideological band of candidates. Because detection is polynomial-time and efficient, it can be embedded in interactive systems to discover indistinguishable alternatives for subgroups of users (Janeczko et al., 2024).

6. Place in the broader theory of clones

Subelection clones inherit their vocabulary from the broader theory of clones in voting. In that theory, the family of all clone sets of a profile,

$E=(C,V)$58

is a clone structure, and clone structures admit an axiomatic characterization and a hierarchical PQ-tree representation. Internal P-nodes allow arbitrary permuting of children, Q-nodes allow only reversal, and clone sets correspond exactly to consecutive sub-frontiers of the tree (Elkind et al., 2011).

That perfect-clone framework is used in several adjacent lines of work. Clone structures themselves are analyzed axiomatically and algorithmically (Elkind et al., 2011). Cloning can be studied as a manipulation of elections, where a candidate is replaced by several clones ranked as a consecutive block by each voter (Elkind et al., 2014). Clone independence becomes an axiom for voting rules: removing a clone should leave the status of non-clones unchanged, and Split Cycle is presented as a Condorcet-consistent method satisfying independence of clones (Holliday et al., 2020). More recently, composition consistency has been shown to be strictly stronger than independence of clones, and its enforcement relies on the hierarchical representation of clone structures via PQ-trees (Berker et al., 24 Feb 2025).

Within this larger landscape, subelection clones are best understood as a relaxation of the perfect-clone condition. They preserve exact contiguity, but only on a designated subelection. This makes them suitable for detecting latent agreement inside electorates that are globally heterogeneous, while keeping a direct formal connection to the classical clone notion from social choice theory (Elkind et al., 2011, Janeczko et al., 2024, Faliszewski et al., 14 Sep 2025).