[Model] ShortestCommonSupersequence

[isPANN]

Motivation

SHORTEST COMMON SUPERSEQUENCE from Garey & Johnson, A4 SR8. A fundamental NP-complete problem in string algorithms. Given a set of strings, find the shortest string that contains each input string as a subsequence (not necessarily contiguous). Different from Shortest Common Superstring (P157), which requires contiguous containment (substring). For two strings, SCS is solvable in polynomial time via the dual relationship with LCS (longest common subsequence). For an arbitrary number of strings, the problem becomes NP-complete even for |Sigma| = 5 (Maier, 1978). Applications include data compression, sequence alignment, and scheduling.

Associated rules:

• R102: MinimumVertexCover -> ShortestCommonSupersequence (as target)

Prerequisite note: The source rule R102 requires MinimumVertexCover which already exists in the codebase.

Definition

Name: ShortestCommonSupersequence
Canonical name: SHORTEST COMMON SUPERSEQUENCE
Reference: Garey & Johnson, Computers and Intractability, A4 SR8

Mathematical definition:

INSTANCE: Finite alphabet Sigma, finite set R of strings from Sigma*, and a positive integer K.
QUESTION: Is there a string w in Sigma* with |w| <= K such that each string x in R is a subsequence of w, i.e., x can be obtained from w by deleting zero or more characters (not necessarily contiguous)?

Variables

• Count: K variables (one per position in the candidate supersequence w), where K is the length bound.
• Per-variable domain: {0, 1, ..., |Sigma|-1} -- index into the alphabet Sigma.
• Meaning: Variable i encodes the symbol at position i of the candidate supersequence w. A satisfying assignment produces a string w of length <= K such that every string in R is a subsequence of w. Alternatively, the problem can be viewed as choosing an interleaving order for the symbols of all input strings.
• dims() specification: vec![alphabet_size; bound] where alphabet_size = |Sigma| and bound = K. Each of the K positions can take any alphabet symbol.

Schema (data type)

Type name: ShortestCommonSupersequence
Variants: none

Field	Type	Description
alphabet_size	usize	Size of the alphabet
strings	Vec<Vec<usize>>	Set R of input strings, each encoded as a vector of alphabet indices
bound	usize	Length bound K

Size fields (getter methods for overhead expressions and declare_variants!):

• alphabet_size() — returns |Sigma|
• num_strings() — returns |R|
• bound() — returns K
• total_length() — returns sum of |x| for x in R

Notes:

• This is a satisfaction (decision) problem: Metric = bool, implementing SatisfactionProblem.
• A string x is a subsequence of w if x can be obtained by deleting characters from w (characters need not be contiguous in w).
• Different from ShortestCommonSuperstring where containment is as a contiguous substring.
• For the optimization version, minimize |w| subject to the subsequence constraint.

Complexity

• Best known exact algorithm:
  • For |R| = 2: O(|x_1| * |x_2|) by DP via the dual LCS relationship: SCS(x_1, x_2) = |x_1| + |x_2| - LCS(x_1, x_2).
  • For fixed |R|: O(product of |x_i| for i=1..|R|) by multi-dimensional DP.
  • For arbitrary |R|: NP-hard. Exact algorithms are exponential. A* search with lower bounds has been applied. Brute force is O(|Sigma|^K) to enumerate all candidate strings.
• Approximation: The greedy algorithm (repeatedly pick the symbol that satisfies the most constraints) gives an O(|Sigma|)-approximation. No PTAS is known.
• NP-completeness: NP-complete (Maier, 1978). Remains NP-complete even if |Sigma| = 5. Also NP-complete over binary alphabet (Raiha and Ukkonen, 1981).
• Polynomial cases: Solvable in polynomial time if |R| = 2 or if all strings have length <= 2.
• declare_variants! complexity string: "alphabet_size ^ bound" (brute-force enumeration of all candidate strings).
• References:
  • David Maier (1978). "The complexity of some problems on subsequences and supersequences". JACM 25(2):322-336.
  • K. J. Raiha and E. Ukkonen (1981). "The shortest common supersequence problem over binary alphabet is NP-complete". Theoretical Computer Science 16(2):187-198.

Extra Remark

Full book text:

INSTANCE: Finite alphabet Sigma, finite set R of strings from Sigma*, and a positive integer K.
QUESTION: Is there a string w in Sigma* with |w| <= K such that each string x in R is a subsequence of w, i.e., w = w0x1w1x2w2 ... xkwk where each wi in Sigma* and x = x1x2 ... xk?
Reference: [Maier, 1978]. Transformation from VERTEX COVER.
Comment: Remains NP-complete even if |Sigma| = 5. Solvable in polynomial time if |R| = 2 (by first computing the largest common subsequence) or if all x in R have |x| <= 2.

How to solve

• It can be solved by (existing) bruteforce -- enumerate all strings w in Sigma* with |w| <= K and check the subsequence condition for each x in R.
• It can be solved by reducing to integer programming -- encode position variables and ordering constraints as integer linear constraints.
• Other: For |R| = 2, DP in O(|x_1||x_2|) via LCS duality. For general |R|, multi-dimensional DP or A search with bounds.

Example Instance

Instance 1 (YES, three strings over ternary alphabet):

• Alphabet: Sigma = {a, b, c} (|Sigma| = 3)
• Strings: R = {"abcb", "bcab", "acba"}
• Bound K = 7
• Candidate supersequence: w = "abcacba" (length 7)
  • "abcb" subsequence at positions 1,2,3,6
  • "bcab" subsequence at positions 2,3,4,6
  • "acba" subsequence at positions 1,3,6,7
• Answer: YES
• Exhaustive enumeration: search space 3⁷ = 2,187. Exactly 42 feasible supersequences exist, including "abcacba". (Used in find_all_satisfying test)

Instance 2 (YES, six strings over binary alphabet):

• Alphabet: Sigma = {0, 1} (|Sigma| = 2)
• Strings: R = {"0110", "1010", "0101", "1100", "0011", "1001"}
• Bound K = 8
• Candidate supersequence: w = "01101001" (length 8)
  • "0110" at positions 1,2,3,4
  • "1010" at positions 2,4,5,6
  • "0101" at positions 1,2,4,5
  • "1100" at positions 2,3,4,6
  • "0011" at positions 1,4,5,8
  • "1001" at positions 2,4,6,8
• Answer: YES

Instance 3 (NO):

• Alphabet: Sigma = {a, b, c}
• Strings: R = {"abc", "bca", "cab", "acb", "bac", "cba"} (all 6 permutations of {a,b,c})
• Bound K = 5
• The minimum SCS of all permutations of 3 symbols has length 7. This follows from the known result that the shortest common supersequence of all n! permutations of n symbols has length n² − 2n + 4 for n ≥ 2 (for n = 3: 9 − 6 + 4 = 7). One such supersequence is "abcabca". Since 7 > 5, no supersequence of length <= 5 exists.
• Answer: NO
• Exhaustive enumeration: search space 3⁵ = 243. 0 feasible supersequences. (Used in find_all_satisfying empty test)

[zazabap]

Issue Quality Check — Model

Check	Result	Details
Usefulness	✅ Pass	Novel string problem not in the reduction graph; connects via VertexCover reduction
Non-trivial	✅ Pass	Distinct subsequence-based satisfaction problem unlike any existing model
Correctness	✅ Pass	All references verified; complexity claims accurate
Well-written	⚠️ Warn	Instance 2 contains messy inline self-correction; complexity string for declare_variants! not provided

Overall: 3 passed, 0 failed, 1 warning

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Usefulness

Pass. ShortestCommonSupersequence does not exist in the codebase. It is a classic NP-complete string problem from Garey & Johnson (A4 SR8) with clear applications in data compression and sequence alignment. The associated rule R102 (MinimumVertexCover -> ShortestCommonSupersequence) would connect it to the existing reduction graph. The motivation clearly explains the problem's significance and distinguishes it from ShortestCommonSuperstring.

Non-trivial

Pass. This is a string-based satisfaction problem with subsequence containment constraints. No existing problem in the codebase operates on strings or has subsequence semantics. It is genuinely distinct from all implemented models.

Correctness

Pass. All references verified:

1. Maier (1978): "The Complexity of Some Problems on Subsequences and Supersequences," JACM 25(2):322-336. Confirmed via ACM Digital Library (DOI: 10.1145/322063.322075). The paper proves SCS is NP-complete for |Sigma| >= 5, matching the issue's claim.

2. Raiha and Ukkonen (1981): "The Shortest Common Supersequence Problem over Binary Alphabet is NP-Complete," Theoretical Computer Science 16(2):187-198. Confirmed via ScienceDirect. Strengthens the NP-completeness result to binary alphabets.

3. Garey & Johnson reference (A4 SR8): The mathematical definition matches the standard formulation. The transformation from VERTEX COVER is correctly attributed.

4. Complexity claims: The polynomial-time solvability for |R|=2 via LCS duality is correct (well-known result). The claim about NP-completeness for |Sigma|>=5 matches Maier's result.

Well-written

Warn. The issue is mostly well-structured, but has some quality issues:

1. Instance 2 inline mess: The verification of "0011" as a subsequence of "01101001" includes visible self-correction and working-out ("Let me recheck...", crossed-out reasoning). This should be cleaned up to show only the final verified positions.

2. Missing declare_variants! complexity string: The Complexity section discusses algorithms but does not provide a concrete expression for implementation (e.g., "alphabet_size ^ bound").

3. Instance 3 (NO) could be stronger: The claim that the minimum SCS of all permutations of 3 symbols has length 7 is correct, but no citation or proof sketch is given. A brief justification would improve this.

All three examples are otherwise correctly verified:

• Instance 1: w="abcacba" contains "abcb" at (1,2,3,6), "bcab" at (2,3,4,6), "acba" at (1,3,6,7). Correct.
• Instance 2: w="01101001" contains all 6 strings at valid increasing positions. Correct.
• Instance 3: K=5 is insufficient for all 6 permutations. Correct.

Recommendations

• Clean up Instance 2's inline self-correction to show only the final verified subsequence positions.
• Add a concrete complexity string for declare_variants!.
• Consider adding a brief justification for the minimum SCS length of 7 in Instance 3.

[zazabap]

Changelog

Two fixes applied to the issue body:

1. Mechanical fix (Instance 2): Standardized the "0011" subsequence verification to position-only format (at positions 1,4,5,8), removing the verbose w[i]=... notation to match the format used by all other entries.

2. Substantive fix (Instance 3): Added justification for the claim that the minimum SCS length of all permutations of 3 symbols is 7. The justification cites the known result that the shortest common supersequence of all n! permutations of n symbols has length n² − 2n + 4 for n ≥ 2, which gives 9 − 6 + 4 = 7 for n = 3, with "abcabca" as a witness.