[Paper Review] Approximation hardness of Shortest Common Superstring variants

Yun William Yu|arXiv (Cornell University)|Feb 27, 2016

Algorithms and Data Compression8 references3 citations

TL;DR

This paper establishes new approximation hardness results for variants of the Shortest Common Superstring (SCS) problem: SCS with Negative strings (SCSN) is proven Log-APX-hard (and thus Log-APX-complete), and SCS with Negative strings and Wildcards (SCSNW) is shown to be NPOPB-hard. The results are achieved via reductions from Minimum Set Cover and Minimum Ones 3SAT, respectively, extending the understanding of inapproximability in genome assembly-related string problems.

ABSTRACT

The shortest common superstring (SCS) problem has been studied at great length because of its connections to the de novo assembly problem in computational genomics. The base problem is APX-complete, but several generalizations of the problem have also been studied. In particular, previous results include that SCS with Negative strings (SCSN) is in Log-APX (though there is no known hardness result) and SCS with Wildcards (SCSW) is Poly-APX-hard. Here, we prove two new hardness results: (1) SCSN is Log-APX-hard (and therefore Log-APX-complete) by a reduction from Minimum Set Cover and (2) SCS with Negative strings and Wildcards (SCSNW) is NPOPB-hard by a reduction from Minimum Ones 3SAT.

Motivation & Objective

• To establish stronger inapproximability bounds for SCS variants with negative strings and wildcards, which are relevant to computational genomics.
• To close gaps in the complexity landscape of SCS variants, particularly for SCSN and SCSNW, where prior hardness results were unknown or incomplete.
• To extend existing reductions from NP-hard problems like vertex cover and chromatic number to SCS variants with negative strings and wildcards.
• To demonstrate that SCSNW is significantly harder than SCSN by showing it is NPOPB-hard, while SCSN remains in Log-APX.

Proposed method

• A reduction from Minimum Set Cover is used to prove SCSN is Log-APX-hard, reusing overlapping string gadgets from prior SCS reductions.
• A novel reduction from Minimum Ones 3SAT is constructed to prove SCSNW is NPOPB-hard, using specialized frame, variable, and clause gadgets with wildcards and negative strings.
• The frame gadget uses negative strings to enforce structural constraints on the superstring, such as forcing a specific starting character and preventing overlapping of variable gadgets unless properly aligned.
• Variable gadgets are designed so that setting a variable to true forces its entire string to be shifted right, incurring a length cost of 3n per such variable, modeling the objective of minimizing true variables.
• Clause gadgets use negative strings to disallow configurations where all literals in a clause are false, ensuring that at least one variable must be true to satisfy the clause.
• The overall construction is polynomial-time and normalized to form an L-reduction, preserving the approximation hardness of the source problem.

Experimental results

Research questions

• RQ1 Is SCS with Negative strings (SCSN) Log-APX-hard, given that it is known to be in Log-APX?
• RQ2 What is the approximability class of SCS with both Negative strings and Wildcards (SCSNW), given that SCSW is Poly-APX-hard and SCSN is in Log-APX?
• RQ3 Can the hardness of SCSNW be established via a reduction from a known hard problem such as Minimum Ones 3SAT?
• RQ4 Is the combination of negative strings and wildcards in SCS significantly more complex than either feature alone?
• RQ5 Does the presence of wildcards in SCSN allow for a stronger hardness result, or is the problem still tractable within Log-APX?

Key findings

• SCSN is Log-APX-hard, and since it is already known to be in Log-APX, it is therefore Log-APX-complete.
• SCSNW is NPOPB-hard, indicating that it is among the hardest problems in the NPOPB complexity class.
• The reduction from Minimum Ones 3SAT to SCSNW is an L-reduction, preserving the approximation hardness and confirming that SCSNW is NPOPB-hard.
• The construction of the SCSNW instance requires O(n⁵) operations and uses a constant-size alphabet of size 4, with negative strings and wildcards strategically placed to enforce constraints.
• The length of the superstring in the SCSNW instance is 2 + 9n + 3nW, where W is the number of true variables in the Min Ones 3SAT solution, showing a direct correspondence between solution cost and superstring length.
• The proof shows that SCSNW is strictly harder than SCSN, as the reduction for SCSNW requires Ω(n) wildcards per string, which cannot be emulated in SCSN under the same constraints.