Clique in 3-track interval graphs is APX-hard

Butman, Hermelin, Lewenstein, and Rawitz proved that Clique in t-interval graphs is NP-hard for t >= 3. We strengthen this result to show that Clique in 3-track interval graphs is APX-hard.

💡 Research Summary

The paper establishes that the Maximum Clique problem on 3‑track interval graphs is APX‑hard, strengthening earlier results that showed NP‑hardness for t‑interval graphs when t ≥ 3. The authors achieve this by constructing an L‑reduction from the well‑known APX‑hard problem 12‑OCC‑MAX‑E2‑CSAT, a restricted version of MAX‑SAT where each variable appears at most twelve times and each clause is a conjunction of exactly two literals.

First, the paper recalls that 12‑OCC‑MAX‑E2‑CSAT itself is APX‑hard via a gap‑preserving reduction from E3‑OCC‑MAX‑E2‑SAT. Using this as a starting point, the authors map any instance (X, C) of 12‑OCC‑MAX‑E2‑CSAT to a 3‑track interval graph G. For each variable xi they create twelve copies of a 3‑track interval representing the positive literal xi and twelve copies for the negative literal ¬xi. Each 3‑track interval consists of three open intervals, one on each track, with integer endpoints in the range