Constant Approximating Parameterized k-SETCOVER is W[2]-hard

Independent Set in k-Claw-Free Graphs: Conditional $$\chi $$-Boundedness and the Power of LP/SDP Relaxations

“…More precisely, in the COMPACT PSP problem, we have that |U | = f (r) • Θ(poly(log |S |)), for some function f (r) ≥ r. Apart from algorithmic motivation, compact instances have recently been used as an intermediate step to show FPT-inapproximibility of (noncompact) classical problems (see, e.g., [25,99] where the compact instances were used in proving FPT-inapproximability of the k-EvenSet and Dominating Set). Recently, [101] showed that the compact version of SET r-COVERING is W[1]-complete, while it is known that the corresponding non-compact problem is W[2]-complete. We hope that studying COMPACT PSP would lead to some ideas that would be useful in proving tight FPT inapproximability of PSP (that is, to weaken the Gap-ETH assumption used in [36]).…”