Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

Breast cancer local recurrence risk in implant-based breast reconstruction with macrotexturized and microtexturized prosthesis: A multicentric retrospective cohort study

Background Nowadays, implant-based breast reconstruction is a common technique after mastectomy. The widespread use of implant employment is prompting significant concerns regarding the oncological safety of prostheses and the potential impact of surface texture on the recurrence of local breast cancer. This article examines the oncological outcomes associated with postmastectomy breast reconstructions using micro- and macrotexturized implants, focusing on the incidence and relative risk (RR).Methods A retrospective cohort study was conducted on patients admitted to Multimedica group (IRCCS, San Giovanni Hospital, Milan) and ICH groups (Humanitas Clinical Institute, Milan) between January 2003 and September 2020. Minimum follow-up considered was of 1 year. Patients submitted to either complete or nipple-spearing mastectomy, who underwent breast reconstruction with macrotexturized or microtexturized prosthesis, were included in group A and B, respectively.Results A total of 646 patients met the basic inclusion and exclusion criteria. Group A included 410 (63.5%) patients and group B included 236 (36.5%). Cancer recurrence absolute risk in group A was 5.6 +/- 2.2% and in group B was of 2.1 +/- 1.8%. RR for breast cancer recurrence in group A compared to group B was of 2.65; confidence interval 95% (1.02; 6.87). Statistical analysis identified a higher local recurrence risk in patients reconstructed with macrotexturized prosthesis (p-value 0.036).Conclusion This study detected a higher risk for local breast cancer recurrence associated to macrotexturized breast implants employment. Further investigations are required to verify these outcomes

Reducing the complexity of reductions

We build on the recent progress regarding isomorphisms of complete sets that was reported in Agrawal et al. (1998). In that paper, it was shown that all sets that are complete under (non-uniform) AC0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a "Gap Theorem", showing that all sets complete under AC0 reductions are in fact already complete under NC0 reductions. The following questions were left open in that paper: ¶1. Does the "gap" between NC0 and AC0 extend further? In particular, is every set complete under polynomial-time reducibility already complete under NC0 reductions? ¶2. Does a uniform version of the isomorphism theorem hold? ¶3. Is depth-three optimal, or are the complete sets isomorphic under isomorphisms computable by depth-two circuits? ¶ We answer all of these questions. In particular, we prove that the Berman-Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions. More precisely, we show that for any class closed under uniform TC0-computable many-one reductions, the following three theorems hold: ¶1. If contains sets that are complete under a notion of reduction at least as strong as Dlogtime-uniform AC0[mod 2] reductions, then there are such sets that are not complete under (even non-uniform) AC0 reductions. ¶2. The sets complete for under P-uniform AC0 reductions are all isomorphic under isomorphisms computable and invertible by P-uniform AC0 circuits of depth-three. ¶3. There are sets complete for under Dlogtime-uniform AC0 reductions that are not isomorphic under any isomorphism computed by (even non-uniform) AC0 circuits of depth two. ¶To prove the second theorem, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.

Stochastic Boolean Satisfiability

. Satisfiability problems and probabilistic models are core topics of artificial intelligence and computer science. This paper looks at the rich intersection between these two areas, opening the door for the use of satisfiability approaches in probabilistic domains. The paper examines a generic stochastic satisfiability problem, SSat, which can function for probabilistic domains as Sat does for deterministic domains. It shows the connection between SSat and well studied problems in belief network inference and planning under uncertainty, and defines algorithms, both systematic and stochastic, for solving SSat instances. These algorithms are validated on random SSat formulae generated under the fixed-clause model. In spite of the large complexity gap between SSat (PSPACE) and Sat (NP), the paper suggests that much of what we've learned about Sat transfers to the probabilistic domain. 1. Introduction There has been a recent focus in artificial intelligence (AI) on solving problems exh..