On Maximum Weight Clique Algorithms, and How They Are Evaluated

Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments

The development of a general algorithmic procedure for university examination timetabling

The problem of scheduling university examinations is becoming difficult for examination officers especially when they have to construct the timetables manually. It is largely due to the increasing number of students and greater freedom in choosing the courses. Examination officers would have to spend a considerable amount of time checking for student conflicts so that no student would have to sit for more than one exam at any one time. There are also other limitations such as the number of examination rooms, the length of the examination period and others. The examination timetabling problem varies between institutions, depending on their particular needs and limited resources. Most of the existing computerised examination timetabling systems found in the literature are developed and used by particular institutions. Therefore, the aim of the research is to produce a general computerised system for timetabling examinations which can be used by most universities. The research is done in two stages; the first stage involves carrying out a survey on the university examination timetabling systems and the second stage is the construction of a university examination timetabler incorporating the common objectives and constraints found in the survey. The survey was carried out to determine the extent to which the computerised examination timetabling procedures are used, to identify the objectives and constraints which are commonly considered when constructing examination timetables and to evaluate the effectiveness of the existing examination timetabling systems in achieving the objectives and satisfying the constraints The construction of the general examination timetabling system is done in two parts. In the first part, a new algorithmic rule is developed to assign exams to the minimum number of sessions without creating conflicts for any student. The rule adopts a clique initialisation strategy as a starting point and a graph colouring approach for assigning the exams. This rule is also quite capable of scheduling exams to the sessions which are as close as to the least number of sessions possible, without having to carry out any backtracking process. The backtracking process can sometimes be time consuming if there are a lot of exams firstly to be scheduled, and secondly clashing with each other. The second part of the work involves minimising the total number of students taking two exams on the same day and scheduling large exams early in the examination period subject to a specified time limit on the overall examination period and a maximum number of students that may be examined in any session. A swapping rule was introduced where exams in one of the sessions in any day with large number of sameday exams are interchanged with exams in other sessions which will reduce the total number of same-day exams. The experimentation showed that if the swapping procedures are repeated three times, the total number of same-day exams will be reduced by 50%. The total number of same-day exams will be reduced even more if some extra sessions can be added to the initial minimum number of sessions. A simple rule was devised to schedule large exams early in the examination period

Certifying Correctness for Combinatorial Algorithms : by Using Pseudo-Boolean Reasoning

Over the last decades, dramatic improvements in combinatorialoptimisation algorithms have significantly impacted artificialintelligence, operations research, and other areas. These advances,however, are achieved through highly sophisticated algorithms that aredifficult to verify and prone to implementation errors that can causeincorrect results. A promising approach to detect wrong results is touse certifying algorithms that produce not only the desired output butalso a certificate or proof of correctness of the output. An externaltool can then verify the proof to determine that the given answer isvalid. In the Boolean satisfiability (SAT) community, this concept iswell established in the form of proof logging, which has become thestandard solution for generating trustworthy outputs. The problem isthat there are still some SAT solving techniques for which prooflogging is challenging and not yet used in practice. Additionally,there are many formalisms more expressive than SAT, such as constraintprogramming, various graph problems and maximum satisfiability(MaxSAT), for which efficient proof logging is out of reach forstate-of-the-art techniques.This work develops a new proof system building on the cutting planesproof system and operating on pseudo-Boolean constraints (0-1 linearinequalities). We explain how such machine-verifiable proofs can becreated for various problems, including parity reasoning, symmetry anddominance breaking, constraint programming, subgraph isomorphism andmaximum common subgraph problems, and pseudo-Boolean problems. Weimplement and evaluate the resulting algorithms and a verifier for theproof format, demonstrating that the approach is practical for a widerange of problems. We are optimistic that the proposed proof system issuitable for designing certifying variants of algorithms inpseudo-Boolean optimisation, MaxSAT and beyond

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum