31.10.2019 KNAPSACK - Algorithms for Knapsack Problems KNAPSACK Algorithms for Knapsack Problems KNAPSACK is a FORTRAN77 library which contains implementations of algorithms for a variety of knapsack problems, by Silvano Martelo and Paolo Toth. Code MT1 MT1R MT2 MTB2 MTU2 MTSL MTC2 Problem 0-1 Knapsack 0-1 Knapsack 0-1 Knapsack Bounded Knapsack Unbounded Knapsack Subset Sum Change Making Type Exact Exact (real data) Exact/Approximate Exact/Approximate Exact/Approximate Exact/Approximate Exact/Approximate MTCB MTM MTHM MTG Bounded Change Making 0-1 Multiple Knapsack 0-1 Multiple Knapsack Generalized Assignment Exact/Approximate Exact/Approximate Approximate Exact/Approximate MTHG Generalized Assignment Approximate MTP Bin Packing Exact/Approximate Languages: KNAPSACK is available in a FORTRAN77 version. Related Data and Programs: BIN_PACKING, a dataset directory which contains examples of the bin packing problem, in which a number of objects are to be packed in the minimum possible number of uniform bins; CHANGE_MAKING, a dataset directory which contains test data for the change making problem; COMBO, a FORTRAN90 library which includes many combinatorial routines. GENERALIZED_ASSIGNMENT, a dataset directory which contains test data for the generalized assignment problem; KNAPSACK_01, a FORTRAN77 library which uses brute force to solve small versions of the 0/1 knapsack problem; KNAPSACK_01, a dataset directory which contains test data for the 0/1 knapsack problem; KNAPSACK_MULTIPLE, a dataset directory which contains test data for the multiple knapsack problem; LAMP, a FORTRAN77 library which solves linear assignment and matching problems. LAU_NP, a FORTRAN90 library which implements heuristic algorithms for various NP-hard combinatorial problems. https://people.sc.fsu.edu/~jburkardt/f77_src/knapsack/knapsack.html 1/4 31.10.2019 KNAPSACK - Algorithms for Knapsack Problems PARTITION_PROBLEM, a FORTRAN77 library which seeks solutions of the partition problem, splitting a set of integers into two subsets with equal sum. SUBSET, a FORTRAN77 library which enumerates combinations, partitions, subsets, index sets, and other combinatorial objects. SUBSET_SUM, a FORTRAN77 library which seeks solutions of the subset sum problem. SUBSET_SUM, a dataset directory which contains examples of the subset sum problem, in which a set of numbers is given, and is desired to find at least one subset that sums to a given target value. TOMS632, a FORTRAN77 library which solves the multiple knapsack problem, by Silvano Martello and Paolo Toth. This is ACM TOMS algorithm 632. Author: Silvano Martello, Paolo Toth. Reference: 1. Silvano Martello, Paolo Toth, MKP: 0-1 multiple knapsack problem, ACM Transactions on Mathematical Software, Volume 11, Number 2, June 1985, pages 135-140. 2. Silvano Martello, Paolo Toth, Knapsack Problems: Algorithms and Computer Implementations, Wiley, 1990, ISBN: 0-471-92420-2, LC: QA267.7.M37. Source Code: knapsack.f, the source code. knapsack.sh, commands to compile the source code. Examples and Tests: knapsack_prb.f, a sample calling program. knapsack_prb.sh, commands to compile and run the sample program. knapsack_prb_output.txt, the output file. List of Routines: BLD explicitly determines set a - a1 or set a - a1 - a2 when lev .gt. 1 . BLDF explicitly determines set a1 or set a - a1 or set a - a1 - a2 when lev .eq. 1 . BLDSR1 explicitly determines set a1 when lev .gt. 1 . CHMT1 checks the input data. CHMT1R checks the input data. CHMT2 checks the input data. CHMTB2 checks the input data. CHMTC2 checks the input data. CHMTCB checks the input data. CHMTG checks the input data. CHMTHG checks the input data. CHMTHM checks the input data. CHMTM checks the input data. CHMTP checks the input data. https://people.sc.fsu.edu/~jburkardt/f77_src/knapsack/knapsack.html 2/4 31.10.2019 KNAPSACK - Algorithms for Knapsack Problems CHMTSL checks the input data. CHMTU2 checks the input data. CMPB solves a bounded change-making problem through the branch-and-bound algorithm. CORE determines the core problem. COREC determines the core problem. CORES determines the core problem. DEFPCK ? DETNS1 computes the cardinality of set A1. DETNS2 computes the cardinality of set A2. DINSM determines the dynamic programming lists. DMIND defines array MIND to contain the pointers to the sorted items ENUMER performs a branch-and-bound search. FEAS checks for infeasibility. FFDLS performs a first-fit decreasing heuristic and initializes LS and LSB. FIXRED fixes the variables after a local reduction. FMED computes median of the ratios of the first 2 and the last item. FORWRD performs statements 1-9. GHA applies the approximate algorithm gh with function (a). GHBCD applies the approximate algorithm gh with functions (b), (c) and (d). GHX applies the approximate algorithm gh with function (b) or (c) or (d). GR1 reduces a maximization gap. GR2 reduces a maximization gap. HBFDS performs a best-fit decreasing heuristic. HEUR determines the best initial heuristic solution. IMPR1: first improvement. IMPR2: second improvement. INSERT inserts item i in bin m and updates fs, x, ifp, k. KP01M solves, through branch-and-bound, a 0-1 single knapsack problem. KPMAX solves a 0-1 single knapsack problem using an initial solution. KPMIN solves a 0-1 single knapsack problem in minimization form. KSMALL finds the k-th smallest of n elements in o(n) time. L2 computes the lower bound. L3 reduces the current problem, and compute lower bound l3 and a new upper bound nub . LCL2 computes a local lower bound and execute a preprocessing for reduction. MAXT determines the three items of maximum weight. MGR1 finds an initial solution (quick algorithm). MGR2 finds an initial solution (accurate algorithm). MPSORT rearranges a(i1:i2) so a(it+i1-1) contains the it-th smallest element. MT1 solves the 0-1 single knapsack problem. MT1R solves the 0-1 single knapsack problem with real parameters. MT2 solves the 0-1 single knapsack problem. MTB2 solves the bounded single knapsack problem MTC1 solves a change-making problem through the branch-and-bound algorithm. MTC2 solves the unbounded change-making problem MTCB solves the bounded change-making problem MTG solves the generalized assignment problem MTHG heuristically solves the generalized assignment problem MTHM heuristically solves the 0-1 multiple knapsack problem MTM solves the 0-1 multiple knapsack problem MTP solves the bin packing problem MTS solves a small subset sum problem. MTSL solves the subset-sum problem MTU1 solves the unbounded single knapsack problem MTU2 solves the unbounded single knapsack problem MWFDS performs a modified worst-fit decreasing heuristic. NEWBimproves on the current upper bound iubf0 by taking into account the PAR does a parametric computation of the upper bounds. PEN0 computes the penalty for an item j which was assigned to no knapsack. https://people.sc.fsu.edu/~jburkardt/f77_src/knapsack/knapsack.html 3/4 31.10.2019 KNAPSACK - Algorithms for Knapsack Problems PEN1 computes the penalty foro an item j which was assigned more than one knapsack. PI computes a feasible solution to the current problem. PREPEN determines pak , kap and pakl (pointers for computing penalties) PRESP defines the core problem. REARR re-arranges the initial solution. REDNS reduces, without sorting, the items not in core. REDS reduces the original problem. REDU reduces an unbounded knapsack problem (po,wo) through dominance relations. RESTOR restores the situation preceding the reduction of level k and update lastw . SEARCH finds largest NL such that R < W(NL). SIGMA computes an upper bound ub on the best final solution which SKP1 solves a 0-1 single knapsack problem. SKP2 solves the 0-1 single knapsack problem SOL determines the solution vector x for the original problem. SORT7 sorts in increasing order the elements from SORTI sorts the integer array a by decreasing values (derived from SORTI2 sorts the integer array a by increasing values (derived from SORTR sorts the real array a by decreasing values (derived from subroutine TAB builds the new dynamic programming list tdb from the current TERMIN terminates the execution. TIMESTAMP prints out the current YMDHMS date as a timestamp. TRANS transforms a bounded knapsack problem (n, p, w, b) into TRIN transforms an gap in minimization form to an equivalent instance in maximization form. UBFJV computes the fisher-jaikumar-van wassenhove upper bound. UBRS computes the improved ross-soland upper bound. UPDATE updates the optimal solution after a local heuristic. UPSTAR updates the current optimal solution. USEDIN determines, through binary search, the location loc of td YDEF sets y(l,i,j) = ny . YUSE sets ny = y(l,i,j) . You can go up one level to the FORTRAN77 source codes. Last revised on 06 December 2009. https://people.sc.fsu.edu/~jburkardt/f77_src/knapsack/knapsack.html 4/4
