Supplementary files of the article ''Finding small feedback arc set on large graphs''

The supplementary files are organized as follows.

In the first section, we describe the sheets located under the results folder, which contains detailed results of our experiments.

In the second section, we describe how to obtain the datasets involved in our experiments.

In the third section, we describe the usage of source codes provided with this document.

Experimental Results

The sheets in Unweighted.xlsx and Weighted.xlsx are described as follows.

• GraphStats: Basic statistics of the graphs we tested. For the weighted case, we use random seed fixed from 1 to 10 to generate random positive integral weights for each arc, and the sheet lists the sum of arc weights of the generated weighted digraphs. This sheet is also provided in the .csv format(UnweightedStats.csv,WeightedStats.csv).
• SolRaw: Raw experimental result for heuristics. This sheet is also provided in the .csv format(UnweightedSol.csv ,WeightedSol.csv).
• SolPct: The size of the solutions obtained by the heuristics measured by the percentage to the size(sum of weights) of the arc set, aggregated per (graph, algorithm).
• SolPctTbl: Another representation of SolPct.
• SolTime: The running time of heuristics, aggregated per (graph, algorithm).
• RedRaw: Raw experimental result for reduction algorithms. This sheet is also provided in the .csv format (UnweightedRed.csv).
• RedPct: The size of the reduced graph measured by the percentage to the arc set size, aggregated per (graph, algorithm).
• RedPctTbl: Another representation of RedPct.
• RedTime: The running time of reduction algorithms, aggregated per (graph, algorithm).

There are two sheets in Gap.xlsx, Raw and Tbl. The first is the same as SolRaw for Unweighted.xlsx, and the second is the same as SolPctTbl for Unweighted.xlsx, with additional columns that contains graph statics. We provide the .csv format of the first table(Gap.csv).

We are able to open the new .xlsx sheets with LibreOffice 7.3.7.2 30(Build:2) on WSL 2 Ubuntu 22.04.

Datasets

There are four datasets included by our experiments, ISCAS, ISPD, LAW, and SNAP.

The format conversion of the first two datasets was done by Ali Dasdan, which are publicly available at https://github.com/alidasdan/graph-benchmarks.

For LAW and SNAP, you can find all graphs we tested in https://law.di.unimi.it/datasets.php and https://snap.stanford.edu/data/index.html respectively.

Converting graphs with WebGraph format

Our algorithm cannot read graphs with compressed WebGraph format directly.

Therefore, we provide converter scripts(conv.bat for Windows and conv.sh for Linux) and associated java libs under the folder webgraphconv.

We tested our scripts on a PC running the following OSs.

1. Windows 10 Pro 21H2 with openjdk version "11.0.15" 2022-04-19 LTS installed.
2. WSL Ubuntu 20.04 with openjdk version "11.0.16" installed.

You can follow these steps to convert a graph in WebGraph format into a graph that can be read by our algorithms.

1. Download .graph and .properties files from LAW. For example, you can find the graph word-assoc in https://law.di.unimi.it/webdata/wordassociation-2011/ .
2. Put .graph and .properties files directly with our script.
3. Run conv.bat <GraphNameBeforeExtension>. For example, execute conv.bat wordassociation-2011 in windows cmd or ./conv.sh wordassociation-2011 in linux bash.
4. The program outputs wordassociation-2011.txt in the same folder.

Read the graphs

You need to provide InputGraphType parameter in command line to our programs in order to specify the file format of the input graph.

We provide the this parameter for each graph we tested in the third column of datasets.csv.

You can check \src\include\graph\io.hpp for more details.

Codes

We provide our code under the src folder.

We successfully compile our codes(fas_alg.cpp, fas_red.cpp, generate_synthetic_graphs.cpp) with the following compilers(flags -O3 -I"include"):

• gcc version 9.3.0 (Ubuntu 9.3.0-17ubuntu1~20.04)
• gcc version 9.4.0 (Ubuntu 9.4.0-1ubuntu1~20.04.1)
• gcc version 7.3.0 (x86_64-posix-seh-rev0, Built by MinGW-W64 project)
• gcc version 12.1.0 (x86_64-posix-seh-rev3, Built by MinGW-W64 project)

Heuristics

You can use the following command lines to test the heuristic algorithms we implemented.

fas_alg InputGraphPath InputGraphType AlgName  [-r Seed] [-p] [-w MaxWeight] [-o OutputSolutionPath]

Parameter InputGraphPath denotes the path of the input graph file.

Parameter InputGraphType is described above.

You need to specify the heuristic algorithm you want to run according to the following table:

Algorithm	InputGraphType
The Greedy algorithm proposed by Eades, et al.	Greedy
Preprocess the input with SCC Decomposition, then solve with Greedy.	scc+Greedy
Preprocess the input with two-cycle elimination first, then SCC Decomposition, then solve with Greedy.	scc+2cyc+Greedy
Preprocess the input with SCC Decomposition, two-cycle elimination, the chain contraction, then solve with Greedy.	scc+2cyc+chain+Greedy
Preprocess the input with the non-recursive version of our Reduce algorithm, then solve with Greedy.	RED+Greedy
Preprocess the input with our Reduce algorithm, then solve with Greedy.	REDstar+Greedy
Preprocess the input with HCS reduction algorithm, then solve with Greedy.	HCS+Greedy
SimFASDC algorithm with parameter $r=1/q$.	RASq
The FASDC algorithm with parameter $r=1/3$.	RASstar

P.S. The implementation of our FASDC algorithm are templated, therefore you may only chose $q=2,3,4,6,8,12,16,24,32,48$. For more options, you will need to modify our source code.

The following list describes the usage of the flags.

• The flag -r Seed assigns a random seed for the following flags. The seed is fixed to zero by default. In our experiments, we fixed the seeds from 1 to 10 for each graph, each algorithm.

• The flag -p instructs the program to permute the label of vertices and the order of arcs in the input. This will not affect the vertex labels in the output.

• The flag -w MaxWeight instructs the program to assign random positive integral weights that uniformly distributes in [1, MaxWeight]. The maximum weight is set to one by default.

• The flag -o OutputSolutionPath instructs the program to output the linear ordering found by the algorithm into OutputSolutionPath.

Example:

fas_alg ../Datasets/ISCAS/s27.d DU Greedy -r 2023 -p -w 5 -o out.txt

The expected output is(the last column might vary with different machines):

s27.d_Greedy_5_2023,4,0

The first column denotes the task that has been executed, in the format of [GraphName]_[AlgName]_[MaxWeight]_[Seed].

The second column denotes the weight of the minimum feedback arc set found by the algorithm.

The last column denotes the time consumed by the algorithm, in milliseconds.

In the out.txt, you may expect the following content:

9 2 0 18 4 54 10 3 1 19 53 5 50 32 31 37 41 30 12 13 21 39 14 22 26 6 43 42 40 15 25 35 29 28 27 17 16 34 48 38 33 20 52 51 45 44 36 47 24 23 8 7 46 11 49

The $i$-th number of the output denotes to the position of the vertex with number $i-1$ in the corresponding linear ordering. i.e., vertex $0$ is ranked at the 10-th place in the result linear ordering, since the first integer in the file equals to 9.

Reductions

You can use the following command lines to test the reduction algorithms we implemented.

fas_red InputGraphPath InputGraphType AlgName  [-r Seed] [-p] [-w MaxWeight]

Parameter InputGraphPath denotes the path of the input graph file.

Parameter InputGraphType is described above.

You need to specify the heuristic algorithm you want to run according to the following table:

Algorithm	InputGraphType
Reduce the input with SCC Decomposition.	scc
Reduce the input with two-cycle elimination first, then SCC Decomposition.	scc+2cyc
Reduce the input with SCC Decomposition, two-cycle elimination, the chain contraction.	scc+2cyc+chain
Reduce the input with the non-recursive version of our Reduce algorithm.	RED
Reduce the input with our Reduce algorithm.	REDstar
Reduce the input with HCS reduction algorithm.	HCS

Example:

./fas_red ../Datasets/ISPD/ibm18.d DU REDstar -r 2023 -p -w 1

The expected output is(running time in the last column might vary with different machines):

ibm18.d_REDstar_1_2023,85227,105451,1343

The first column denotes the task that has been executed, in the format of [GraphName]_[AlgName]_[MaxWeight]_[Seed].

The second column denotes the number of arcs in the reduced digraph.

The third column denotes the sum of arc weights in the reduced digraph.

The last column denotes the time consumed by the algorithm, in milliseconds.

Generating synthetic graphs

Source code: gen_pa.cpp and gen_er.cpp.

You may get the synthetic graphs under Erdos-Renyi(gen_er.cpp) model and the Preferential Attachment(gen_pa.cpp) model directly by compile and run.