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Greedy_USM.m
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Greedy_USM.m
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function [X_max, fvalues, itertime] = Greedy_USM(f, V, randomized)
% Unconstrained submodular maximization: max_S F(S)
% f: submodular function of class SetFct
% V: ground set
% if randomized = 0
% We use the "Deterministic USM" algo from SMunconstrained(journal)
% which achieves a 1/3 approximation
% if randomized = 1
% We use the "Randomized USM" algo from SMunconstrained(journal)
% which achieves a 1/2 approximation in expectation
% if randomized = 2 (TODO: modify code to use SetFct interface for this option)
% We use the "Deterministic Unconstrained" algorithm from
% "Deterministic Algorithms for Submodular Maximization Problems"
% which achieves 1/2 approximation
n = length(V);
fvalues = zeros(n,1);
itertime = zeros(n,1);
if nargout>=3
% Start the clock.
tic;
end
if randomized ~= 2
X = [];
Y = V;
[fX_val, fX_obj] = f(X);
[fY_val, fY_obj] = f(Y);
for i = V
[fXi_val, fXi_obj] = add(fX_obj, X, i);
f_marg_X = fXi_val - fX_val; %f_marg(X, i);
[fYmi_val, fYmi_obj] = rmv(fY_obj, Y, i);
f_marg_Y = fYmi_val - fY_val; %-f_marg(setdiff(Y,i),i);
if randomized
a = max(f_marg_X,0);
b = max(f_marg_Y,0);
if a == 0 && b == 0
p = 1;
else
p = a/(a + b);
end
if binornd(1,p)
X = union(X, i);
fX_val = fXi_val;
fX_obj = fXi_obj;
else
Y = setdiff(Y,i);
fY_val = fYmi_val;
fY_obj = fYmi_obj;
end
else
if f_marg_X >= f_marg_Y
X = union(X, i);
fX_val = fXi_val;
fX_obj = fXi_obj;
else
Y = setdiff(Y,i);
fY_val = fYmi_val;
fY_obj = fYmi_obj;
end
end
fvalues(i) = max(fX_val, fY_val);
if nargout>=3
itertime(i) = toc;
end
end
X_max = X;
else
m = 1;
p = 1;
X = {[]};
Y = {V};
for i = 1:n
a = zeros(m,1);
b = zeros(m,1);
for j = 1:m
a(j) = f_marg(X{j},i);
b(j) = -f_marg(setdiff(Y{j},i),i);
end
%% find extreme point of (P) by solving fractional knapsack problem (see Claim A.1)
v = p.*(a - 3*b);
s = p.*(b - 3*a);
B = p'*(b - 2*a);
% cvx_begin
% cvx_precision best
% variable z_cvx(m);
% maximize (z_cvx'*v)
% subject to
% z_cvx'*s <= B
% z_cvx <= 1
% z_cvx >= 0
% cvx_end
% take all items with vj ≥ 0 and sj ≤ 0 & omit all items of vj < 0 and sj ≥ 0
z = zeros(m,1);
z((v >= 0) & (s <= 0))=1;
% update budget
B = B - z'*s;
vs = v./s;
items = 1:m;
% sort the positive items (vj ≥ 0 and sj > 0) in non-increasing order of vj/sj
pos_items = items((v >= 0) & (s > 0));
[~,sorted_pos] = sort(vs(pos_items),'descend');
% sort the negative items (vj ≤ 0 and sj < 0) in non-decreasing order of vj/sj
neg_items = items((v <= 0) & (s < 0));
[~,sorted_neg] = sort(vs(neg_items),'ascend');
j_pos = 1;
while B> 0 && j_pos<= numel(pos_items)
% add positive items until B reaches 0 (or we are out of positive items)
ind = pos_items(sorted_pos(j_pos));
z(ind) = min(B/s(ind),1);
if z(ind) == 1
j_pos = j_pos + 1;
end
B = B - z(ind)*s(ind);
end
j_neg = 1;
while B<0
% add negative items until B reaches 0
ind = neg_items(sorted_neg(j_neg));
z(ind) = min(B/s(ind),1);
if z(ind) == 1
j_neg = j_neg + 1;
end
B = B - z(ind)*s(ind);
end
while vs(j_pos)>= vs(j_neg) && j_pos<= numel(pos_items) && j_neg<= numel(neg_items)
% add positive & negative items in such a way that B stays 0
ind_pos = pos_items(sorted_pos(j_pos));
ind_neg = neg_items(sorted_neg(j_neg));
z(ind_pos) = min(z(ind_pos) - s(ind_neg)/s(ind_pos),1);
if z(ind_pos) == 1
j_pos = j_pos + 1;
end
z(ind_neg) = z(ind_neg) - z(ind_pos)*s(ind_pos)/s(ind_neg);
if z(ind_neg) == 1
j_neg = j_neg + 1;
end
end
w = 1 - z;
%% update distribution
m_old = m;
for j = 1:m_old
if z(j) == 1
X{j} = [X{j},i];
elseif w(j) ==1
Y{j} = setdiff(Y{j},i);
else %fractional sol
p(m+1,1) = w(j)*p(j);
X{m+1} = X{j};
Y{m+1} = setdiff(Y{j},i);
p(j) = z(j)*p(j);
X{j} = [X{j},i];
m = m+1;
end
end
end
%% return set in the support of the distribution with largest f value
f_max = 0;
X_max = [];
for j=1:m
if f(X{j})> f_max
f_max = f(X{j});
X_max = X{j};
end
end
end