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menu="_book"> Reference </span> <span class="devsite-nav-icon material-icons" data-icon="forward" menu="_book"> </span> </a> </li> <li class="devsite-nav-item"> <a href="/optimization/examples" class="devsite-nav-title gc-analytics-event " data-category="Site-Wide Custom Events" data-label="Tab: Examples" track-name="examples" data-category="Site-Wide Custom Events" data-label="Responsive Tab: Examples" track-type="globalNav" track-metadata-eventDetail="globalMenu" track-metadata-position="nav"> <span class="devsite-nav-text" tooltip > Examples </span> </a> </li> <li class="devsite-nav-item"> <a href="/optimization/support" class="devsite-nav-title gc-analytics-event devsite-nav-has-children " data-category="Site-Wide Custom Events" data-label="Tab: Support" track-name="support" data-category="Site-Wide Custom Events" data-label="Responsive Tab: Support" track-type="globalNav" track-metadata-eventDetail="globalMenu" track-metadata-position="nav"> <span class="devsite-nav-text" tooltip > Support </span> <span class="devsite-nav-icon material-icons" data-icon="forward" > </span> </a> </li> </ul> </li> <li class="devsite-nav-item"> <a href="/optimization/service" class="devsite-nav-title gc-analytics-event " data-category="Site-Wide Custom Events" data-label="Tab: OR API" track-name="or api" data-category="Site-Wide Custom Events" data-label="Responsive Tab: OR API" track-type="globalNav" track-metadata-eventDetail="globalMenu" track-metadata-position="nav"> <span class="devsite-nav-text" tooltip > OR API </span> </a> </li> </ul> </div> <div class="devsite-mobile-nav-bottom"> <ul class="devsite-nav-list" menu="_book"> <li class="devsite-nav-item devsite-nav-expandable"><div class="devsite-expandable-nav"> <a class="devsite-nav-toggle" aria-hidden="true"></a><div class="devsite-nav-title devsite-nav-title-no-path" tabindex="0" role="button"> <span class="devsite-nav-text" tooltip>OR-Tools Reference</span> </div><ul class="devsite-nav-section"><li class="devsite-nav-item"><a href="/optimization/reference" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference" ><span class="devsite-nav-text" tooltip>Overview</span></a></li><li class="devsite-nav-item devsite-nav-expandable"><div class="devsite-expandable-nav"> <a class="devsite-nav-toggle" aria-hidden="true"></a><div class="devsite-nav-title devsite-nav-title-no-path" tabindex="0" role="button"> <span class="devsite-nav-text" tooltip>C++ Reference</span> </div><ul class="devsite-nav-section"><li class="devsite-nav-item"><a href="/optimization/reference/index_cpp" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/index_cpp" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/index_cpp" ><span class="devsite-nav-text" tooltip>Overview</span></a></li><li class="devsite-nav-item devsite-nav-expandable"><div class="devsite-expandable-nav"> <a class="devsite-nav-toggle" aria-hidden="true"></a><div class="devsite-nav-title devsite-nav-title-no-path" tabindex="0" role="button"> <span class="devsite-nav-text" tooltip>Algorithms</span> </div><ul class="devsite-nav-section"><li class="devsite-nav-item"><a href="/optimization/reference/algorithms" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms" ><span class="devsite-nav-text" tooltip>Overview</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/dense_doubly_linked_list" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/dense_doubly_linked_list" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/dense_doubly_linked_list" ><span class="devsite-nav-text" tooltip>dense_doubly_linked_list</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/dynamic_partition" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/dynamic_partition" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/dynamic_partition" ><span class="devsite-nav-text" tooltip>dynamic_partition</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/dynamic_permutation" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/dynamic_permutation" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/dynamic_permutation" ><span class="devsite-nav-text" tooltip>dynamic_permutation</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/find_graph_symmetries" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/find_graph_symmetries" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/find_graph_symmetries" ><span class="devsite-nav-text" tooltip>find_graph_symmetries</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/hungarian" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/hungarian" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/hungarian" ><span class="devsite-nav-text" tooltip>hungarian</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/knapsack_solver" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/knapsack_solver" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/knapsack_solver" ><span class="devsite-nav-text" tooltip>knapsack_solver</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/knapsack_solver_for_cuts" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/knapsack_solver_for_cuts" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/knapsack_solver_for_cuts" ><span class="devsite-nav-text" tooltip>knapsack_solver_for_cuts</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/algorithms/sparse_permutation" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/algorithms/sparse_permutation" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/algorithms/sparse_permutation" ><span class="devsite-nav-text" tooltip>sparse_permutation</span></a></li></ul></div></li><li class="devsite-nav-item devsite-nav-expandable"><div class="devsite-expandable-nav"> <a class="devsite-nav-toggle" aria-hidden="true"></a><div class="devsite-nav-title devsite-nav-title-no-path" tabindex="0" role="button"> <span class="devsite-nav-text" tooltip>CP-SAT</span> </div><ul class="devsite-nav-section"><li class="devsite-nav-item"><a href="/optimization/reference/sat" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat" ><span class="devsite-nav-text" tooltip>Overview</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/all_different" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/all_different" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/all_different" ><span class="devsite-nav-text" tooltip>all_different</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/boolean_problem" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/boolean_problem" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/boolean_problem" ><span class="devsite-nav-text" tooltip>boolean_problem</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/circuit" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/circuit" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/circuit" ><span class="devsite-nav-text" tooltip>circuit</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/clause" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/clause" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/clause" ><span class="devsite-nav-text" tooltip>clause</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_constraints" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_constraints" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_constraints" ><span class="devsite-nav-text" tooltip>cp_constraints</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model" ><span class="devsite-nav-text" tooltip>cp_model</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_checker" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_checker" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_checker" ><span class="devsite-nav-text" tooltip>cp_model_checker</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_expand" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_expand" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_expand" ><span class="devsite-nav-text" tooltip>cp_model_expand</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_lns" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_lns" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_lns" ><span class="devsite-nav-text" tooltip>cp_model_lns</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_loader" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_loader" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_loader" ><span class="devsite-nav-text" tooltip>cp_model_loader</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_mapping" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_mapping" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_mapping" ><span class="devsite-nav-text" tooltip>cp_model_mapping</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_objective" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_objective" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_objective" ><span class="devsite-nav-text" tooltip>cp_model_objective</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_postsolve" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_postsolve" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_postsolve" ><span class="devsite-nav-text" tooltip>cp_model_postsolve</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_presolve" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_presolve" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_presolve" ><span class="devsite-nav-text" tooltip>cp_model_presolve</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_search" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_search" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_search" ><span class="devsite-nav-text" tooltip>cp_model_search</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_solver" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_solver" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_solver" ><span class="devsite-nav-text" tooltip>cp_model_solver</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_symmetries" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_symmetries" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_symmetries" ><span class="devsite-nav-text" tooltip>cp_model_symmetries</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cp_model_utils" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cp_model_utils" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cp_model_utils" ><span class="devsite-nav-text" tooltip>cp_model_utils</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cumulative" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cumulative" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cumulative" ><span class="devsite-nav-text" tooltip>cumulative</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cumulative_energy" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cumulative_energy" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cumulative_energy" ><span class="devsite-nav-text" tooltip>cumulative_energy</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/cuts" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/cuts" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/cuts" ><span class="devsite-nav-text" tooltip>cuts</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/diffn" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/diffn" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/diffn" ><span class="devsite-nav-text" tooltip>diffn</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/diffn_util" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/diffn_util" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/diffn_util" ><span class="devsite-nav-text" tooltip>diffn_util</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/disjunctive" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/disjunctive" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/disjunctive" ><span class="devsite-nav-text" tooltip>disjunctive</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/drat_checker" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/drat_checker" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/drat_checker" ><span class="devsite-nav-text" tooltip>drat_checker</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/drat_proof_handler" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/drat_proof_handler" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/drat_proof_handler" ><span class="devsite-nav-text" tooltip>drat_proof_handler</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/drat_writer" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/drat_writer" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/drat_writer" ><span class="devsite-nav-text" tooltip>drat_writer</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/encoding" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/encoding" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/encoding" ><span class="devsite-nav-text" tooltip>encoding</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/feasibility_pump" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/feasibility_pump" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/feasibility_pump" ><span class="devsite-nav-text" tooltip>feasibility_pump</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/implied_bounds" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/implied_bounds" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/implied_bounds" ><span class="devsite-nav-text" tooltip>implied_bounds</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/inclusion" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/inclusion" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/inclusion" ><span class="devsite-nav-text" tooltip>inclusion</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/integer" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/integer" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/integer" ><span class="devsite-nav-text" tooltip>integer</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/integer_expr" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/integer_expr" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/integer_expr" ><span class="devsite-nav-text" tooltip>integer_expr</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/integer_search" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/integer_search" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/integer_search" ><span class="devsite-nav-text" tooltip>integer_search</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/intervals" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/intervals" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/intervals" ><span class="devsite-nav-text" tooltip>intervals</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/lb_tree_search" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/lb_tree_search" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/lb_tree_search" ><span class="devsite-nav-text" tooltip>lb_tree_search</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/linear_constraint" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/linear_constraint" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/linear_constraint" ><span class="devsite-nav-text" tooltip>linear_constraint</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/linear_constraint_manager" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/linear_constraint_manager" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/linear_constraint_manager" ><span class="devsite-nav-text" tooltip>linear_constraint_manager</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/linear_programming_constraint" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/linear_programming_constraint" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/linear_programming_constraint" ><span class="devsite-nav-text" tooltip>linear_programming_constraint</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/linear_relaxation" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/linear_relaxation" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/linear_relaxation" ><span class="devsite-nav-text" tooltip>linear_relaxation</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/lp_utils" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/lp_utils" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/lp_utils" ><span class="devsite-nav-text" tooltip>lp_utils</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/max_hs" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/max_hs" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/max_hs" ><span class="devsite-nav-text" tooltip>max_hs</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/model" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/model" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/model" ><span class="devsite-nav-text" tooltip>model</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/optimization" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/optimization" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/optimization" ><span class="devsite-nav-text" tooltip>optimization</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/parameters_validation" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/parameters_validation" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/parameters_validation" ><span class="devsite-nav-text" tooltip>parameters_validation</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/pb_constraint" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/pb_constraint" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/pb_constraint" ><span class="devsite-nav-text" tooltip>pb_constraint</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/precedences" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/precedences" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/precedences" ><span class="devsite-nav-text" tooltip>precedences</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/presolve_context" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/presolve_context" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/presolve_context" ><span class="devsite-nav-text" tooltip>presolve_context</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/presolve_util" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/presolve_util" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/presolve_util" ><span class="devsite-nav-text" tooltip>presolve_util</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/probing" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/probing" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/probing" ><span class="devsite-nav-text" tooltip>probing</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/pseudo_costs" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/pseudo_costs" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/pseudo_costs" ><span class="devsite-nav-text" tooltip>pseudo_costs</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/restart" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/restart" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/restart" ><span class="devsite-nav-text" tooltip>restart</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/rins" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/rins" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/rins" ><span class="devsite-nav-text" tooltip>rins</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/sat_base" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/sat_base" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/sat_base" ><span class="devsite-nav-text" tooltip>sat_base</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/sat_decision" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/sat_decision" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/sat_decision" ><span class="devsite-nav-text" tooltip>sat_decision</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/sat_inprocessing" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/sat_inprocessing" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/sat_inprocessing" ><span class="devsite-nav-text" tooltip>sat_inprocessing</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/sat_solver" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/sat_solver" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/sat_solver" ><span class="devsite-nav-text" tooltip>sat_solver</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/scheduling_constraints" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/scheduling_constraints" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/scheduling_constraints" ><span class="devsite-nav-text" tooltip>scheduling_constraints</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/scheduling_cuts" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/scheduling_cuts" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/scheduling_cuts" ><span class="devsite-nav-text" tooltip>scheduling_cuts</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/simplification" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/simplification" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/simplification" ><span class="devsite-nav-text" tooltip>simplification</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/subsolver" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/subsolver" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/subsolver" ><span class="devsite-nav-text" tooltip>subsolver</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/swig_helper" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/swig_helper" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/swig_helper" ><span class="devsite-nav-text" tooltip>swig_helper</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/symmetry" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/symmetry" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/symmetry" ><span class="devsite-nav-text" tooltip>symmetry</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/symmetry_util" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/symmetry_util" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/symmetry_util" ><span class="devsite-nav-text" tooltip>symmetry_util</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/synchronization" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/synchronization" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/synchronization" ><span class="devsite-nav-text" tooltip>synchronization</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/table" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/table" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/table" ><span class="devsite-nav-text" tooltip>table</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/theta_tree" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/theta_tree" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/theta_tree" ><span class="devsite-nav-text" tooltip>theta_tree</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/timetable" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/timetable" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/timetable" ><span class="devsite-nav-text" tooltip>timetable</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/timetable_edgefinding" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/timetable_edgefinding" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/timetable_edgefinding" ><span class="devsite-nav-text" tooltip>timetable_edgefinding</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/util" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/util" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/util" ><span class="devsite-nav-text" tooltip>util</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/var_domination" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/var_domination" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/var_domination" ><span class="devsite-nav-text" tooltip>var_domination</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/sat/zero_half_cuts" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/sat/zero_half_cuts" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/sat/zero_half_cuts" ><span class="devsite-nav-text" tooltip>zero_half_cuts</span></a></li></ul></div></li><li class="devsite-nav-item devsite-nav-expandable"><div class="devsite-expandable-nav"> <a class="devsite-nav-toggle" aria-hidden="true"></a><div class="devsite-nav-title devsite-nav-title-no-path" tabindex="0" role="button"> <span class="devsite-nav-text" tooltip>Network Flow and Graph</span> </div><ul class="devsite-nav-section"><li class="devsite-nav-item"><a href="/optimization/reference/graph" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph" ><span class="devsite-nav-text" tooltip>Overview</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/assignment" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/assignment" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/assignment" ><span class="devsite-nav-text" tooltip>assignment</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/christofides" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/christofides" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/christofides" ><span class="devsite-nav-text" tooltip>christofides</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/cliques" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/cliques" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/cliques" ><span class="devsite-nav-text" tooltip>cliques</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/connected_components" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/connected_components" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/connected_components" ><span class="devsite-nav-text" tooltip>connected_components</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/ebert_graph" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/ebert_graph" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/ebert_graph" ><span class="devsite-nav-text" tooltip>ebert_graph</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/eulerian_path" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/eulerian_path" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/eulerian_path" ><span class="devsite-nav-text" tooltip>eulerian_path</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/graph" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/graph" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/graph" ><span class="devsite-nav-text" tooltip>graph</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/graphs" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/graphs" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/graphs" ><span class="devsite-nav-text" tooltip>graphs</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/hamiltonian_path" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/hamiltonian_path" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/hamiltonian_path" ><span class="devsite-nav-text" tooltip>hamiltonian_path</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/io" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/io" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/io" ><span class="devsite-nav-text" tooltip>io</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/iterators" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/iterators" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/iterators" ><span class="devsite-nav-text" tooltip>iterators</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/linear_assignment" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/linear_assignment" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/linear_assignment" ><span class="devsite-nav-text" tooltip>linear_assignment</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/max_flow" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/max_flow" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/max_flow" ><span class="devsite-nav-text" tooltip>max_flow</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/min_cost_flow" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/min_cost_flow" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/min_cost_flow" ><span class="devsite-nav-text" tooltip>min_cost_flow</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/minimum_spanning_tree" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/minimum_spanning_tree" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/minimum_spanning_tree" ><span class="devsite-nav-text" tooltip>minimum_spanning_tree</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/one_tree_lower_bound" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/one_tree_lower_bound" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/one_tree_lower_bound" ><span class="devsite-nav-text" tooltip>one_tree_lower_bound</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/perfect_matching" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/perfect_matching" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/perfect_matching" ><span class="devsite-nav-text" tooltip>perfect_matching</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/shortestpaths" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/shortestpaths" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/shortestpaths" ><span class="devsite-nav-text" tooltip>shortestpaths</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/strongly_connected_components" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/strongly_connected_components" track-type="bookNav" track-name="click" track-metadata-eventdetail="/optimization/reference/graph/strongly_connected_components" ><span class="devsite-nav-text" tooltip>strongly_connected_components</span></a></li><li class="devsite-nav-item"><a href="/optimization/reference/graph/topologicalsorter" class="devsite-nav-title gc-analytics-event" data-category="Site-Wide Custom Events" data-label="Book nav link, pathname: /optimization/reference/graph/topologicalsorter" track-type="bookNav" track-name="click" 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